Cool Math for Hot Music.pdf

Computational Music Science

Guerino Mazzola Maria Mannone Yan Pang

Cool Math for Hot Music A First Introduction to Mathematics for Music Theorists

Computational Music Science

Series Editors Guerino Mazzola Moreno Andreatta

More information about this series at http://www.springer.com/series/8349

Guerino Mazzola • Maria Mannone • Yan Pang

Cool Math for Hot Music A First Introduction to Mathematics for Music Theorists

Guerino Mazzola School of Music University of Minnesota Minneapolis, Minnesota, USA

Maria Mannone School of Music University of Minnesota Minneapolis, Minnesota, USA

Yan Pang School of Music University of Minnesota Minneapolis, Minnesota, USA

ISSN 1868-0305 ISSN 1868-0313 (electronic) Computational Music Science ISBN 978-3-319-42935-9 ISBN 978-3-319-42937-3 (eBook) DOI 10.1007/978-3-319-42937-3 Library of Congress Control Number: 2016956578 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover illustration: Cover image designed by Maria Mannone Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

All enjoyment is musical, consequently mathematical. (Novalis)

Preface

Fig. -1.1. Maria Mannone, Guerino Mazzola, and Yan Pang. Photo and © 2015 by A.J. Wattamaniuk.

The idea for this book came from Yan Pang, a PhD student taking the course “Mathematics for Music Theorists” at the School of Music of the University of Minnesota. She was not in love with mathematics at all—bad experiences, bad teachers, the usual story. Fortunately, Maria Mannone, another PhD student taking that course who had studied theoretical physics, helped Yan get acquainted with mathematical rigor and beauty. Soon, Guerino Mazzola, the teacher, learned how to teach math using thorough musical motivation

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and avoiding abstract nonsense in favor of concrete conceptual development of theory. One day, Yan confessed that she had become enthusiastic about mathematics for music in theory and composition (using both composition software and classical score writing), and she suggested that Guerino should consider writing a textbook in this inspiring style. He thought about her idea and in fact recalled that there was no first introduction to mathematical music theory. Guerino’s book Geometrie der Töne [73]—the most elementary among his music theory books—was written in German and not conceived in a style that would meet the criteria of a first introduction. Given the enthusiastic experiences with Maria and Yan, Guerino approached his publisher, Springer, with a proposal to write this book with the two co-authors for a maximal advantage from the students’ perspective. Springer did not hesitate a single moment, and we could immediately delve into this important project. Accordingly, this book is not intended to present a dry mathematical text about tools that may be used in music. Rather, we want to develop a discourse full of pleasure and fun that in every moment motivates concepts, methods, and results by their musical significance—a narrative that inspires you to create musical thoughts and actions. We want to offer a presentation abundant in images, scores, and compositional strategies and enriched by audio examples from music theory and composition so that you can not only view the concepts but also experience them. However, to be handed tools with no opportunity to use them can be frustrating. Therefore, we also describe our concepts, methods, and results to help you apply them to your own unfolding skills in musical creativity. The wonderful advantage of a mathematical concept framework is its universal applicability, and this also includes its social dimension. The outdated ideology of a lonely genius who finds new creations in the trance of drugs and existential borderline experience is replaced by a collaborative and relaxed environment of global communication. This can accommodate any direction of musical creativity in the world of digital media and augment its power to shape the future of the beautiful truth of music. Of course, nobody is forced to accept our offer, and it is true: The payoff will not show up immediately. If you want to challenge yourself with this colorful book, you will be given a tool for creativity and discipline for your whole life. But if you prefer to enjoy an easy life without any challenge beyond flat consumption, we wish you all the best in your cage of nothingness. However, please consider this book in case you change your mind and come back to the challenge of true beauty. Mathematical examples and exercises are headed by √ √ Example, Exercise, whereas musical examples and exercises are headed by

ˇ “*

Example,

ˇ “*

Exercise.

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The reference to mathematical examples or exercises is “Example, Exercise”, the reference to musical examples or exercises is “Musical Example, Exercise”. The exercises are intended to be challenges for the reader to solve a problem by applying the concepts and results that have been presented in the text. Solutions to the exercises have been provided, but the reader should not consult them without first having tried his or her own approach. For this reason, the solutions can be found at the end of the book, in Chapter 34. The numbers of the solutions match those of the corresponding mathematical or musical exercises. Mathematical theorems and propositions always need to be proved. This is mandatory in science whenever we claim the truth of a statement. All mathematical results that are shown in this book in fact do have a proof, but it does not always serve our purposes in style and depth to include the proof in our text. Therefore, we include references to published text where proofs can be found. We also sometimes give a hint to a proof and leave it to the reader to fill in details as an exercise. Original illustrations, both computer and hand-made drawings have been created by the authors. The music examples in this book are available as MIDI, Sibelius, and MP3 files. They are all accessible via www.encyclospace.org/special/MMBOOK. So if you look of the file XX.mid, you define the address www.encyclospace.org/special/MMBOOK/XX.mid. As in the previous books of this Springer series on performance theory and musical creativity, Emily King has been an invaluable help in transforming our text to a valid English prose; thank you so much for your patience with non-native English. We are pleased to acknowledge the strong support for writing such a demanding treatise by Springer’s science editor Ronan Nugent. Minneapolis, September 2015

Guerino Mazzola, Maria Mannone, Yan Pang

Contents

Part I Introduction and Short History 1

The ‘Counterpoint’ of Mathematics and Music . . . . . . . . . . . . . 1.1 The Idea of a Contrapuntal Interaction . . . . . . . . . . . . . . . . . . . . . . 1.2 Formulas and Gestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematics and Technology for Music . . . . . . . . . . . . . . . . . . . . . 1.4 Musical Creativity with Mathematics . . . . . . . . . . . . . . . . . . . . . . .

2

Short History of the Relationship Between Mathematics and Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Artes Liberales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Zarlino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Zaiyu Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Mathematics in Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 An Example for Music Theorists . . . . . . . . . . . . . . . . . . . . . 2.6 Athanasius Kircher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Leonhard Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Joseph Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Hermann von Helmholtz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Wolfgang Graeser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Iannis Xenakis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Pierre Boulez and the IRCAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 American Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1 Genealogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 David Lewin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Guerino Mazzola and the IFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.1 Preparatory Work: First Steps in Darmstadt and Zürich (1985-1992) . . . . . . .

1 1 2 2 3 5 5 7 8 8 10 11 14 17 19 20 21 23 24 25 25 28 29 30 30

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2.15.2 The IFM Association: The Period Preceding the General Proliferation of the Internet (1992-1999) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.3 The Virtual Institute: Pure Virtuality (1999-2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.4 Dissolution of the IFM Association (2004) . . . . . . . . . . . . . 2.16 The Society for Mathematics and Computation in Music . . . . . .

31 31 32 33

Part II Sets and Functions 3

The Architecture of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Some Preliminaries in Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pure Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Xenakis’ Herma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 45 46

4

Functions and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Ordered Pairs and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Equipollence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 52 56 57

5

Universal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Final and Initial Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Cartesian Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Coproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Subobject Classifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Cartesian Product of a Family of Sets . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 63 64 64 65

Part III Numbers 6

Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Ordinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 73 75

7

Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8

Natural Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9

Euclid and Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.1 The Infinity of Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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10 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10.1 Arithmetic of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 11 Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 11.1 Arithmetic of Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 12 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 13 Roots, Logarithms, and Normal Forms . . . . . . . . . . . . . . . . . . . . . 107 13.1 Roots, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 13.2 Adic Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 14 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Part IV Graphs and Nerves 15 Directed and Undirected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 15.1 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 15.2 Undirected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 15.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 16 Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 16.1 A Nervous Sonata Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 16.1.1 Infinity of Nervous Interpretations . . . . . . . . . . . . . . . . . . . . 136 16.1.2 Nerves and Musical Complexity . . . . . . . . . . . . . . . . . . . . . . 137

Part V Monoids and Groups 17 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 18 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 19 Group Actions, Subgroups, Quotients, and Products . . . . . . . . 151 19.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 19.2 Subgroups and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 19.2.1 Classification of Chords of Pitch Classes . . . . . . . . . . . . . . . 157 19.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 20 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 20.1 Two Composition Methods Using Permutations . . . . . . . . . . . . . . 166 20.1.1 Mozart’s Musical Dice Game . . . . . . . . . . . . . . . . . . . . . . . . . 166 20.1.2 Mannone’s Cubharmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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21 The Third Torus and Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . 171 21.1 The Third Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 21.1.1 Geometry on T3×4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 21.2 Music Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 21.2.1 Chord Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 21.2.2 Key Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 21.2.3 Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 22 Coltrane’s Giant Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 22.1 The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 22.2 The Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 23 Modulation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 23.1 The Concept of a Tonal Modulation . . . . . . . . . . . . . . . . . . . . . . . . 192 23.2 The Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 23.3 Nerves for Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 23.4 Modulations in Beethoven’s op. 106 . . . . . . . . . . . . . . . . . . . . . . . . . 199 23.5 Quanta and Fundamental Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Part VI Rings and Modules 24 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 24.1 Monoid Algebras and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 206 24.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 25 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 26 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 26.1 Generalities on Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 26.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 26.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 27 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 27.1 Affine Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 27.2 Free Modules and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 27.3 Sonification and Visualization in Modules . . . . . . . . . . . . . . . . . . . 234 27.3.1 Creative Ideas from Math: A Mapping Between Images and Sounds . . . . . . . . . . . . . . . 235 28 Just Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 28.1 Major and Minor Scales: Zarlino’s Versus Hindemith’s Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 28.2 Comparisons between Pythagorean, Just, and 12-tempered Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 28.3 Chinese Tuning Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

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28.3.1 The Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 28.3.2 A System that Is Completely Based on Fifths . . . . . . . . . . 247 29 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 29.1 The Yoneda Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Part VII Continuity and Calculus 30 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 30.1 Generators for Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 30.2 Euler’s Substitution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 31 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 32 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 32.1 Mathematical and Musical Precision . . . . . . . . . . . . . . . . . . . . . . . . 268 32.2 Musical Notation for Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 268 32.3 Structure Theory of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 32.4 Expressive Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 33 Gestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 33.1 Western Notation and Gestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 33.2 Chinese Gestural Music Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 33.3 Some Remarks on Gestural Performance . . . . . . . . . . . . . . . . . . . . 276 33.4 Philosophy of Gestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 33.5 Mathematical Theory of Gestures in Music . . . . . . . . . . . . . . . . . . 281 33.6 Hypergestures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 33.7 Hypergestures in Complex Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Part VIII Solutions, References, Index 34 Solutions of Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 34.1 Solutions of Mathematical Exercises . . . . . . . . . . . . . . . . . . . . . . . . 289 34.2 Solutions of Musical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Part I

Introduction and Short History

1 The ‘Counterpoint’ of Mathematics and Music

Summary. Joining mathematics and music for most of us creates a love-hate relationship, although historically, with the Pythagorean origin, these two fields of human knowledge and activity were united. In this book we don’t want to enforce their unification for two reasons: different evolution of these fields, and major creative interaction. –Σ–

1.1 The Idea of a Contrapuntal Interaction First: the history of mathematics and music proves that these partners are different in methodology, language, and existential position, so identifying them would no longer be possible now (more about the Pythagorean position in Section 2.1). Second: the difference in their perspectives is a major force for creative interaction, and this is a major reason for the historically important mutual inspiration mathematics and music have given to each other. The recent inspiration of the great mathematician Alexander Grothendieck (1928-2014) for his highest intellectual challenge, the idea of a theory of motives, was understood as a musical idea of fundamental mathematical structures that act like musical motives in the great symphony of mathematics. For these reasons we want to propose a picture of “mathemusical” interaction that expresses the general atmosphere of their interaction. This picture is that of a counterpoint of two voices, the cantus firmus of mathematics and the discantus of music, which interact in a consonant harmony but move in a creative fashion on the axis of time through history and unfold in a contrapuntal tension of autonomous but deeply connected voices.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_1

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1 The ‘Counterpoint’ of Mathematics and Music

1.2 Formulas and Gestures Similar to counterpoint, mathematics and music present different perspectives of the total picture. There are two fundamental components: formulas and gestures. Musical formulas are well known—for example, the ternary song form A − B − A, or the cadence formula I − IV − V − I in harmony. But music cannot be reduced to such form(ula)s; it needs to deploy them in its sounds’ time and space. The aim of this deployment is the gestural action of musicians. In other words, music transfers formulas into gestures when performers interpret the written notes, and when the composers unfold formulas into the score’s gestures. Similarly, mathematicians do mathematics; they don’t just observe eternal formulas. In algebra, they move symbols from one side of an equation to the other. Mathematics thrives by intense and highly disciplined actions. You will never understand mathematics if you do not “play” with its symbols. However, the mathematical goal is not a manipulatory activity; it is the achievement of a formula that condenses and compacts your manipulatory gestures. Mathematics, therefore, shares with music a movement between gestures and formulas, but it moves in the opposite direction of the musical process. Let us show this graphically: formulas 

music

- gestures

mathematics

The famous music theorist Eduard Hanslick in his book Vom MusikalischSchönen [49] defines musical content as “tönend bewegte Formen,” not just forms, but “forms that are moved in sound.” In fact, the formal aspect—the formula—of a cadence, for example, is not sufficient to generate content. The form(ula) needs to be moved, and so it is deployed in a gestural dynamics. And Hanslick illustrates his idea with the kaleidoscope, a dynamic arrangement of forms that receive their aesthetic value in a self-referential internal relationship.

1.3 Mathematics and Technology for Music In light of the preceding characterization of the musical movement from formulas to gestures, it is not surprising that music has always been realized by playing instruments, making sounds on interfaces between gestures and their sounding output. For the school of Pythagoras, the instrument was the monochord, with one string to hear the musical intervals associated with vibrating strings of variable length. Nowadays, musical instruments are often constructed using digital information technology, as typically available on mobile computers, smartphones and the like. Music technology has always reflected the style and methodology of musical theories and formalisms. The classical European musical notation, for example, is highly adapted to the traditional keyboard instruments that enable discrete sets of notes to be played. We shall discuss such developments in the short historical Chapter 2.

1.4 Musical Creativity with Mathematics

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1.4 Musical Creativity with Mathematics Obviously, mathematical skills are mandatory for any kind of musical technology, and therefore mathematics supports musical creativity enabled by its instrumental arsenal. But mathematical structures, formulas, and methodologies have always played a crucial role in the creative construction of music. Bach used the symmetries of retrograde and inversion; Mozart invented the musical dice game; Bartók applied Fibonacci numbers to organize time, Messiaen, Boulez, Pousseur, Eimert, Stockhausen, and others applied group theory to create their serial compositions, extending the thoroughly mathematical dodecaphonic ideas of Schönberg and Hauer [98]. Working with mathematical formulas does not guarantee good musical results. This is also true for a good piano: You can always play bad music on any instrument, even on digital music players. But instruments can help shape musical thoughts in a compact and precise way. For example, in the spectral music approach, very precise mathematical representations of complex sound colors (timbres) were needed and applied to define those compositions. Spectralism originated in France in the early 1970s, and techniques were developed at the Paris-based IRCAM (Institut de Recherche et Coordination Acoustique/Musique) with its computers and with the Ensemble l’Itinéraire, by composers such as Gérard Grisey and Tristan Murail. The style of musical creativity can be very different, working with abstract harmonic or rhythmic structure, gestural continuous dynamics, or else with probability theory and statistics. For each style, there are mathematical languages, theories, and often software that can help shape one’s creative fantasies. In this book, we shall show how your knowledge of moving from formula to gesture and vice versa can generate inspiration for a creativity that transcends pure romantic dreams (which usually don’t make sound)—or nightmares. Let us recall that the great novelist Hermann Hesse (1877-1962) was awarded the Nobel Prize for Literature in 1946 for his novel Das Glasperlenspiel [50] (The Glass Bead Game), which essentially describes a sophisticated futuristic game that exchanges mathematical formulas and musical compositions.

2 Short History of the Relation Between Mathematics and Music

Summary. This chapter is a short overview of some important persons and movements in the history of the interaction of mathematics and music. It is far from complete, but should give the reader a first impression of this traditional and deeply “mathemusical” culture. –Σ– Some technical terms will be used in this historical chapter. All terms will be thoroughly explained in later chapters. Please use the book’s index to find references to these terms if needed.

2.1 Pythagoras For an account on the philosopher and mathematician Pythagoras and his school see [114]. Pythagoras was born on the island of Samos. After having traveled to Egypt and probably India, he moved to Croton in Magna Graecia around 530 BC and founded his school. The school was also a kind of sect, and it is reported that they were vegetarians, but this is not a historically firm fact as it is also reported that the members of his school were allowed to eat every kind of meat, except from oxen. He heavily influenced Plato with the idea that mathematics in its abstraction was a secure basis of all Fig. 2.1. Pythagoras (ca. 571-ca. 497 BC). philosophy and science. The Greek root μ´ αθησις (mathesis), meaning knowledge, testifies to this understanding. According to Bertrand Russell [99], he should be considered the most influential Western philosopher. He was also far ahead of his time, being the first to believe that the Earth is a sphere and it orbits a central fixed spot! © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_2

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A more reliable fact is that his school had to make an oath on the tetraktys, the cosmological symbol for which Pythagorean philosophy and cosmology is known, see Figure 2.2.

Fig. 2.2. The tetractys, the cosmological symbol for which Pythagorean philosophy and cosmology.

The tetractys is a triangular symbol built from ten points, ten being a sacred number in ancient Greece. The points are piled in decreasing groups of 4,3,2,1 points. This generates a sequence of fractions 2 : 1, 3 : 2, 4 : 3, which were considered as basic consonances when played on the Pythagorean experimental device, the monochord, see Figure 2.3.

Fig. 2.3. The monochord has one string. Its pitch is doubled by one octave when the string length is halved, 1/2, plays a fifth higher when its length is taken 2/3, and it plays a fourth higher when taken 3/4.

The musical aspect of the Pythagorean approach was not to make any compositions in the modern sense of the word. They would try to hear the hidden harmony of the universe that was thought be represented by the tetractys symbol. This symbol played the role of what contemporary physics would call a “world formula.” The tetractys was the Pythagorean world formula that ultimately described the universe in a mathematical shape, the numerical tetractys triangle. In this sense, Pythagorean music was an experimental science, and the monochord was the experimental instrument/machine/testing ground (like the role of the Large Hadron Collider (LHC) particle accelerator for the Centre European de Recherche Nucléaire (CERN) today).

2.2 Artes Liberales

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Therefore the Pythagorean school was the attempt to unify mathematics and physics in a sounding paradigm. Music was the physical expression of a cosmological principle (the tetractys) of mathematical nature. The idea of individual artistic expression in music was not part of the Pythagorean school.

2.2 Artes Liberales

Fig. 2.4. Artes Liberales

Artes liberales, the liberal arts, were a medieval canon of education for the free persons (as opposed to slaves and bondservants). They were seven in

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number, divided into two groups: the quadrivium, the “fourfold path,” comprising music, arithmetic, geometry, and astronomy (called astrology at that time), and the trivium, the “threefold path,” comprising grammar, logic (called dialectic), and rhetoric, see Figure 2.4. The remarkable point here is that music and the mathematical sciences, arithmetic and geometry, were grouped together. This is due to the Pythagorean tradition to view music as a mathematical science. The humanities, as grouped in the trivium, were separated from music. The tradition to group music with the humanities was introduced much later, essentially due to Decartes’ psychological interpretation of music. In the 19th and 20th centuries, music has been redirected to the mathematical sciences, mainly due to the development of acoustics and the mathematical nature of modern physics.

2.3 Zarlino Gioseffo Zarlino was one of the Renaissance’s most important music theorists, composers, and musicians (singer and organist). He was born in Chioggia near Venice. He was educated by Franciscans and later joined their ranks. He was maestro di cappella of St. Mark’s, a most prestigious position in Italy. His theoretical work Le istitutioni harmoniche [118] and later Dimostrationi harmoniche [119] established new insights into meantone, tempered, and just tunings. He established a harmony that emphasized C-major (the ionian mode) and recognized the basic role of the major and minor triad, reFig. 2.5. Zarlino lating them by a symmetry in the just-tuning space. He (1517-1590). also developed just tuning from the Pythagorean tuning, adding the major third interval because the Pythagorean tuning created difficult ratios for certain intervals. This was a logical extension also of the tetractys construction, adding a fifth row with five points to the Greek construction, see Figure 2.6. He also sought, as one of the first theorists, an explanation for the forbidden parallels of fifths and octaves in counterpoint.

2.4 Zaiyu Zhu Despite the difference between Chinese and Western music development, the basis of the equal-temperament scale (twelve equal semitone intervals per octave, we call this the 12-tempered in Section 13.1) was first mathematically calculated by the Chinese mathematician and musician Zaiyu Zhu (1536-1611) in 1584 (Figure 2.7). But this was never widely used in composing the indigenous music of China until 1685. Before Zhu’s tuning system was conceived,

2.4 Zaiyu Zhu

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Fig. 2.6. The extension of Pythagorean tetractys by amplification with a fifth row, yielding the new ratio 5 : 4 of the just-tuned major third interval.

there wasn’t a standard tuning system between different instrument types such as strings, winds, and keyboards. The formula he discovered allowed pitch-pipes in a equal-tempered scale of twelve equal-ratio semitones per octave, which led to a revolution of music and physics. The problem of how to modulate between different keys bothered many theorists, until Zhu solved it. This solution shows that music should not be separated from mathematics. Independently, only 150 years after Zhu’s theory, in the 18th century, the equal-temperament became the basis of Western composition. Equal-tempered tuning is widely used today because it is the best tuning system for modulation in performance. Zhu com- Fig. 2.7. Zaiyu Zhu (1536mented on his theory of equal-temperament tun- 1611). ing that scholars would have to be well acquainted both with acoustics and with mathematical calculation. Fifty-two years after Zhu published this equaltempered tuning system, theorists Père Mersenne discovered exactly the same principles, using knowledge of mathematics to solve music problems. Zhu inherited his peerage “Prince Zheng” in 1553 as the first son of Zheng, when Emperor Zheng died. Although he could have had power, instead, he decided to live in self-imposed retirement to focus on his research. Zhu’s famous books include (乐律全书) On the Equal Temperament, 1584, [120], (律吕精义) A Clear Explanation of That which Concerns the Equal Temperament, 1595/96 [121], and (算学新说) Reflection on Mathematics, 1603, [122]. Theorist Fritz A. Kuttner describes him as “one of the most important historians of his nation’s music” [57]. This system is a great invention, specifically for tonal modulation. However, in our contemporary practice it is still used in combinations of different tuning systems depending on the instrumental setup.

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2.5 Mathematics in Counterpoint

Fig. 2.8. Johann Sebastian Bach (1685-1750) with the score of canon triplex in six voices, written using only three of them.

In the classical and baroque periods, composers already used mathematical techniques to create complex music in a short time. For example, Johann Sebastian Bach used such techniques to compose and quickly write his works, see Figure 2.8. For example, such techniques allow composers to focus on musical expressivity of melodic lines, instead of wasting time in exhibiting parallel fifths and octaves in classical counterpoint. We will briefly describe a classical technique used by composers. In this sense, the use of mathematics in art, and in particular in music, has ancient roots. As indicated in classic texts about counterpoint, e.g., [42] and [30], there are many technical constraints, such as the prohibition of parallel fifths and octaves1 , hidden fifths and octaves between external voices (soprano and bass), not more than three consecutive parallel intervals of thirds, no intervals of ascending major sixths in the same melodic line and so on, as defined by Zarlino’s theory2 [118]. To create musical variety, contrario motu (when one voice raises, another voice goes down) is required. 1

2

The traditional reason behind prohibiting parallel octaves and fifths, as well as hidden octaves and fifths between external voices, is to avoid the perception of one voice (instead of two), since it can be difficult to identify two voices that are an octave or fifth apart. The composer Giacomo Puccini, referring to the parallel fifths in the beginning of Act III of La Bohème, said: “Two parallel fifths are prohibited, more than two are allowed.”

2.5 Mathematics in Counterpoint

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Writing polyphonic music that follows strict counterpoint rules requires a lot of practice and time. However, one might wonder how composers of the past were able to write high-quality music quickly when they did not have access to artificial light. This is because composers of the past already used mathematics to simplify their working process. By using matrices, permutations and sequences of numbers, they had a valid tool to write down counterpoint compositions. The knowledge of classical rules was a distinctive sign of a well-trained musician. For example, the Italian opera composer Giuseppe Verdi, well known for popular popular, world-renowned melodies, used to say that he was able to write a fugue in three hours to be esteemed by rigorous critics. This skill is only possible if mathematical rules are known. 2.5.1 An Example for Music Theorists Let us now consider an example (taken from [30]). The Italian composer Luigi Cherubini wrote a double choir with perfect imitation inverse and contrario motu, where the first choir is mirrored into the second choir, as shown in Figure 2.9.








 

       





   













 


























  


 








 




 





















  








 



 












   












       







  







Fig. 2.9. The first four measures of a double choir with perfect imitation inverse and contrario motu, composed by Luigi Cherubini. The sound example is Cherubini.

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Such a structure is very difficult to achieve, unless we use a clever permutation of notes. The notes of the diatonic C-major scale are numbered by c → 1, d → 2 . . . b → 7. The first note of the soprano of the first choir, g in the soprano clef (an ancient clef with the c on the first line, see Figure 2.11), corresponds to number 5 in the diatonic C-major scale. It is mirrored into a (sixth note in diatonic Cmajor scale) of the bass line in the second choir. Then, the first correspondence is 5 to 6. The g of contralto in the first choir (written in alto clef, another ancient clef with c on the third line), position 5, is mirrored again in a of tenor in the second choir (tenor clef, with c on the fourth line). Putting together all these correspondences, we can obtain the permutation   1765432 (2.1) 3456712 These sequences accurately avoid two vertical correspondences: (1) the 1 − 4 interval, which is the prohibited fourth interval, and (2) a 5 − 1 interval, whose inversion is the fourth. Moreover, numbers in the first and second sequence are consecutive except in one point: 3, 4, 5, 6, 7, and again 1 and 2. If we choose a different permutation, as   1765432 (2.2) 6712345 we can compose an original mirror double-choir, as shown in Figure 2.10. Another known mathematical tool is the usage of the tabula mirifica by Athanasius Kircher [53], whom we discuss in Section 2.6. In fact, the tabula mirifica omnia contrapunctisticae artis arcana revelans (a wonderful table that reveals all hidden techniques of counterpoint art), contains a matrix (introduced in Chapter 26), see Figure 2.13. Bach also used this table while composing A Musical Offering. A famous example of a retrograde canon in Bach’s composition is shown in Figure 2.12. It is written in the soprano clef (Figure 2.11). The following is a simple example: a matrix with five rows and columns, also symmetric with respect to the main diagonal, to construct a simple canon. ⎛ ⎞ 12345 ⎜2 1 2 3 4⎟ ⎜ ⎟ ⎜3 2 1 2 3⎟ (2.3) ⎜ ⎟ ⎝4 3 2 1 2⎠ 54321 Diagonal(s) in the square correspond(s) to identical notes. Elements of codiagonal(s) belong to the same consonant chord. For example, 5 − 3 − 1 if written as 1 − 3 − 5 yields c − e − g. Numbers 2 − 4 indicate the minor third d − f . Numbers 1 − 3 mean major third c − e, and 2 − 2 the unison d − d, another consonant interval.

Soprano (I)

     6

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Contralto (I)

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  

Soprano (II)



Contralto (II)

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2.5 Mathematics in Counterpoint

 

6 Risposta inversa contraria.

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

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Fig. 2.10. A mirror double-choir. The sound example is Nuovo_doppio_coro.

Fig. 2.11. The first two (treble clefs) specify g above middle c, the next ones central c, except the last three (bass clefs), which specify f below central c.

By going through the first row, we get c − d − e − f − g, corresponding to 1 − 2 − 3 − 4 − 5. By going through the first row and then down the fifth column, we get 1 − 2 − 3 − 4 − 5 − 4 − 3 − 2 − 1, corresponding to the sequence c − d − e − f − g − f − e − d − c, which constitutes our first melodic line. To write a counterpoint, it is necessary to insert other voices that produce consonant intervals between them and the first melodic line (cantus firmus). To find consonant notes with respect to the first voice, we can move along the corresponding co-diagonal. For example, if the second voice starts when the first voice is playing e, it means, starting on 3 (element on the first row and third column), then the first note of the second voice belongs to the secondary

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 7

 12

 16



 

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    



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 

            

               

                                                      

Fig. 2.12. A retrograde canon in Bach’s A Musical Offering. The sound example is Bach. The sound example contains also the solution.

co-diagonal 3 − 1 − 3. If we choose 1, we have c, which is consonant with respect to e. To insert a third voice, for example starting when the first one is playing g (5), we choose a number from the main co-diagonal 5 − 3 − 1 − 3 − 5, for example 3, e. We get a simple canon, see Figure 2.14. In this way we obtain a canon, a particular kind of polyphonic music where the same melody is repeated by different voices. The case we have constructed here is an octave canon in contario motu, where the notes are exactly the same, without any transposition. Mathematically it can immediately be defined as a time-translation of the same pitch pattern. Other similar games are palindromes and retrograde canons. A palindrome (Figure 2.15) is a sequence that is symmetric with respect to onset-time inversion (retrograde). A retrograde canon is built by a single line, played by a first musician from left to right, and by the second one from right to left, i.e., in retrograde motion. See Figure 2.16 for a retrograde canon and Figure 2.17 for the retrograde canon with its solution. More details about canons are given in [28].

2.6 Athanasius Kircher Athanasius Kircher is believed to be the last uomo universale, the Renaissance Man who knew everything at his time. He was born in the German village Geisa near Fulda and educated in a Jesuit college in Fulda. He learned Hebrew from a rabbi and studied philosophy and theology at Paderborn. He became professor at the University of Würzburg, where he taught Hebrew, Syriac, and mathematics. Kircher published around forty important works, from religion, geology, medicine, knowledge science (encyclopedias), chemistry, physics, Sinology, and Egyptology. He was also active as an inventor of technology and medical devices. Figure 2.19 shows the design of a steam-powered organ. He was one of

2.6 Athanasius Kircher

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Fig. 2.13. Kircher’s tabula mirifica omnia contrapunctisticae artis arcana revelans.

  

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5

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Fig. 2.14. A simple canon.

the first to observe microbes on a microscope and set forth the now valid theory that plagues were caused by infectious microorganisms. For our musical concerns, his work in knowledge science is most important [36], also, because music to Kircher was the model of any order. In the tradi-

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  

 



        

 



Fig. 2.15. An original example of another similar game is the palindrome. The sound example is Palindromo.

   

   

     





   





  



Fig. 2.16. A retrograde canon.

    

   

                     





    

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  

          

Fig. 2.17. A retrograde canon together with its retrograde solution. The sound example is retrograde, first the retrograde canon, then the canon with its solution.

tion of Lullus of a formalization of the encyclopedic process, Kircher referred to the Lullian topology of a three-dimensional field of possibilities of conceptualization. In his book Ars Magna Sciendi (The Great Art of Knowledge) [54], Kircher develops an arithmetic of 36 basic concepts that he combines into a amount of 1067 combinations—of course a size that by far surpassed the possibilities of his time. His combinatorics had no grammatical rules (such rules were only introduced later by Gottfried Wilhelm Leibniz with his binary methods of deduction). This was a foreshadow of what is known as “big data” nowadays. The size of his combinatorics was only one problematic aspect; the second was the claim that his system had a universal language character. In his work [55], he tried to solve the combinatorial complexity by “machine-readable” formalization. He conceived a translation box with drawers and tables of vocables. In the 17th century, such a task was far from realistic. This proposal of a three-dimensional universal dictionary was far beyond the limits of scientific feasibility – in fact, one could say the same about Kircher’s entire program, according to philosopher Wilhelm Schmitt-Biggeman [102]. Fig. 2.18. Athanasius Kircher (1602-1680).

Kircher also worked on a combinatorial theory of music. For him, every field of knowledge was ultimately a big data system. His approach was less mathematical than combinatorial in the sense of modern computer science. He was concerned with a “machine-readable” repertory of concepts, in music and elsewhere. This is a very modern perspective in light of the present music soft-

2.7 Leonhard Euler

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Fig. 2.19. Kircher’s design of a steam-powered organ.

ware for composition, notation, and analysis. Figure 2.20 shows Kircher’s arca musarythmica, which he described in his work Musurgia Universalis [53]. Recall that Wolfgang Amadeus Mozart with his “musikalisches Würfelspiel” (musical dice game) also conceived a combinatorial construction of musical compositions from elementary components (see Sectiondicegame. Kircher’s approach to music follows the Pythagorean tradition that music is strongly related to the technology of its physical realization and also demonstrates that in music the progress of technology is always reflected in the musical realm.

2.7 Leonhard Euler Leonhard Euler is one of the greatest mathematicians of all times. His formula eiπ + 1 = 0 is considered to be the most beautiful formula in mathematics as it combines the basic numbers e, π, 0, 1 as well as equality, addition, multiplication, and exponentiation. His work comprises seventy-six quadro format (12-inch high) volumes. Euler was born in Basel, Switzerland, and first studied theology, Greek, and Hebrew because his father wanted him to become a priest. But the famous mathematician Johann Bernoulli, who taught Euler mathematics, persuaded father Euler to let Leonhard study mathematics. He completed his studies in Basel with a dissertation about sound, De sono, in 1726. He applied for a position as professor of physics at the University of Basel but did not get the position and became professor of mathematics at the Imperial Russian Academy of Sciences in Saint Petersburg in 1727. He moved to Berlin in 1741

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Fig. 2.20. Kircher’s arca musarythmica. Its purpose was to develop a large number of musical compositions by combining elementary components.

and was professor of mathematics at the Berlin Academy until 1766, when he returned to Saint Petersburg, where he spent the last part of his live. Euler’s work is very broad, reaching from theoretical to applied mathematics, from celestial to fluid mechanics, from formal logic to mathematical music theory. The latter field is why we are particularly interested in Euler’s work. He wrote three treaties on music [31, 32, 33]. The second deals with consonances, and we shall see that Euler has made an important contribution to the definition of a musical consonance with his prime-number-based gradus suavitatis function (see Musical Example 64). But the most important contribution to music theory is Euler’s definition of the geometric space of pitch classes in his third Fig. 2.21. Leonhard Eu- work, where a two-dimensional space is spanned by ler (1707-1783). the axis of fifths and the axis of major thirds. This space is now known as the Euler space (see Musical Example 63). Figure 2.22 shows Euler’s geometry of pitch classes. Euler’s success with his mathematical music theories was limited at his time because it apparently was too mathematical for musicians and too musical for mathematicians—that sounds familiar to us, doesn’t it?

2.8 Joseph Fourier

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Fig. 2.22. Leonhard Euler’s two-dimensional space of pitch classes that is spanned by the axes of fifths (horizontal) and major thirds (vertical).

2.8 Joseph Fourier Jean-Baptiste Joseph Fourier was a French mathematician and physicist. He has become one of the most influential mathematicians of all times because of his theory of periodic functions that can be represented as (usually infinite) sums of sinusoidal functions (the function’s partials) that have frequencies that are natural multiples of the basic frequency defined by the functions’ periods. Born in Auxerre, Fourier was educated by the Benedictine Order of the Covent of St. Mark. After his engagement in the French Revolution, he was appointed in 1795 to the École normal supérieure (ENS) in Paris. In 1797 he became the successor of Joseph-Louis Lagrange for calculus Fig. 2.23. Joseph Fourier and mechanics at the ENS. (1768-1830). In 1802, Napoleon appointed Fourier prefect of the department of Isère in Grenoble. In 1807, Fourier wrote his first essay, On the Propagation of Heat in Solid Bodies. In 1816, Fourier moved to England but returned to Paris in 1822, where he was the permanent secretary of the French Academy of Sciences. In the same year, he wrote the most important work Théorie analytique de la chaleur (The Analytic Theory of Heat). In this treatise, he stated his famous theorem on the sinusoidal decomposition of periodic functions. His proof was not correct, however, and Peter Gustav Lejeune Dirichlet succeeded as the first to give a mathematically complete demonstration. The Fourier Theorem is extremely important for music since vibrations of the air that are perceived with a pitch by humans are periodic functions of the air pressure (at least for the usually short time of their duration). The Fourier theory represents such functions as sums of sinusoidal vibrations, and this is what music theorists call the partials or overtones. This partial decomposition is to some degree responsible for the sound color of different instruments (other contributions being the amplitude envelope and also changes in the overtone

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amplitudes during the sound’s process). Music theorists have tried to view the Fourier partials as being intrinsic components of a sound, but this is erroneous. There are infinitely many decompositions in the form Fourier described, and this is a mathematical field called functional analysis. The Fourier decomposition was also thought to be the analysis that the cochlear inner ear performs. This is also not the case, but see [75, Appendix B] for details. The digital technology has also found algorithms for fast calculation of partials, the Fast Fourier Transform (FFT). Without FFT, modern sound technology would be infinitely slow. Therefore, Fourier’s theorem has become a fundamental tool for the present sound technology.

2.9 Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist. His contributions reach from physiological theories and experiments (he proved first that nervous signal propagation was not infinite as scientists had believed previously!), experimental and theoretical physics (conservation of energy, thermodynamics, etc.). He is also known as a philosopher of science and a messenger for the civilizing power of science. Helmholtz’s father urged him to

Fig. 2.24. Helmholtz constructed his resonator (i) to measure partials of complex sounds, here within his instrumentation to mimic vowel sounds.

study medicine, but the son was more interested in natural science. He became associate professor of physiology at the Prussian University of Königsberg in 1849, was full professor of anatomy and physiology at the University of Bonn in 1855, moved to the University of Heidelberg in 1858 and was professor of physics at the University of Berlin from 1871 through the rest of his life.

2.10 Wolfgang Graeser

21

We are interested in Helmholtz’s work because he applied the Fourier Theorem to two topics. First, he constructed a machine, called the Helmholtz resonator, that could isolate the partials of a complex sound, see Figure 2.24. The second application of Fourier’s Theorem was Helmholz’s beat theory of consonance. His idea was that the partials of two complex sounds that build a musical interval would interact in our ear and produce beats. Beats occur when you superimpose two sinusoidal waves of frequencies f, g, the result is a wave whose frequency is the average f +g 2 while its amplitude (loudness) has frequency f −g 2 ; i.e., the loudness pulsates with that difference frequency, which is called beat in music acoustics. Helmholtz thought that beats of 16 Hz (Hertz, a unit of frequency) were maximal contributions to what he called roughness of the interval. This is what Fig. 2.25. Herrmann he supposed as being the reason for the perception of von Helmholtz (1821consonance and dissonance. This is interesting since 1894). Helmholtz’s dissonance concept depends on the instrumental sound color, which sounds reasonable. However, Heinrich Husmann later proved experimentally that human perception of dissonance also persists if the two sounds of the interval are presented to the two ears separately. This puts Helmholtz’s idea into question as this situation would not produce beats in our ears. Also, the roughness does not define the strict dichotomy consonance/dissonance that is required in musical counterpoint. It is a function that defines continuous degrees of consonance, not strict separation.

2.10 Wolfgang Graeser Wolfgang Graeser was a German-Swiss violinist, music theorist, and mathematician. He was born in Zürich. At the age of ten, he learned to play the violin. At the young age of seventeen, he studied music, mathematics, and Oriental languages in Berlin and Zürich. His contributions to the group-theory in music and musical gesture theory were far ahead of his time. When the twentytwo-year-old Graeser, caught by an attack of depression, hung himself at home in Berlin on June 13, 1928, he had already published two major works: (1) At the age of seventeen, the treatise Bachs “Kunst der Fuge” [46], which revolutionized music theory in general and our understanding of Bach’s opus posthum in particular. Following Graeser we may ask whether the incessant structural fascination of Bach’s music cannot be explained from a network of locally present symmetries. Here is his summary: Die Eigenschaft der Symmetrie spielt in der Musik eine so ungeheure Rolle, daß sie verdient, an erster Stelle betrachtet zu werden. Wir

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werden in der “Kunst der Fuge” ihre fast uneingeschränkte Herrschaft besonders deutlich erkennen. (In music, the property of symmetry plays such an enormous role that it merits priority consideration. In the “Art of Fugue” we shall recognize its virtually unlimited dominance with abundant evidence.) Greaser also was one of the first theorists to describe explicitly the global structure in music, the fact that musical compositions are the result of a complex “gluing” process of small parts. He describes a contrapuntal form as follows [46, p.17]: Bezeichnen wir die Zusammenfassung irgendwelcher Dinge zu einem Ganzen als eine Menge dieser Dinge und die Dinge selber als Elemente der Menge, so bekommen wir etwa das folgende Bild einer kontrapunktischen Form: eine kontrapunktische Form ist eine Menge von Mengen von Mengen. Das klingt etwas abstrus, wir wollen aber gleich sehen, was wir uns darunter vorzustellen haben. Bauen wir einmal ein kontrapunktisches Werk auf. Da haben wir zunächst ein Thema. Dies ist eine Zusammenfassung gewisser Töne, also eine Menge, deren Elemente Töne sind. Aus diesem Thema bilden wir eine Durchführung in irgendeiner Form. Immer wird dies Durchführung die Zusammenfassung gewisser Themaeinsätze zu einem Ganzen sein, also eine Menge, deren Elemente Themen sind. Da die Themen selber Mengen von Tönen sind, so ist die Durchführung eine Menge von Mengen. Und eine kontrapunktische Form, ein kontrapunktisches Musikstück ist die Zusammenfassung gewisser Durchführungen zu einem Ganzen, also ein Menge, deren Elemente Mengen von Mengen sind, wir können also sagen: eine Menge von Mengen von Mengen. (If we call the collection of some objects a set, and these objects the elements of the set, we get the following image of a contrapuntal form: a contrapuntal form is a set of sets of sets. This sounds somewhat abstruse, but we will see in a moment what we have to imagine. Let us build a contrapuntal work. We first of all have its theme. This is a collection of sounds, i.e., a set whose elements are the sounds. From this theme we build a development in a determined form. This development always will be the collection of certain instances that build a whole. Therefore the development is a set of sets. And the contrapuntal piece of music is the collection of developments, i.e., a set whose elements are sets of sets, that’s why we may say: a set of sets of sets.) (2) One year before his suicide, the book Der Körpersinn [47], which opened Graeser’s vision of what he called a more “faustian dynamic” thinking and understanding of music, dance, and other arts. The latter work was inspired by Graeser’s strong experience of a dancing class that followed the main theme of Bach’s “Kunst der Fuge” and thereby incited an existential connection

2.11 Iannis Xenakis

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between his abstract symbolic investigations and these symbols’ real dancing embodiment [123]. In a nutshell, Graeser’s legacy testifies the immense tension that causes music to vibrate between facticity of formulaic compression and unfolding gesturality in the making. And it asserts the belief and insight of an ingenious talent in the utopia of coherence between these ontological poles. Despite his profiled discoveries, Graeser was never really understood by musicians, music theorists, or music philosophers. His literally suicidal tension somehow hindered academic minds to approach this body of knowledge, much as it has taken nearly a century for philosophers to seriously deal with Nietzsche’s “philosophy with the Fig. 2.26. Wolfgang hammer.” Graeser (1906-1928).

2.11 Iannis Xenakis Iannis Xenakis is a Greek-French architect, engineer, music theorist, and composer. He is most famous for his mathematically inspired compositions and for integrating electronic technology, in particular his computer system for graphical composition UPIC (Unité Polyagogique Informatique du CEMAMU, where CEMAMU is Xenakis’ Centre d’Études de Mathématique et Acoustique Musicales in Paris). Xenakis was born in Brăila, Romania. He moved to Athens in 1938 and studied civil engiFig. 2.27. Iannis Xenakis neering at the National Technical University of (1922-2001). Athens. He graduated in 1947 but, having been part of left-wing resistance, had to leave Greece and arrived in Paris later in 1947. He was in fact sentenced to death in absentia by the right-wing regime; this decision was revoked 1974 after the fall of the Regime of the Colonels. In Paris, Xenakis started collaborating with the famous architect Le Corbusier. This experience with architectural spaces strongly influenced Xenakis’ spatial approach to musical composition. Xenakis was always interested in music and wanted to take lessons with Nadia Boulanger and Arthur Honegger, but it was a failure as Xenakis was interested in everything but classical composition techniques. When Xenakis was presented to composer Olivier Messian, Messian understood that this student was something special [71]: I understood straight away that he was not someone like the others. (...) He is of superior intelligence. (...) I did something horrible which I

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should do with no other student, for I think one should study harmony and counterpoint. But this was a man so much out of the ordinary that I said (...) No, you are almost thirty, you have the good fortune of being Greek, of being an architect and having studied special mathematics. Take advantage of these things. Do them in your music. In 1954, Xenakis was accepted into Pierre Schaeffer and Pierre Henry’s Groupe de Recherches de Musique Concrète. In 1957, Xenakis received his first composition award. In 1959 he left Le Corbusier’s studio and started making a living as a composer and teacher. He founded the CEMAMU in Paris and in 1963 wrote his main work of mathematical music theory: Musiques formelles [117], later extended and translated to English as Formalized Music: Thought and Mathematics in Composition (1971) [116]. In this work he explained the mathematical techniques he used for music composition: set theory, stochastics, game theory, probability theory, and group theory of permutations. With his computer-aided composition tool, the UPIC, he was able in 1979 to design compositions on the basis of a two-dimensional (plane) graphical input (Figure 27.2). It is interesting that despite his strongly spatial approach to musical composition, Xenakis never applied linear algebra, matrix calculus, or even differential geometry. It is reported that his problem with not being able to apply affine transformations to his configurations was solved only by the graphical composition software presto invented by one of the authors (Guerino Mazzola), but see Section 2.15.

2.12 Pierre Boulez and the IRCAM Pierre Boulez is a French composer, music theorist, conductor, and organizer of computeraided technologies in music, especially as a former director (1977-1992) of the IRCAM (Institut de Recherche et Coordination Acoustique/Musique) in Paris. Boulez was born in Montbrison, France, and in his early childhood demonstrated talents for Fig. 2.28. Pierre Boulez music and mathematics. He studied music at the (1925- ). Paris Conservatoire under Olivier Messiaen and Andrée Vaubourg. Boulez learned from Messiaen about integral serialism (series in all parameters, not only pitch, but see Musical Example 68). His radical experiments, such as the Structures pour deux pianos that we discuss in the Musical Example 68, also led him to question the strictly mathematical formalism of serial approaches. However, he has been a strong advocate of the mathematical methodology, as is reflected in his writing in the second volume of his important book Musikdenken heute [23, p.71]:

2.13 American Set Theory

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Weil die mathematische Methode die Wissenschaft ist, die zur Zeit die am weitesten entwickelte Methodologie besitzt, war mir daran gelegen, sie zum Vorbild zu nehmen, das uns helfen kann, unsere gegenwärtigen Schwachstellen zu beheben. (Because the mathematical method is the science that presently has the most developed methodology, I was interested to take it as a model that can help us eliminate our present weaknesses.) Boulez also applied mathematical tools in his compositions—for example what is known as “pitch multiplication,” used in Le marteau sans maître (19531955). The founding of the IRCAM by president Georges Pompidou has a remarkable background story. In Germany’s Max-Planck-Society, they have all kinds of scientific institutes, but in 1966, they wanted to add an institute for music research3 , they wanted to add an institute for music research. They asked Boulez about his concept of such an institute. He answered that he would see it as a place where composers and orchestras would have a creative environment of limitless freedom. And it seems that Germany had a good amount of money to help Boulez realize such an institute. But the society directors were not satisfied with Boulez’s suggestions—they wanted a really scientific concept, not just some superb artistic venue. It was after this disagreement that Pompidou approached Boulez and offered him the IRCAM near the Centre Pompidou. The IRCAM became one of the world’s leading centers for computer-aided musical creativity. It also included (and still does) a special seminar, MaMuX (Mathematics, Music and X, anything to be added freely), of which one of the authors (Mazzola), Gérard Assayag, Moreno Andreatta, and François Nicolas were the co-founders. A number of music software, such as OpenMusic or Max, have been developed at the IRCAM.

2.13 American Set Theory The purpose of this section is to account for and discuss some crucial contributions of the American tradition to the emergence and proliferation of what American composer and theorist Milton Babbitt termed “professional music theory” [10]. 2.13.1 Genealogy The impossibility of giving even a partial (ordered) description of the topics dealt with in the American music-theoretical literature since the 1950s leads us to look for historically and methodologically pertinent ‘segmentations’ in 3

Information relayed to one of the authors (Mazzola) by a former society director, Valentin Braitenberg

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the domain of contemporary music theory. Perhaps one of the most fruitful approaches is based on the underlying dichotomy between an apparently more compositional attitude (Babbitt) and a radically analytical perspective (Allen Forte) toward music theory. Both of these approaches are divided into two formal parts: meta-theory and methodology/compositional theory, and historical and theoretical essays/analytical studies, respectively. The former work points to “the engagement by composers in fundamental music-theoretical explications” [21, p.vii], while the latter suggests that “the very fact that Forte is not himself a composer has changed the field of theory considerably” [15, p.50]. We may also suggest here that this distinction not only is relevant for an historical discussion on pitch-class set theory, but it also helps in understanding how this theory successively enlarged its field of applicability thanks to important works by John Rahn [95], Robert Morris [89], and David Lewin [60, 61]. As pointed out by George Perle in his comprehensive study on serial and atonal music, “the most important influence of Arnold Schönberg’s method is not the 12-note idea in itself, but along with it the individual concept of permutation, inversional symmetry and complementation, invariance under transformation, aggregate construction, closed systems, properties of adjacency as compositional determinants (...)” [94, p.x]. This ‘Babbittian’ presentation of 12-tone problematics constitutes perhaps the most appropriate introduction to Babbitt. It is widely accepted that he first provided “twelve-tone theory with a consistent technical vocabulary” [94, p.xiv] and suggested that the relevance and “the force of any ‘musical system’ was not as universal constraints for all music but as alternative theoretical constructs, rooted in a commonality of shared empirical principles and assumptions validated by tradition, experience, and experiment” [21, p.ix]. One cannot emphasize enough that, for Babbitt, a “set” is an ordered collection of pitch classes, and it is used as a perfect synonym for row and series. In contrast, the very predecessor of Forte’s “pitch-class set” is Babbitt’s “source set,” a set “considered only in terms of the content of its hexachords, and whose combinatorial characteristics are independent of the ordering imposed on this content” [11, p.57]. A synonym for it is “collection,” first introduced by Lewin in [62] and widely discussed for its analytical pertinence in [63]. Subsequently, a vast body of American literature was devoted to the study of the specific properties of sets and collections, particularly combinatoriality and partitioning [69, 51, 43, 16, 106, 107]. Partition problems connected with Babbitt’s original idea have also largely proved their relevance to mathematics with their natural embedding into the theory of groups [60, 89]. But even the idea of applying the mathematical concept of group for modeling musical systems can be regarded as one of Babbitt’s most fruitful intuitions4 , provided that “the rules of formation and transformation of the twelve-tone system are 4

However anticipated in 1924 by Wolfgang Graeser in his study on Bach’s “Kunst der Fuge” [46].

2.13 American Set Theory

27

interpretable as defining a group element (a permutation of order of set numbers) and a group operation (composition of permutations)” [12, p.20]. This equivalence of structures, first introduced in [13], has important compositional consequences that are “directly derivable from the theorems of finite group theory” [14, p.8]. It is perhaps no exaggeration to consider the introduction of groups by Graeser and then Babbitt as the ‘Copernican Revolution’ of modern music theory. Suggestions for further reading in this area may be found in Rahn’s review of Lewin’s “Generalized Musical Intervals and Transformation” [96]. The most important representative of the analytical approach in the American music-theoretical literature is Forte who is the author of a theory of set complexes [38] and of a book primarily devoted to the atonal music of Arnold Schönberg, Anton von Webern, and Alban Berg in the first twenty years of the 20th century [39]. Forte’s main purpose is to “provide a general theoretical framework, with reference to which the processes underlying atonal music may be systematically described” [39, p.ix]. Forte’s starting point is “firmly analytical, springing from a truly fervent desire to uncover the secrets of an (...) enigmatic repertoire” [19, p.50]. In his most recent article, Forte particularly emphasized this aspect, stating “The structure of atonal music (is) above all the study of a musical repertoire rather than a theoretical presentation” [40, p.83]. However, one of his most striking merits was the introduction of a “consistent terminology for pitch-class collections based on the mathematical properties of the set” [19, p.49] together with the elaboration of the “set complex,” a topic recently developed by Robert Morris [90]. But it is probably true that “Allen Forte’s real success lies in the developments he inspired: beyond his theorization of atonal music, his work convinced many of the interest of a formal study of chromatic space” [88, p.90]. An example of this is Rahn’s pedagogically oriented introduction to some problematics common to the atonal and serial repertoire [95]. The book also prepares “its reader for the professional literature in the field” [95, p.v] and gives accurate references for further specialized topics including advanced serialism and combinatoriality. Babbitt’s “general formative role”, together with Forte’s terminological heritage, are widely recognized in the American sphere, and the book has become a standard reference for further discussions in set theory. Many of the topics Morris deals with refer to the theory of local compositions, see our Musical Example 72. Morris’ recent formalization of his so-called “compositional spaces” provides a new theoretical tool that compensates for some of the weaknesses of his original compositional model. As defined by Morris, “compositional spaces are out-of-time structures from which the more specific and temporally oriented compositional design can be composed” [91, p.330].

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2.13.2 Comments We want to conclude this section with three short comments—mathematical, conceptual, and model-oriented—on the American Set Theory (AST) . Mathematically, the AST is a very unusual achievement. Its concepts are thoroughly out of date from the point of view of 20th-century mathematical conceptualization. Even the most standard concepts in group theory are ignored such as the index of a subgroup and the concept of a group action and corresponding elementary facts such as orbit cardinalities. Although the theory of categories has been around since the early 1940s and is even recognized by computer scientists, there is no apparent attempt in AST to deal with morphisms between pcsets (pitch-class sets), for example. We have seen certain germs of this direction in the definition of abstract subsets, but this is not what leads to a powerful theory of relations between local or global musical objects. It would be important to adopt the findings of AST to workable mathematical formalism such as it has been used by Fripertinger in Pólya and de Bruijn enumeration theory, for example. It would also be necessary to confront the AST approach with the many other parameters that define musical events, such as onset, duration, loudness, glissando, and crescendo, just to name a few important ones. The AST has never dealt with all these parameters in a global conceptual framework. The work of Dan Tudor Vuza or Anatol Vieru does heavily favor such an extension. However, many valid lists of isomorphism classes, such as chord classes under transposition, or transposition together with inversions have been established. We also have to recognize that the mathematical conceptualization can make these musicological concepts operational for computer programs and algorithms. The conceptual comment must address a dramatic need for precise musicological concepts as tools for dodecaphonic analysis and of its extension in atonal theory. European music theory has only very rarely shown up in this domain, perhaps best in the work of Herbert Eimert and Iannis Xenakis. But the mainstream of post-World War II European musicology had turned toward dialectic mumbo-jumbo and far-out aesthetics and transcendental black-box theories. So Americans had to start from scratch with precise conceptualization of even the simplest concepts such as pcsets (pitch-class sets), pcsegs (pitchclass segments), and their classes. Whatever the status of an infant theory the AST concept framework might be, it is an indispensable reset of a rotten conceptualization in musicology where even the most elementary things are blurred. The question of theoretical modeling is a difficult one. It appears that modeling has predominantly been oriented toward and useful for compositional strategies. Morris’ composition designs further our ability to understand the complex construction of precise sound aggregates when starting from pcsets and similar elementary local compositions. Also the analytical use of the AST language is a considerable one. We are happy that it is finally possible to speak in defined terms about analytical programs of atonal and tonal music

2.14 David Lewin

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(e.g., in John Amuedo’s work). But there remains a big lack of models in the sense that beyond descriptive tasks, the AST language has very seldom led to musicological modeling. Most theorems of AST are of strictly combinatorial nature. So we could summarize the AST achievement as a necessary but far from sufficient attempt to escape decadent and impotent European musicology.

2.14 David Lewin David Lewin was a music theorist, critic, and composer, and many consider him to be one of the most innovative American scholars of his generation. He was born in New York, graduated in 1954 with a degree of mathematics from Harvard, and returned to Harvard in 1985 after several positions elsewhere. In 1961, he was the first professional musician to compose a computergenerated work at Bell Labs. We are interested in Lewin’s work because he initiated a change of paradigm in mathematical music theory from the traditional American Set Theory to what is known as Transformational Theory. His new approach was displayed in his book Generalized Musical Intervals Fig. 2.29. David Lewin (1933-2003). and Transformations [60]. His Generalized Interval Systems (GIS) were a description of a simply transitive group action (see our discussion in Musical Example 46). The new paradigm was to use arrow diagrams between musical objects (such as pitch classes) instead of abstract sets. Lewin also imagined this passage, which he believed to be different from the purely objective and passive “cartesian” approach, to be a gestural approach. In [60], he states the now famous question: “If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there?” This new attitude was only an intuitive idea, however, and Lewin’s mathematical tools remained within simple group or monoid actions. Later, Lewin and his student Henry Klumpenhouwer developed more sophisticated arrow diagrams, called Klumpenhouwer nets (K-nets), which realized as a special case what in modern mathematics is known as a limit. See [80] for a discussion of K-nets from the perspective of modern mathematics. Mazzola set forth the GIS idea in 1980 in his mathematical music theory that was based on the more general category theory of modules, see Figure 15.1 for an example. Lewin and Klumpenhouwer’s mathematical approach was a special case of the new theory of categories that Samuel Eilenberg and Sounders Mac Lane introduced in 1945. This is a significant example of how important it has become for music theorists to learn about the new mathematical achievements in the second half of the 20th century.

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2.15 Guerino Mazzola and the IFM Guerino Mazzola (one of the authors) was born in Dübendorf, near Zürich, Switzerland. He has been characterized by Elaine Chew as the “godfather of mathematical music theory.” He founded what his student Thomas Noll has dubbed the “Zürich School of Mathematical Music Theory,” which can be viewed as the European counterpoint of American Mathematical Music Theory (comprising American Set Theory). Mazzola qualified as a professor5 for algebraic geometry and representation theory in 1981 at the Fig. 2.30. Guerino Maz- University of Zürich and is a free jazz pianist (havzola (1947- ). ing published twenty-four LPs and CDs), music critic, and composer. His first book Gruppen und Kategorien in der Musik [72] already used sophisticated modern mathematics—module theory, category theory, homotopy theory, and algebraic geometry to build models of modulation as well as the classification of local and global musical structures. He applied these tools to analyze Beethoven’s “Hammerklavier” Sonata op. 106, and to compose a new sonata [76] from the analysis. He also designed and programmed the music software presto (around 1989) and RUBATO (released in 1992 and still in use) and composed a concert for piano and percussion [74]. His main work, The Topos of Music, is a broad presentation of the geometric logic of theory and performance that is built upon a conceptual framework of denotators and forms and uses sophisticated techniques of topos theory, the synthesis of geometry and logic initiated by the great mathematician Alexander Grothendieck. In 2003 Mazzola qualified as a professor in computational sciences at the University of Zürich, having been a part of the Computer Science Department there since 1992. He has been a professor at the School of Music of the University of Minnesota since 2007. The following sections describe the historical development of the Zuürich School that eventually led to the foundation of the Society of Mathematics and Computation in Music that is described in Section 2.16. 2.15.1 Preparatory Work: First Steps in Darmstadt and Zürich (1985-1992) From 1984 to 1986, Mazzola was the scientific director of the symmetry exhibition and symposium in Darmstadt, where he first achieved visualization and popularization of concepts and results from Mathematical Music Theory—as he elaborated in his 1981 university lectures and his first more sketchy book “Gruppen und Kategorien in der Musik” [72]. The work of popularization was performed in collaboration with the Technische Universität Darmstadt, and in 5

Called habilitation in Europe.

2.15 Guerino Mazzola and the IFM

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particular with Georg Rainer Hofmann, at that time a PhD student of professor José Encarnaçao. From 1987 to 1989, upon recommendation by conductor Herbert von Karajan, Mazzola and Hofmann undertook a joint project at the FraunhoferIGD in Darmstadt. The project, sponsored by private investors, produced a composition software for ATARI called MDZ71. Later, several hundred copies were commercially distributed. The software also was presented to von Karajan in 1988. As these efforts matured, the idea of a “Big Science in Music,” paired with its realization in the form of a corresponding institute, became virulent. During their countryside walk, Hofmann and Mazzola pleaded for the creation of an association for the advancement of an Institute for Fundamental Research in Music (IFM). Meanwhile, the first systematic treatment of Mathematical Music Theory (MaMuTh), Geometrie der Töne, was published in 1990 [73]. This gave a theoretical background to the ideas promoted thus far. 2.15.2 The IFM Association: The Period Preceding the General Proliferation of the Internet (1992-1999) In 1992 an interest group was formed with the goal of finding people who support the idea and the foundation of an IFM association. In summer 1993, such an association, named “Verein zur Gründung eines Instituts für Grundlagenforschung in der Musik,” was established. Among the initial members were Mazzola and his wife, Christina, the statistician and composer Jan Beran, and the mathematician and musicologist Daniel Muzzulini. The next step was to win over members of a prominent patronage: (Wolfgang Auhagen, Valentin Braitenberg, Manfred Eigen, Heinz Götze, Walther von Hahn, Michael Leyton, Ernst Lichtenhahn, Helga de la Motte, Hellmuth Petsche, Roland Posner, Peter Stucki, Ernst Terhardt, Walter Thirring, and Heinz-Gregor Wieser). The association’s board succeeded to manage the periodic information of the members of the association and the members of the patronage about the association’s achievements, to organize annual meetings and symposia with scientific and artistic musical themes. The last such event took place in 2002 in Zürich during the Third Seminar on Mathematical and Computer-Aided Music Theory, where two pianists, a music performer, and an interactive multimedia environment were engaged. 2.15.3 The Virtual Institute: Pure Virtuality (1999-2003) The association gained recognition, which culminated on the one hand in a collaboration with the MultiMedia Lab of the Computer Science Department

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of the University of Zürich, directed by professor Peter Stucki, a patronage member, and on the other in the successful funding by the VW Foundation of a MaMuTh research group at the Technische Universität Berlin, directed by the association’s vice president, Thomas Noll. Moreover, IFM collaboration with the IRCAM in Paris intensified and resulted in the MaMuPhi and MaMuX seminars. As the Internet became more and more omnipresent, the IFM association also recognized that the physical presence of an institute was no longer of primordial importance. Instead, the IFM association succeeded in the foundation of a virtual institute, named i2musics (internet institute for music science). One should note that these synergies also led to a number of academic success stories, for example the habilitation of Joachim Stange-Elbe in a computergenerated analysis and performance of Bach’s Art of Fugue at the University of Osnabrück, just to name one case. Perhaps the most successful result of IFM collaboration is the book The Topos of Music [75]. Mazzola wrote it with the aid of nineteen collaborators and contributors from Mexico, the U.S., France, Germany, Austria, and Switzerland—many of them IFM members. Nonetheless, the i2musics has progressively developed an existence of exclusive online virtuality. This means that our efforts to build a working virtual institute were not paralleled by a successful implementation of software for collaborative virtual music science, although a prototype of such a platform was implemented by two very gifted students. 2.15.4 Dissolution of the IFM Association (2004) So, is the termination of the IFM association a “mission accomplished” or is it a failure of the bold idea for a Big Science in Music? We are proud of the many relationships, connections, synergies, and careers that had been created over ten years. But it must also be admitted that music science is still a difficult job when its fundamental role in our society is pursued on a serious level of exact argumentation and conceptualization and not merely on the feuilletonistic thread of an ornamental descriptive narration. What came after this turning point? First of all, one had to let the germs develop globally and also in the many different facets of knowledge science, of spiritualization of science towards the synthesis that author Hermann Hesse dubbed “Glasperlenspiel” (Glass Bead Game) in one of his famous novels. And then, we needed citizens who were willing and able to think and act beyond ignorant and brutal economic dimensions, because brute survival will never create symphonies, will never need lullabies, will never need free jazz, but only gun and rocket sounds and execution rhythm. Perhaps we really needed to understand and to make others understand that life is all about music, that this art of time is infinitely more important than money, because it is not an extension, not a simple quantity. It’s life, and music is about life. But this was

2.16 The Society for Mathematics and Computation in Music

33

a hard thing to sell. Well, the next section offers a solution to this challenge: the Society for Mathematics and Computation in Music. Last but not least, this book, being co-authored by Mazzola and two excellent PhD students at the School of Music of the University of Minnesota, Maria Mannone and Yan Pang, is a marvelous proof of the viability of Mathematical Music Theory for a creative and scientifically based future of music.

2.16 The Society for Mathematics and Computation in Music The first half of the 21st century was germinal for the globalization of mathematical music theory. Four International Seminars on Mathematical Music Theory and Music Informatics were organized: in Saltillo, Mexico (2000, by the mathematician Emilio Lluis Puebla, during the an- Fig. 2.31. The Society for nual conference of the Mexican Mathematical So- Mathematics and Computaciety), Sauen, Germany (2001, by Thomas Noll tion in Music from Technische Universität Berlin), Zürich, Switzerland (2002 at the Computer Science Department, by Guerino Mazzola and the IFM), and Huatulco, Mexico (2010 by Lluis Puebla), see [81]. In 2003 at the IRCAM in Paris, Andreatta organized a memorable seminar around American Set Theory [7], which was a first gathering of many American and European music theorists, including Moreno Andreatta, Jean-Michel Bardez, Célestin Deliège, Joseph Dubiel, Jason Eckardt, Allen Forte, Xavier Hascher, Guerino Mazzola, Andrew Mead, Robert Morris, Jean-Jacques Nattiez, Paul Nauert, John Rahn, André Riotte, and Luigi Verdi.

Fig. 2.32. The 2015 MCM conference participants at Queen Mary University in London.

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2 Short History of the Relationship Between Mathematics and Music

This germinal power eventually helped create the Society for Mathematics and Computation in Music (MCM). Music theorist Robert Peck from Louisiana State University in Baton Rouge formally brought this society to life in 2006. In June 2007, the Society’s first biannual conference took place in Berlin at the National Institute for Music Research, followed by the 2009 MCM conference at Yale University, the 2011 MCM conference at the IRCAM in Paris, the 2013 MCM conference in Montreal at McGill University Schulich School of Music & CIRMMT, and the 2015 MCM conference at the Queen Mary University in London, see Figure 2.32 for a group photograph. The conference in 2017 will be held at the National Autonomous University of Mexico (UNAM) in Mexico City. The proceedings of all these conferences are documented by Springer International Publishing. The Society also has its own journal, the Journal of Mathematics and Music, published by Taylor & Francis, and the Springer book series Computational Music Science (editors: Andreatta and Mazzola). Mazzola is the MCM’s founding president, see [86] for more information. Summarizing, the long historical process of the counterpoint of mathematics and music has now led to a synthesis of art and science that rewards many passionate and often desperate efforts to reunite beauty and truth.

Part II

Sets and Functions

3 The Architecture of Sets

Summary. We present the theory of pure sets—the basis of all classical mathematics—in the form of the Zermelo-Fraenkel Axiomatics. This theory is illustrated with Iannis Xenakis’ composition Herma. –Σ–

3.1 Some Preliminaries in Logic This is not an introduction to formal logic, but an absolutely minimal presentation of logical sentences that describe how logical statements—statements that are either true or false, and nothing more—can be combined. We shall only look at such sentences, which are either true or false, nothing else, and only one of these two values. This is classical logic. For example, the sentence A: “Two plus three equals five.” is true, and the sentence B: “The Earth is a disc.” is false. In classical logic, sentences can be combined into new sentences as follows, where A, B are sentences: 1. Negation: The sentence NOT A is true if A is false, it is false if A is true. 2. Conjunction: The sentence A AND B is true if both A and B are true, it is false in all other cases. 3. Disjunction: The sentence A OR B is false if both A and B are false, it is true in all other cases. 4. Implication: The sentence A IMPLIES B is false if A is true and B is false, it is true in all other cases. Here is a tabular representation of these four logical functions: A NOT A F T F T

A F F T T

B A AND B F F T F F F T T

A F F T T

B A OR B F F T T F T T T

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_3

A F F T T

B A IMPLIES B F T T T F F T T 37

38

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3 The Architecture of Sets

Exercise 1 Show that the logical values of the two combinations A IMPLIES B

and

(NOT A) OR B

coincide. In mathematical prose, we often encounter the double sentence (A IMPLIES B) AND (B IMPLIES A). This combination is abbreviated by writing (and saying) A iff B, representing the wording “if and only if” for the double implication.

ˇ “*

Example 1 Here are some musical examples of such logical combinations: 1. If A = “Any fifth interval (7 semitones, such as c → g) is a consonant interval.” is true, its negation NOT A is false and means NOT A = “Any fifth is not a consonant interval.” 2. If A = “Composers need to be inspired.” and B = “Composers need to have strong creative tools.”, then A AND B is only true if both A and B are true, i.e., if “Composers are inspired and have strong creative tools.” is true. The conjunction is false if, for example, composers are inspired but don’t have strong creative tools. 3. If A = “The composer is very musical.” and B = “The composer has excellent compositional skills.”, then A OR B is true if at least one of these sentences holds. That is, it is false if the composer is neither very musical nor has excellent compositional skills. 4. If A = “The composer is very creative.” and B = “The composer has success.”, then A IMPLIES B is only true if either B is true or both A and B are false. It is false if the fact that the composer is very creative implies that he or she does not have any success.

Remark 1 We shall make many examples to illustrate ideas, concepts, and results in this book. However, in the early stage of this discourse, it is often not possible to make examples that only use the material we have developed so far. Therefore, some examples will not be logically consistent, anticipating some details that will be developed later. However, no example will be involved in circular arguments. All examples will be an illustration of our arguments without violating the logical process of our discourse. We hope the readers can accept such illustrations in favor of easier understanding of the matter being unfolded.

3.2 Pure Sets In the first decades of the 20th century, when set theory was developed by Georg Cantor, the initial enthusiasm was soon dampened by some serious logical catastrophes. The initial impetus of set theory was that sets were thought

3.2 Pure Sets

39

to be very simple concepts: A set was conceived as being a kind of container that could contain some objects, similar to a bag when you buy food in a market. If an object x is contained in a container X, one says “x is an element of X” and usually writes x ∈ X. For example, a set that has no elements is an “empty bag”, denoted by ∅.

ˇ “*

Example 2 Musical scores are containers of notes, see Figure 3.1. In music, there are also containers of musical parts: Song form A-B-A, sonata form: {exposition, development, recapitulation, coda}, etc.

Fig. 3.1. A score is a container of note objects. It is constructed from the empty score container by adding a number of notes, rests, etc. The empty score to the right, representing a real composition, is a pure container. This marche funèbre was composed by Alphonse Allais in 1897 for the funeral of a deaf friend.

In this undisciplined generality, absurd constructions were possible. Here is a famous one, Bertrand Russell’s antinomy: Consider the set S, which is defined such that its elements are all sets that are not elements of themselves. For example ∅ ∈ S.

Fig. 3.2. An impossible set.

Moreover, when we face the question of whether S ∈ S or not, a problem arises: If S ∈ S, then, by definition of S, S should not be an element of S, a

40

3 The Architecture of Sets

contradiction. Else, if S is not element of S, then it must, again by definition of S, be an element of itself, and this is another contradiction. In other words, such an S does not exist. A second problem was included in this undisciplined approach. It was not clear what kinds of objects would be allowed to build sets or be included as elements of sets: violins, flowers, ideas, musicians, whatnot! All of this forced mathematicians to restart their enterprise and work out a set theory that was as pure as Hanslick’s absolute music: pure sets. Pure sets are very simple concepts. They have a name, which is simply a symbol, a word, or the like, such as ∅, A, Ch, X2 , etc. Next, a set has elements. This is expressed by the symbol we already used: x ∈ X stating that the set named x is an element of the set named X. But what is x? It is again a set, much like X. In other words, sets have sets as their elements. A set is determined by pointing to its elements. That’s all—a set is a multiple pointer to sets. Therefore: A set is identified by its name and its elements. Of course, this simple and circular conceptualization is no guarantee that Russell’s antinomy is eliminated. We have to impose a list of rules of how to build sets. This set of rules is called the Zermelo-Fraenkel-Choice set theory, ZFC for short (after mathematicians Ernst Zermelo and Abraham Fraenkel, and the axiom of choice). It is built from only eight axioms—rules that define which sets can be constructed. This system is not absolutely foolproof, but can guarantee the elimination of antinomies of Russell type. It is, however, nearly miraculous that the entire mathematical language, all its concepts, and all proof techniques can be deduced from this tiny list of initial statements. We use a notation of sets with curly brackets. If a, b, ..., z is list of sets that are the elements of a set X, we write X = {a, b, ..., z}. If F (a) is an attribute that a set can or cannot have, then the set of all elements having this attributes (if it exists) is denoted by X = {a|F (a)}. Before we get started with these axioms and their consequences, let us briefly explain the terminology of different types of statements in mathematics: • An axiom is a statement that is supposed to be true, but no proof thereof is given. • A theorem is an important true statement (that needs a usually involved proof). • A proposition is a true statement, requiring a proof, but is less important than a theorem. • A lemma is a true statement, requiring a proof, but which is only an auxiliary result used for proving a proposition or a theorem. • A corollary is a true statement that follows easily from a proposition or theorem. • A definition is the introduction of a new concept that uses previously constructed concepts. Some concepts might be introduced in a circular way, re-

3.2 Pure Sets

41

ferring to themselves in the definition, but in this book, only the set concept will be introduced in a circular way. When introducing a new mathematical concept N ew, its definition, if it is given by an equation, is written as N ew := Old, or Old =: N ew, where the Old part contains already defined concepts. • A sorite is a collection of easy statements that follow directly from given definitions. Here is the list of the eight ZFC axioms: Axiom 1 (Axiom of Empty Set) There exists the empty set, denoted by ∅, which has no elements, i.e., for all sets x, x ∈ ∅, or ∅ = {}.

ˇ “*

Example 3 The empty set is the formal concept that represents the empty container. Figure 3.3 shows a famous composition by John Cage that—similar to the above example—is an empty container. Only the duration, 4 minutes and 33 seconds, is determined for Cage’s composition [25].

Fig. 3.3. John Cage’s composition 4’33", an empty set of notes. Only the container is set—this is the musical analogue of the empty set ∅. The score is restricted to the command to be quiet, tacet in Latin, for 4’33”. It was composed in 1952 [25].

Here, x ∈ y means that x ∈ y is false. For the second axiom, we need to define equality of sets. Definition 1 If a, b are sets, we say that a is a subset of b, in symbols a ⊂ b, if x ∈ a implies x ∈ b for every element x ∈ a. See Figure 3.4 for an example of subsets in music. We say that the sets a, b are equal, in symbols a = b, if a ⊂ b and b ⊂ a. The next axiom is purely technical to deal with equality.

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3 The Architecture of Sets

Fig. 3.4. The left hand score (part below) is a typical subset of the total score for both hands (upper part). We write the subset symbol vertically, so do not confuse it with the union symbol. The sound example is Yan_set.

Axiom 2 (Axiom of Equality) For sets a, x, y, if x ∈ a and x = y, then y ∈ a. Axiom 3 (Axiom of Union) If a is a set, there is the set {x| there is an element b ∈ a such that x ∈ b}

This set is denoted by a and is called the union of a.

Fig. 3.5. The total score of a sonata is the union of the set a = {A, B, C, D} of its subscores A, B, C, D.

3.2 Pure Sets

43

ˇ “* Example 4 If we think of a sonata a = {A, B, C, D} as being a set with the elements A = exposition, B = development, C = recapitulation, D = coda,

then a is the entire score, i.e., the container that contains all elements of A, B, C, and D, see Figure 3.5. Notation 1 If a = {b, c} or a = {b, c, d}, we write respectively.



a = b∪c or



a = b∪c∪d,

Axiom 4 (Axiom of Pairs) If a and b are two sets, then there is the pair set {a, b}. Notation 2 If Φ is an attribute of sets, we simply write Φ(x) instead of “ Φ(x) is true.” Axiom 5 (Axiom of Subsets for Attributes) For any set a, if Φ is an attribute, then there is the subset a|Φ of a that is defined by a|Φ = {x|x ∈ a AND Φ(x)}.

ˇ “*

Example 5 If a is a score, i.e., a set of notes, rests, etc., then taking Φ(x) for “x = C# OR x = G” extracts the subset a|Φ = {C#, G} if both, x = C# and x = G are elements of the score a. If Φ(x) stands for “duration of x=”, then a|Φ is the subscore of all eighth notes in a. Axiom 6 (Axiom of Powersets) For any set a, there is the powerset 2a of a, which is defined by 2a = {x|x ⊂ a}.

ˇ “*

Example 6 Think of a rehearsal situation. If a is a score, the conductor may choose to hear any subscore of a—strings, strings and winds, winds and brass, etc. The set that contains all the possible subscores is the powerset 2a of that score. Lemma 1 For any set a, there is the set a+ = a∪{a}, it is called the successor of a. Proof 1 We construct the successor a+ of a as follows. First, by Axiom 6, since a ⊂ a, a ∈ 2a . Second, by Axiom 5, take the attribute Φ(x) meaning x = a. Then we get {a} = 2a |Φ. Third, by Axiom

4, the pair set b = {a, {a}} exists. Finally, by Axiom 3, we get the set a+ = b = a ∪ {a}. This is a typical procedure for a proof: We use axioms or already proved facts and deduce new results. Axiom 7 (Axiom of Infinity) There is a set w with ∅ ∈ w and such that x ∈ w implies x+ ∈ w.

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3 The Architecture of Sets

Fig. 3.6. In a score with two sets, a for the cantus firmus (CF) voice and b for the discantus (D) voice, we have the intersection a ∩ b of the common notes.

This reminds us of the Chinese proverb, “One creates two, two creates three, three creates everything.” The axiom of infinity guarantees that there is one set container that for every element it also contains its successor. Definition 2 For two sets a, b, the set a ∩ b = {x|x ∈ a AND x ∈ b} (which exists by Axiom 5) is called the intersection of a and b. If a ∩ b = ∅, we say that a and b are disjoint. See Figure 3.6 for an example. Axiom 8 (Axiom of Choice) Let d be a set whose elements are all nonempty, and such that any two different elements x, y ∈ d are disjoint, then there is a subset c ⊂ d such that for every x ∈ d there is exactly one element in c ∩ x.

Fig. 3.7. The axiom of choice guarantees that we may create a set c that has exactly one element in each of the elements of a set a. Here we take the first note of each of the three measures in the set a.

Here are the first immediate facts about sets. Proposition 1 For three sets a, b, c we have (i) (Commutativity of union) a ∪ b = b ∪ a.

3.2 Pure Sets

45

(ii) (Associativity of union) a ∪ (b ∪ c) = (a ∪ b) ∪ c. We therefore write a ∪ b ∪ c for this set. Proposition 2 If a = ∅, then the set {x|x ∈ z for all z ∈ a} 

exists and is denoted by a.  If a = ∅, then a doesn’t exist  since its defining attribute “x ∈ z for all z ∈ a” would hold for all sets and a would be the set of all sets, which does not exist. Definition 3 If a, b are sets, their difference is denoted by a − b and defined by a − b = {x|x ∈ a AND x ∈ b}. In the example of Figure 3.6, the difference a − b consists of all CF notes that are not shared with the D notes. Sorite 1 For any three sets a, b, c we have (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

c − a ⊂ c, If a ⊂ c, then c − (c − a) = a, c − ∅ = c, c − c = ∅, a ∩ (c − a) = ∅, If a ⊂ c, then a ∪ (c − a) = c, c − (a ∪ b) = (c − a) ∩ (c − b), c − (a ∩ b) = (c − a) ∪ (c − b), c ∩ (a − b) = (c ∩ a) − (c ∩ b).

3.2.1 Boolean Algebra With these constructors of sets, one can define a calculus on 2a called Boolean algebra of a. We first present the theory and then discuss a composition, Herma by Xenakis, that was composed using this theory. The Boolean algebra of 2a is defined as follows. Definition 4 If x, y ∈ 2a , then we define x + y = (x ∪ y) − (x ∩ y). We further define x.y = x ∩ y. Proposition 3 For set a, and for any three elements x, y, z ∈ 2a , we have: (i) (commutativity) x + y = y + x, x.y = y.x. (ii) (associativity) x + (y + z) = (x + y) + z, x.(y.z) = (x.y).z; we therefore also write x + y + z and x.y.z, respectively. (iii) (neutral elements) We have x + ∅ = x, x.a = x. (iv) (distributivity) x.(y + z) = x.y + x.z. (v) (idempotency) x.x = x.

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3 The Architecture of Sets

(vi) (involution) x + x = ∅. (vii) (solutions of equations) the equation x + y = z has exactly one solution w, i.e., there is exactly one set w ⊂ a such that w + y = z. Remark 2 This structure will later be discussed as the crucial algebraic structure of a commutative ring, see Chapter 24. √

Exercise 2 Let a = {r, s, t, w} with four different elements r, s, t, w. In the Boolean algebra 2a , calculate the solution x of x + y = z within 2a for y = {r, s, w}, z = {s, w}.

ˇ “*

Exercise 1 Let a = {♩, , }. In the Boolean algebra 2a , calculate the solution w of w + y = z within 2a for y = {, }, z = {♩, }. √

Exercise 3 Let a = {∅}. Calculate the complete table of sums x + y and products x.y for x, y ∈ 2a , using the symbols 0 = ∅, 1 = a. 3.2.2 Xenakis’ Herma The following example has been discussed in [4]. Here, we give a short summary of that discussion. Xenakis was one of the first composers to use advanced mathematical procedures to compose music. His method was described in his book Formalized Music [116]. The composition Herma for piano uses Boolean algebra to create its detailed structure. Xenakis calls his method “symbolic music.” In [4], the authors also provide us with a computer-aided implementation of the composition, using their music software OpenMusic. The Boolean algebra on 2a that Xenakis uses starts from a frame set a = R, which, according to the composer, is the set R “of all the sounds of a piano.” That means all the pitches of the piano (usually 88 in number). He then selects three subsets A, B, C of R, as shown in Figure 3.8, and creates new subsets using the Boolean operations of union, intersection, and complementation. We have to learn why these operations are all Boolean. The intersection is the Boolean product. For the union x ∪ y of x, y ∈ 2a , observe that a − x ∪ y = (a − x) ∩ (a − y) = (a − x).(a − y). But the complementation a − x is the solution z of the equation z + x = a. Therefore, after taking differences a − x, a − y, we can construct a − x ∪ y and then its difference set x ∪ y = a − (a − x ∪ y). The core set needed by Xenakis is denoted by F . It is achieved by a flow chart of preliminary operations as shown in Figure 3.9. This flow chart shows two construction modes of F : plane (1), associated with dynamics signs f and ff , and plane (2), associated with dynamics signs ff and ppp. This flow chart is the basis of the display of the note sets associated with the different operations in musical time. It has been observed in [4] that Xenakis here works on two different time levels: the symbolic or

3.2 Pure Sets

47

Fig. 3.8. Xenakis selects three subsets A, B, C of the set R of all piano sounds.

logical “outside-of time” realm of imaginary time of thoughts, and the musical “in-time” of musical events in real time. We shall come back to this distinction when discussing performance theory in Chapter 32.

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3 The Architecture of Sets

Fig. 3.9. Two construction modes define F : plane (1), associated with dynamics signs f and ff , and plane (2), associated with dynamics signs ff and ppp.

4 Functions and Relations

Summary. The elements of a set can be written in any order, and we don’t yet have a way to define ordered structures. So we now develop the theory of ordered structures. It also allows us to define the central concept of a function. –Σ– So far, the elements of a set can be displayed in any order, e.g., {a, b, c} = {c, b, a}. But the order of a collection may be relevant for the description of a mathematical situation. We will take care of this problem in this chapter.

4.1 Ordered Pairs and Graphs Definition 5 If x, y are two sets, the ordered pair (x, y) is defined to be the following set: (x, y) = {{x}, {x, y}} Observe that the set (x, y) always exists; it is a subset of the powerset of the pair set {x, y}. Here is the essence of this definition: Lemma 2 For any four sets a, b, c, d, we have (a, b) = (c, d) iff a = c and b = d. Therefore, one may speak of the first and second coordinate a and b of the ordered pair (a, b).

ˇ “* Example 7 In music theory, they often speak about “ordered sets” in the sense that an “ordered set” {c, e}, representing an interval, is different from an “ordered set” {e, c}. But this is an abuse of language. In mathematics, sets are always unordered collections of their elements. An “ordered set” {c, e} would then be represented as an ordered pair (c, e), which is different from the ordered pair (e, c). © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_4

49

50

4 Functions and Relations

√

Exercise 4 Defining (x, y, z) = ((x, y), z), show that (x, y, z) = (u, v, w) iff x = u, y = v, z = w. Definition 6 For sets a, b, their cartesian product is the set a × b = {(x, y)|x ∈ a, y ∈ b} √

Exercise 5 Check with the Powerset Axiom that a × b always exists.

Definition 7 A graph g is a subset g ⊂ a × b of a cartesian product.

Fig. 4.1. A drawing of Antoni Gaudí’s Sagrada Família church in Barcelona (left) as a graph in the cartesian product space of height and width can be represented also as a graph of notes, a composition living in the cartesian product space of onset times and pitches. We come back to questions of visualization and sonification for music in Section 27.3.

√

Example 1 The diagonal is the graph Δa ⊂ a × a of all ordered pairs (x, x), x ∈ a (Figure 4.2). For a graph g ⊂ a × b, the inverse graph g −1 ⊂ b × a is the graph of all ordered pairs (y, x) such that (x, y) ∈ g. Definition 8 For a graph g ⊂ a × b, we have two projections, pr1 (g) = {x|there exists a (x, y) ∈ g} ⊂ a and pr2 (g) = {y|there exists a (x, y) ∈ g} ⊂ b.

4.1 Ordered Pairs and Graphs

51

Fig. 4.2. The diagonal graph of set a.

Definition 9 (Composition of Graphs) For graphs g ⊂ a × b, h ⊂ b × c, their composition is the graph h ◦ g = {(x, z)|there exists y ∈ b such that (x, y) ∈ g and (y, z) ∈ h}. Lemma 3 (Composition of Graphs is Associative) If g ⊂ a × b, h ⊂ b × c, k ⊂ c × d are three graphs, then k ◦ (h ◦ g) = (k ◦ h) ◦ g, which we therefore denote by k ◦ h ◦ g.

Fig. 4.3. The composition graph g in the cartesian product of onsets and pitches, together with its two projections, which show the onsets and pitches that are involved in this composition.

ˇ “*

Exercise 2 Consider the graph g shown in Figure 4.3. Show that the composition g −1 ◦ g is the graph in the cartesian product Onsets × Onsets whose pairs (s, t) are the pairs of times that show repeated pitches.

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4 Functions and Relations

4.2 Functions

Fig. 4.4. A melody is a typical functional graph in the onset-pitch space. For every onset, there is at most one associated pitch in the graph. The flute and the monochord are instruments on which one plays only melodies, never two sounds at the same time.

Definition 10 A graph g ⊂ a×b is called functional if (x, y), (x, z) ∈ g implies y = z, see Figure 4.4. A function is a triple f = (a, b, g), where g ⊂ a×b is a functional graph and pr1 (g) = a. The set a is called the function’s domain and denoted by dom(f ), while the set b is called the function’s codomain and denoted by codom(f ). We often write f : a → b for f . For every x ∈ a there is a unique y ∈ b such that (x, y) ∈ g. We write y = f (x). See Figure 4.5 for an example.

Fig. 4.5. If the onset times are eight multiples of one eighth (), then this score shows a melodic function: It is a functional graph that has values for all available onset times.

√

Example 2 For every set a we have the identity function Ida = (a, a, Δa). Denote 1 = {∅}. Then, for every set a, there is a unique function ! : a → 1. And

4.2 Functions

53

there is a unique function ! : ∅ → a. We use the symbol ! of functions whenever there is a unique function in given context. Definition 11 If we have pr2 (g) = b for a function f = (a, b, g), we say that f is surjective (epi) and also write f : a  b. If f (x) = f (y) implies x = y for any two arguments x, y ∈ a, we say that f is injective (mono) and also write f : a  b. If f is epi and mono, we say that f is bijective (iso) and also write ∼ f : a → b.

Fig. 4.6. Illustration of three function types: injective, surjective, and bijective.

If f : a → b and g : b → c are functions then we have the composition of functions g ◦ f : a → c that is defined by the composition of these functions’ graphs. Clearly, the composition of two epi/mono/iso functions is again epi/mono/iso. And a function f is iso iff the inverse of its graph is again functional, and the function that is defined by the inverse graph is denoted by f −1 . For an iso f we therefore have f −1 ◦ f = Ida , while f ◦ f −1 = Idb .

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Exercise 3 In musical composition, there is a simple method to create prototypes of melodies: dodecaphonism, which was invented by composer and theo∼ rist Arnold Schoenberg around 1921. He conceived 12-tone series s : O12 → P12 , where O12 is a set of 12 onset times O12 = {o0 , o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 , o9 , o10 , o11 },

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Fig. 4.7. Note names for keyboard keys and note and rest durations.

and where P12 = {p0 , p1 , p2 , p3 , p4 , p5 , p6 , p7 , p8 , p9 , p10 , p11 } is the set of 12 pitch names on the piano, typically named by p0 = C, p1 = C , p2 = D, p3 = D , p4 = E, p5 = F, p6 = F , p7 = G, p8 = G , p9 = A, p10 = A , p11 = B, where the symbol denotes the black key to the right of the white key it follows, e.g., C is the black key to the right of the white key C. See also Figure 4.7, where we show the note names corresponding to keys, as well as durations of notes and the corresponding rests.

Fig. 4.8. 12-tone series for Anton von Webern’s compositions op. 17.2 and op. 30. ∼

For any two such 12-tone series g, h : O12 → P12 , one may consider their ∼ composition h−1 ◦ g : O12 → O12 . Explain the meaning of pairs of times in this

4.2 Functions

55

composed bijection on the time set. Calculate the resulting bijection for the two series composed by Anton von Webern in his op. 17.2 and op. 30 as shown in Figure 4.8. Definition 12 If f : a → b, g : c → d are two functions, their cartesian product function is the function f × g : a × c → b × d that is defined by f × g(x, y) = (f (x), g(y)). √

Exercise 6 Show that if f, g are both epi/mono/iso, then so is their cartesian product f × g.

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Example 8 In serial music, Schönberg’s idea to consider a bijection s : ∼ O12 → P12 was generalized to other parameters beyond pitch. For example, serial composers also consider 12 durations in a set D12 = {d0 , d1 , d2 , d3 , d4 , d5 , d6 , d7 , d8 , d9 , d10 , d11 }. ∼

∼

Given a pitch series s : O12 → P12 as well as a duration series t : O12 → D12 , ∼ they then take the cartesian product series s×t : O12 ×O12 → P12 ×D12 . To get a new series that would determine pitch as well as duration for each onset time, they compose it with the diagonal injection Δ : O12  O12 × O12 : o → (o, o). This yields the desired function s × t ◦ Δ : O12 → P12 × D12 .

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Example 9 In music theory, there is a theory of harmony that was proposed by Hugo Riemann, the (musical) function theory. It is a funny coincidence that there is also a mathematical function theory (same name!) that was developed by Bernhard Riemann. But beyond these similarities, the two theories have nothing in common. Hugo Riemann’s musical function theory was the program for attributing to every possible chord one of three possible harmonic functions, Tonic (T ), Dominant (D), and Subdominant (S) [29]. This idea should have defined a harmonic syntagmatics, i.e., a procedure to attribute to compositions sequences of harmonic functions to represent the meaning of the harmonic movement through time. A classical syntagm in this spirit is the sequence IC = {c, e, g}, IVC = {f, a, c}, VC = {g, b, d}, IC = {c, e, g} of triadic degree chords (here in C major), the standard tonality of white keyboard keys when starting at the tonic key C. This candential sequence is understood as a shorthand for the harmonic identification of the given tonality (here, in fact, one octave of C-tonality is identified as the union IC ∪ IVC ∪ VC ). In such a cadence, degree I was thought to have the function value T , degree V value D, and degree IV value S. Riemann’s idea was to attribute such function values to all chords, effectively defining a tonality by such a function. So tonality C major would be defined as a function C − T onality : Ch → T DS = {T, D, S} whose domain Ch is the set of all chords, which are by definition all the finite sets in 2P itches , where P itches represents the set of all pitches. We shall define “finiteness” of a

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set in Section 6.3, but here, just take it as the property that one can count a chord’s elements and end up after a number of steps. This means that Riemann wanted to define all possible tonalities X as functions X − T onality : Ch → T DS. He however didn’t intend to allow general tonality functions; he intended to extend functions X − T onality|T estSet : T estSet → T DS defined on a small subset T estSet ⊂ Ch of well-known chords to the total function. In particular, his intention was to get the values C − T onality(IC ) = T, C − T onality(VC ) = D, C − T onality(IVC ) = S. He also set the condition that values for other triadic degrees IIC = {d, f, a}, IIIC = {e, g, b}, V IC = {a, c, e}, V IIC = {b, d, f } should be determined according to special relations of these degrees to the three degrees I, V, IV . We shall discuss in Chapter 16 the reasons why Riemann’s project could not work. 4.2.1 Equipollence In set theory, a set a is considered to be “essentially the same as” or equipollent ∼ to b if there is a bijection f : a → b. Equipollence has these three properties: Lemma 4 Let a, b, c be sets. (i) (Reflexivity) a is equipollent to a. (ii) (Symmetry) If a is equipollent to b, then b is equipollent to a. (iii) (Transitivity) If a is equipollent to b, and b is equipollent to c, then a is equipollent to c. The class of all sets that are equipollent to a is called a’s cardinality. Cardinality groups all sets that are related by bijections among one another. We want to briefly summarize the question of whether there are “arbitrary large” sets. To begin with, observe that we have an injection a  2a : x → {x}. ∼ But there is no bijection a → 2a , 2a is strictly “larger” than a (see Figure 4.9). In fact, suppose we have f : a  2a . Then take the subset n = {x|x ∈ f (x)} of a. For n, there is no y ∈ a such that f (y) = n. If y ∈ f (y) = n, we have y ∈ n by definition of n, and if y ∈ f (y) = n, then y ∈ n, so nothing works here. This argumentation is similar to the one used for Russell’s antinomy. From this, the Bernstein-Schröder theorem implies that there is also no injection 2a  a. This theorem states that for two injections a  b, b  c and a, c being equipollent, the middle set b must also be equipollent to a, c. Therefore an injection 2a  a would imply that a and 2a are equipollent, what we just disproved. See [77, Vol. I, Section 4.1] for details.

4.3 Relations

57

Fig. 4.9. The powerset of set a is larger than a, as is seen here for subsets of a that are not hit by the map x → {x}.

4.3 Relations Relations among elements of a given set are essential for the description of musical configurations. Definition 13 Given a set a, a binary relation on a is a graph R ⊂ a × a. We often write xRy for (x, y) ∈ R. In specific situations, many symbols are used, such as ≤, p4 > p5 > p6 > p7 > p8 > p9 > p10 > p11 . On the value set 2 the ordering 0 < 1 is conserved. Then, a chord (ci ) precedes a chord (di ) if the first index j from the right where they differ has cj < dj . This means that chords with longer sequences of zeros from the right precede others. For example, (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1) > (0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1) or (0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0) > (0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0). Using this well-ordering and a specific method to define equivalence classes of chords, the so-called prime form of a chord [108] is defined. We come back to this topic in Chapter 19.

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Example 16 A practical example of lexicographic ordering is the ordering among notes in a score. Suppose that notes of a score are given as elements of a fourfold cartesian product O × P × L × D, where O, P, L, D are linearly ordered sets of onset, pitch, loudness, and duration. Then the lexicographic ordering

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on O × P × L × D is linear and works as follows. A note n1 = (o1 , p1 , l1 , d1 ) precedes note n2 = (o2 , p2 , l2 , d2 ) iff the first coordinate where they differ is smaller for n1 than for n2 . For example, if both, n1 , n2 , are in a chord, then the lower pitch defines the preceding note.

Part III

Numbers

6 Natural Numbers

Summary. Natural numbers are the first topic studied by all students in first years of elementary school. Here, the classic definitions of ordinal and natural numbers are entirely derived from set theory. The well-known five Peano axioms that define natural numbers are now presented as a theorem. –Σ–

Fig. 6.1. The number domains within the class of sets.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_6

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The famous mathematician Leopold Kronecker (1823-1891) said, “God created the natural numbers, everything else is human work.” In fact, natural numbers (usually meaning the numbers 1, 2, 3, . . .) are the basis of the entire mathematical reasoning and structural architecture. In view of Kronecker’s statement, the achievements of set theory are also remarkable since they disprove Kronecker: We are now capable of constructing the natural numbers (and, of course, all the others) without direct reference to a divine creator, but simply to ZFC set theory, see Section 3.2. We shall not give all the proofs here, but state some theorems to get to the natural numbers as fast as possible. In the first section, we want to present the construction of ordinal numbers, a type of sets where general arithmetic can be performed. In the second section, extract the set N of natural numbers, which are very special ordinal numbers. In the third section we shall start working with finite sets that use natural numbers as basic concepts. This chapter is the basis for the entire arithmetic of numbers (integer, rational, real, and complex numbers) that will be unfolded in the sequel. The general idea is this: Start with the empty set zero, 0 = ∅, then take its successor 1 = 0+ = {0}, then 2 = 1+ = {0, 1}, then 3 = 2+ = {0, 1, 2}. We see that the earlier numbers b are always set-theoretic elements of the later ones, e.g., 1 ∈ 3. The main task now is to define this construction in a rigorous way and look for a set that contains all these sets. Section 6.1 is necessary to extract a basic type of sets, the ordinal numbers, from which natural numbers can be constructed. Ordinal numbers are characterized by three simple attributes, and the corresponding propositions are easy to prove. They have no direct meaning for music so far; therefore the reader may skip this section or look it up when characteristic properties of ordinal numbers are at stake. However, it is important to observe that without the concept of ordinal numbers, no clean theory of numbers is feasible. Ordinal numbers are that missing link that led Kronecker to believe that mathematicians cannot succeed on their own.

6.1 Ordinal Numbers Definition 21 A set a is an ordinal number iff it is transitive, alternative, and founded. This means the following: (i) a is transitive iff x ⊂ a for all x ∈ a. (ii) a is alternative iff x = y or x ∈ y or y ∈ x for all x, y ∈ a. (iii) a is founded iff for every non-empty subset b ⊂ a, there is x ∈ b such that x ∩ b = ∅. √

Exercise 10 Prove that 0, 1, 2, 3 are ordinal numbers. Suppose there is a set J = {J}. Prove that J is not ordinal.

6.2 Natural Numbers

73

Here are the most important properties of ordinal numbers, which we state without (the quite simple) proof, but refer to [77, Section 5.1]. Theorem 5 (Characteristic Properties of Ordinal Numbers) (i) If a is ordinal, then x is ordinal for every x ∈ a. (ii) If a, b are ordinal, then (exclusively) a = b or a ∈ b or b ∈ a. In other words, ∈ is linear order between ordinals (although no set of all ordinals is at stake here, but see below). (iii) If d is ordinal, and a ⊂ d is non-empty, then there is x ∈ a such that x ∈ y or x = y for all y ∈ a. One can see this as a well-ordering among ordinals (although no set of all ordinals is at stake here). (iv) If a, b are ordinals with a ∈ b, then either a+ ∈ b or a+ = b. And a+ ∈ b+ . There is no x such that a ∈ x ∈ a+ . So the relation a ∈ a+ is a “minimal” one in the ordering among ordinals. (v) If a, b are ordinals, then a = b iff a+ = b+ . (vi) If a, b are ordinals, then a is equipollent to b iff a+ is equipollent to b+ . (vii) A set a is ordinal iff a+ is so. (viii) Suppose an attribute of sets Φ is such that for any ordinal a, Φ(a) if Φ(x) for all x ∈ a. Then Φ(b) for all ordinal numbers b. As an immediate consequence, we can prove that there is no set AllOrd of all ordinal numbers. In fact, suppose that it exists. Then it clearly is transitive since each of its elements is so. It is alternative by the Theorem’s statement (ii). It is founded since for a non-empty b ⊂ AllOrd, take x ∈ b and take the minimal ordinal in b ∩ x. This is also minimal in b. But then, AllOrd would be ordinal, therefore AllOrd ∈ AllOrd, which is impossible for ordinals again by statement (ii) of Theorem 5.

6.2 Natural Numbers The collection of all ordinals is not a set, but a clever choice of a good subset of this collection yields the natural number set. Definition 22 A natural number is an ordinal number n such that (i) Either n = 0 (ii) or n = m+ (where m is automatically an ordinal); (iii) and every x ∈ n is either 0 or x = y + (and y is automatic ordinal). √

Exercise 11 Show that the elements of a natural number n are natural numbers. Here is the classical definition of natural numbers as given by the Italian mathematician Giuseppe Peano (1858-1932). This definition now has the shape of a theorem (that’s why Kronecker was wrong):

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Theorem 6 The Peano Axioms (i) (ii) (iii) (iv) (v)

0 is a natural number. If n is natural, then so is n+ . 0 is not a successor, 0 = n+ , of any natural number n. For natural numbers n, m, n = m iff n+ = m+ . (Proof by induction) If Ψ is a property of natural numbers such that Ψ (0), and Ψ (n+ ) whenever Ψ (n), then Ψ (n) for every natural number n.

The proof is an easy exercise in view of what we have learned about ordinal numbers. The fifth axiom is fundamental for many proofs since it makes it to show that some attribute is true for all natural numbers as soon as one has the “step-wise” condition Ψ (n) ⇒ Ψ (n+ ). As a first application of this proof scheme we can now exhibit the set N of all natural numbers: Proposition 5 (i) There is a set N which consists exactly of all natural numbers, i.e., n ∈ N iff n is natural. (ii) The relation n ∈ m between natural numbers defines a well-ordering on N. Proof 2 To prove (i), take the set w that is guaranteed by ZFC axiom 7. Take the subset N ⊂ w of the natural numbers in w. We claim that every natural number n is in N. In fact, by definition of w, 0 ∈ w, and if a natural number n is in w, then, again by the definition of w, n+ ∈ w. Therefore, by Peano’s fifth axiom, all natural numbers must be in w, and we are done. Statement (ii) follows directly from the general ordinal number Theorem 5. √

Example 6 Let us denote the first natural numbers to get a small number of standard symbols. We show a sequence of numbers where each subsequent number in the list is the successor n+ of the preceding number n in the sense of set theory: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

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Example 17 Observe that in mathematics, physics, and computer science, natural numbers start at 0, not at 1. Natural numbers are omnipresent in music, too. Here are some examples. • We count measures, usually starting at measure 1. But as the initial onset should be 0, starting at measure 0 would not be a bad idea. However, if the initial measure is incomplete, we number the first complete measure as one, so the incomplete measure would be number zero. • Within a given score, we count the beats, e.g., the number of eighth notes (also called quavers) at a determined position in the score. Pitch is also often given a natural number value, for example in the digital MIDI code1 , where middle C on a keyboard has number 60. 1

MIDI is the acronym for Musical Instrument Digital Interface, a standard code for the exchange of digital performance data between computers and electronic musical instruments. See http://www.midi.org and [78] for more information.

6.3 Finite Sets

75

• For a musical interval, one usually starts with the pitch of the lower note and counts the number of semitone steps to reach the upper note. The prime interval means to count 0, the minor second has 1, the major second 2, the minor third 3, the major third 4, the fourth 5, the tritone 6, the fifth 7, the minor sixth 8, the major sixth 9, the minor seventh 10, and the major seventh 11 pitch semitone steps. • For strings, one counts the open string and the first or second position for fingering instructions. • The works of a composer are counted by natural numbers. • The number of instruments in an orchestra is counted by natural numbers, e.g., a quartet is an orchestra with four string instruments (first and second violin, viola, violoncello).

6.3 Finite Sets Definition 23 A set x is finite iff it is equipollent to a natural number. Proposition 6 A subset of a finite set is finite. The proof is an easy exercise. To begin with, we may suppose that x is a natural number n. Then, the proof goes by induction on n. Proposition 7 This proposition identifies natural numbers within finite sets. (i) Two natural numbers are equipollent iff they are equal. (ii) The natural numbers are the finite ordinal numbers. Again, the proof goes by induction, and we omit it.

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Example 18 Referring to Musical Example 3, our pitch name set P12 is equipollent to the set 12 = {0, 1, 2, . . . 11} of natural numbers, and therefore is finite. The same is valid for the onset time set O12 . A dodecaphonic series ∼ ∼ s : O12 → P12 is therefore represented as a bijection s : 12 → 12. The first important application of this theory relates to permutations, i.e., ∼ bijections F → F of finite sets F . We denote by Sn the set of permutations of the natural number n. For example, we can look at a dodecaphonic series   0 1 2 3 4 5 6 7 8 9 10 11 s= 3 11 2 0 4 6 7 5 8 9 10 1 that sends every number in the top row to the number in the bottom row at the same column, e.g., 3 → 0. This set has an additional structure that later will turn out to be a central structure in group theory. But here we just use what’s immediately feasible. To begin with, any two permutations π, ψ ∈ Sn are functions that can be composed to a new permutation π ◦ ψ ∈ Sn . This

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composition is associative but generally not commutative. There is a special permutation Idn , the identity on n, that is “neutral” for composition: Idn ◦ π = π ◦ Idn = π for all π ∈ Sn . Finally, every permutation π ∈ Sn has an inverse π −1 since it is a bijection on n. Call this structure the group of permutations of n. Definition 24 For a natural number n and a set X, a sequence of length n in X is an injection g : n  X. The set of sequences of length n in X is denoted by X n .

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Example 19 If X = P12 , a sequence t : 3  P12 defines a three-element chord, called a triad in music theory. Of course, the order of pitches, e.g., t = (c, g, e), isn’t important when choosing a set {c, e, g} for illustrative purposes. But the order in the sequence is relevant if we care about the set’s members, for example, if we want to conceptualize chord inversions. Starting at e in {c, e, g}, then taking g, then c, we get the first inversion of {c, e, g}, or starting at g, then taking c, then e, defines the second inversion of {c, e, g}. This formalism of sequences was introduced in 1981 by one of the authors (Mazzola) in his university course on mathematical music theory and then in [72, Section I.2] for the classification of general musical structures; it has recently been applied under the catchword “orbifold” to topological considerations about voice leading [110]. We now define an equivalence relation on X n . Let g, h ∈ X n . We set g ∼ h iff there is π ∈ Sn such that g ◦ π = h. Check that it is an equivalence relation. Denote by X n /Sn the set of equivalence classes. ∼

Proposition 8 The sets Xn = {Y |Y ⊂ X AND Y → n} and X n /Sn are equipollent. ∼

Proof 3 If Y ∈ Xn , then we have a bijection g : n → Y that extends to an injection in X n , and then its equivalence class, by composing it with the ∼ inclusion Y ⊂ X. If we take another bijection h : n → Y , then they differ by the −1 permutation π = h ◦ g ∈ Sn , therefore they define the same equivalence class [g] = [h]. Conversely, if [h] ∈ X n /Sn , then its image Y = Im(h) is in Xn , and it is independent of the representative h since the difference of representatives is only a permutation of n.

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Exercise 4 Given the sequence t = (c, e, g), find the permutation π ∈ S3 such that t ◦ π = (e, g, c), the first inversion of t.

7 Recursion

Summary. Recursion is a technique to define concepts that depend on natural numbers. First, such a concept is defined for n = 0. Then, the concept for n is supposed to be defined, and we use the concept for n to define the concept for n + 1. Therefore, it is defined for all natural numbers n. This is the idea of recursion, namely the definition by induction. We will prove that this mathematical process is possible. We then apply recursion to create musical compositions. –Σ– “Proof by induction” is a proof technique involving statements about all natural numbers. Recursion also deals with all natural numbers, but it does not deal with a statement. It deals with defining sets, functions, and the like that are a function of natural numbers, i.e., concepts that are “parametrized” by natural numbers. Let us give an intuitive example of such a conceptualization. Suppose for a moment that we have defined product n · m of natural numbers. We would like to define the so-called factorial function n! that is defined by the formula n! = 1 · 2 · 3 · . . . (n − 1) · n. The critical part of such a definition is “. . .”. This is not mathematically acceptable language. The recursive idea here is to say, ok, we know what (n − 1)! means, and then set n! = (n − 1)! · n. Recursion is based upon a theorem that guarantees that such constructions yield well-defined objects. So what is the situation? We are given a type of objects f that are parametrized by natural numbers and have values in a given set X, i.e., f : N → X, or else f ∈ X N . In our example above, we would take X = N and f (n) = n! (where by definition f (0) = 0! = 1). The recursive construction is based on the idea that values of f (n) for n < m imply the value f (m). Here is the precise conceptualization. We need projections for every n ∈ N, |n : X N → X n : f → f |n , where f |n = f ◦ [n is the restriction of f to n and [n : n  N the inclusion. Also write |nm : X n → X m for the projection of X n onto X m for n ≥ m. We then have for every pair of natural numbers m ≤ n © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_7

77

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the following commutative diagram. XN

|n m

-

? Xn

|m

|n

- Xm

Definition 25 A recusive function on set X is a function r : X N → X N such that for all n ∈ N and f, g ∈ X N , f |n = g|n implies r(f )(n) = r(g)(n). And here is the Recursion Theorem. Theorem 7 (Recursion Theorem) If r : X N → X N is a recursive function on X, then there is a unique fixpoint F ixr ∈ X N of r, which means r(F ixr ) = F ixr . First of all, the proof needs an easy lemma: Lemma 6 If r : X N → X N is a recursive function, then for all n ∈ N and f, g ∈ X N , f |n = g|n implies r(f )|n+ = r(g)|n+ . Proof 4 This goes by induction on n. For n = 0, we have f |0 = g|0 for any two functions, since this is the restriction to the empty set 0. Then r(f )(0) = r(g)(0), hence r(f )|1 = r(g)|1. Suppose the statement is true for m and take n = m+ . Since f |n = g|n implies f |m = g|m, we have r(f )|n = r(g)|n by induction. But also r(f )(n) = r(g)(n), hence r(f )|n+ = r(g)|n+ . Corollary 1 Denote the set Rec(X) of recursive functions r : X N → X N . Then Rec(X) is a semigroup, i.e., a monoid without neutral element, under composition of set functions. Monoids are defined later in Chapter 17. Here we just have to verify that composition is defined. √

Exercise 12 Give a proof of this corollary.

This implies that for a recursive function r : X N → X N and for any n ∈ N, we have a restricted recursive function r|n : X n → X n such that the following diagram commutes: r X N −−−−→ X N ⏐ ⏐ ⏐| ⏐ |n  n r|n

X n −−−−→ X n And also, for m ≤ n, we have the commutative diagram

7 Recursion r|n

X n −−−−→ ⏐ ⏐ |n m

79

Xn ⏐ ⏐ |n m

r|m

X m −−−−→ X m We now want to show by induction on n that each of the restrictions r|n has a unique fixpoint, i.e. F ix(r|n) ∈ X n with r|n(F ix(r|n)) = F ix(r|n). If this is proved, the theorem follows immediately since then by the commutativity of diagram (7), the fixpoint F ix(r|m) is the restriction of the fixpoint F ix(r|n). This allows us to define F ixr as the unique function in X N that restricts to F ix(r|n) for every n ∈ N. For n = 0, X 0 is a singleton, so everything is clear. Suppose n = m+ , and the claim is proved for m. So we have a unique fixpoint Fm of r|m: r|m(Fm ) = Fm . Take any element G ∈ X m+ with G|m = Fm . Take then Qm+ := r|m+ (G). We claim that this element is the fixpoint for r|n. We have Qm+ |m = r|m(G|m) = r|m(Fm ) = Fm = G|m. Therefore by the above lemma, r|m+ (Qm+ ) = r|m+ (G) = Qm+ , and we are done.

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Example 20 Fibonacci numbers (invented by Italian mathematician Leonardo da Pisa, known as Fibonacci) are a classic example of a recursive construction. We take X = N and define rF ibonacci (x) = (1, 1, . . . rF ibonacci (x)i = xi−2 + xi−1 , . . .). This produces the fixpoint F ixrF ibonacci = (1, 1, 2, 3, 5, 8, 13, 21, . . .). Fibonacci numbers have been used by many composers. For example, Béla Bartók has implemented Fibonacci numbers in many compositions [59], e.g., in the number of measures of his composition Allegro Barbaro. The ostinato F -minor chords occur in groups of exactly 3, 5, 8, and 13 measures [59]. Also, the Chinese composer Mingzhu Song has created a composition The Scene of Sichuan Opera in which the note-group lengths are defined from Fibonacci numbers [105], see Figure 7.1. One should add that Fibonacci numbers are important also because of the fractions F ixrF ibonacci ,i+1 /F ixrF ibonacci ,i which tend to a real number 1.6180339887 . . . that is known as the golden ratio in the arts. We come back to this number when discussing real numbers in Chapter 12.

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Example 21 This musical example an illustration of the recursive method, but it also adds a general new method of musical composition. It was discovered while one of the authors (Mazzola) presented recursion in his course about mathematics for music theorists. A very talented student, Ben Klein, asked whether one could make sounds using recursion. And yes, it works. Here is the result.

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Fig. 7.1. In the composition The Scene of Sichuan Opera, Mingzhu Song constructs note-group lengths according to Fibonacci numbers. The sound example is Yan_Fibonacci.

The full recursion theorem deals with recursive functions r : X N → X N , but it essentially builds upon the recursive functions on finite powers r|n : X n → X n . On such a nth power, the fixpoint is not reached after infinite iterations of r, but after n powers (r|n)n . Therefore we may take any initial element x = (xi )i ∈ X n and calculate its trajectory (r|n)k (x) = 0, 1, 2, . . . n in X n that terminates in the fixpoint F ixr|n . The general composition method runs as follows: We take a set X n , whose elements can be viewed as sound events. For example, we may select the set P of pitches on the piano and interpret an element x ∈ P n as a sequence of pitches x = (x0 , x1 , . . . xn−1 ) in a musical melody. Then a trajectory (r|n)k (x), k = 0, 1, 2, . . . n can be read as an n + 1-length sequence of melodies (r|n)k (x). We may also view the trajectory in a retrograde movement, i.e., as a sequence of melodies in the time ordering (r|n)n (x), (r|n)n−1 (x), . . . (r|n)1 (x), (r|n)0 (x) = x. Denote by (r, n, x, +), (r, n, x, −) the original and retrograde trajectories, respectively. Given this formalism for a selected number n and “musical” space X, we may now select a sequence (x, ri , yi ), i = 1, 2, . . . v of triples with ri ∈ Recn (X), x, yi ∈ X n . This defines the following sequence of direct or retrograde trajectories: T ra((x, ri , yi )i ) = (r1 , n, x, +), (r1 , n, y1 , −), (r2 , n, y1 , +), (r2 , n, y2 , −), . . . (rv , n, yv−1 , +), (rv , n, yv , −). Figure 7.2 illustrates this compositional construction. In the following example we have taken a very simple configuration: We set n = 4, X the pitch space (in natural numbers). We take a first recursive function r1 to move from x, the motive of Beethoven’s Fifth Symphony 5th Symphony. We add y1 to be a four-note motive from O Fortuna in Karl Orff’s Carmina Burana, then conclude with y2 , a motive from Thelonious Monk’s Blue Monk. The first recursion function is r1a with a = 3, r1a (x) = (0, x1 + a, x2 + a, . . . xi + a, . . .) with F ixr1a = (0, a, a + a, a + a + a, . . .), the successive addition

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Fig. 7.2. The trajectory sequence associated with a sequence x, y1 , . . . yv of start points and recursive functions r1 , . . . rv .

of a’s, in our case F ixr13 = (0, 3, 6, 9, . . .), a sequence of minor thirds. The second recursion function r2 is the composition r14 ◦ r13 of r13 and r14 . Therefore it converges to its fixpoint with double speed—we only need two steps to and from its fixpoint. The fixpoint here is a sequence (0, 4, 7, 11, 14, . . .) of major and minor thirds. If we had taken the composition r13 ◦ r14 , we would have obtained a sequence (0, 3, 7, 10, 14, . . .) of minor and major thirds. Figure 7.3 illustrates this compositional construction.
 


     


  



  






 

 

   






  

  


 



  




  





  


                             





Fig. 7.3. A composition (One for Ben) generated according to the recursive method described above. The sound example is oneforben.

8 Natural Arithmetic

Summary. In the previous chapter, we defined natural numbers. Now, we are interested in how to combine these numbers to obtain other natural numbers. In this chapter, we will define three operations: addition, multiplication, and exponentiation. –Σ– Now we are able to define the classical operations: addition, multiplication, and exponentiation of natural numbers, all of these being defined using the Recursion Theorem. We take X = N for this setup. For addition, we take natural number a and define the function a + (n) = a + n for n ∈ N by the recursive function ra+ : NN → NN with ra+ (f )(0) = a and ra+ (f )(n+ ) = f (n)+ . For multiplication, we take natural number a and define the function a · (n) = a · n for n ∈ N by the recursive function ra· : NN → NN with ra· (f )(0) = 0 and ra· (f )(n+ ) = f (n) + a. For exponentiation, we take natural number a and define the function n N N a exp(n) = a for n ∈ N by the recursive function ra exp : N → N with + ra exp (f )(0) = 1 and ra exp (f )(n ) = f (n) · a. These are the most important properties of these operations, which we state without (mostly trivial and by induction) proofs: Sorite 3 Let a, b, c be natural numbers. We have these laws: (i) (Additive neutral element) a + 0 = 0 + a = a. (ii) (Additive associativity) a + (b + c) = (a + b) + c, which is therefore written as a + b + c. (iii) (Additive commutativity) a + b = b + a. (iv) (Multiplicative neutral element) a · 1 = 1 · a = a. (v) (Multiplicative associativity) a · (b · c) = (a · b) · c, which is therefore written as a · b · c. (vi) (Multiplicative commutativity) a · b = b · a. © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_8

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(vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi)

8 Natural Arithmetic

(Multiplication distributivity) a · (b + c) = a · b + a · c. (Exponential neutral element) a1 = a. (Exponentiation, + distributivity) ab+c = ab · ac . (Exponentiation, · distributivity) (a · b)c = ac · bc . (Additive monotony) If a < b, then a + c < b + c. (Multiplicative monotony) If c = 0 and a < b, then a · c < b · c. (Exponential base monotony) If c = 0 and a < b, then ac < bc . (Exponential exponent monotony) If c = 0, 1 and a < b, then ca < cb . (Ordering of operations) If a, b > 1, then a + b ≤ a · b ≤ ab . If a, b are natural numbers such that a ≤ b, then there is exactly one natural number x such that a + x = b. It is now easy to define sums or products of many numbers. If (s0 , s1 , . . . sn , sn+1 , . . .)

0 n+1 is a sequence of natural numbers, we define s = s0 and i=0 si = 0i=0 i n+1 n ( i=0 si ) + sn+1 , similarly for the product: i=0 si = s0 and i=0 si = n ( i=0 si ) · sn+1 .

ˇ “* Example 22 In music, natural arithmetic is basic. One may add numbers of instruments. One also has to calculate times—for example, after 28 quavers, one has to solve the equation 28 + x = 40 to know how many quavers are necessary to add up to 40 quavers. The duration of a concert is calculated by adding the durations of the concert’s movements. If we have measures that consist of eight quavers, how long is the rest in a measure after 5 quavers? One has to solve the equation 5 + x = 8. All such natural operations are elementary but indispensable for any precise information about natural numbers in music.

9 Euclid and Normal Forms

Summary. Romans used letters to denote natural numbers. Language of modern computers utilizes binary representation. Here we describe different ways to represent natural numbers, ending the chapter with an important theorem about prime numbers, already known to Euclid (300 BC). –Σ– This chapter deals with standard representations of natural numbers. The most important theorem, due to Euclid, is this one: Theorem 8 (Division Theorem) If a, b ∈ N with b = 0, then there is a unique pair r, q ∈ N with r < b, and such that a = q · b + r. Proof 5 The proof is by induction on a. For a = 0 it is trivial. So let a = b + 1. Suppose we have r , q  such that b = q  · b + r with r < b. Then a = b + 1 = q  · b + r + 1. If r + 1 < b we are done, else r + 1 = b, and we may set q = q  + 1, r = 0 and we are done. This means that one can now define uniquely that remainder r < b after division by b.

ˇ “*

Example 23 When dealing with pitch in semitone units, as typically realized by the keys of a piano, and which is quantified by natural numbers, one considers octaves between pitches, which means that one adds multiples of 12 semitones. In music theory, one often doesn’t deal with pitch, but with pitch up to octaves. This corresponds to the situation in the Division Theorem, where b = 12. For a given pitch a one then wants to know the non-negative remainder after adding up octaves. This means that one has to solve the equation a = q · 12 + r in the Division Theorem. The remainder r is the called the pitch class (number) of pitch a. For example, if a = 27, one gets 27 = 2 · 12 + 3, © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_9

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yielding pitch class 3. The Division Theorem guarantees that this class number is uniquely defined. The ability to define pitch classes is a consequence of this theorem. Our previous usage of a 12-element pitch-class set P12 is based on this technique. Theorem 9 Let a, b ∈ N with b > 1 and a = 0. Then there are unique natural numbers c, s, r with 0 < s < b and r < bc such that a = s · bc + r. The proof is similar to the preceding one, so we omit it. The next theorem is the well-known representation of natural numbers using a “basis” number, usually known for decimal numbers, i.e., b = 10. Let us first state the theorem: Theorem 10 (Adic Normal Form) Let a, b ∈ N with b > 1 and a = 0. Then there exists a unique sequence s0 , s1 , . . . sn of natural numbers, all si < b and sn = 0 such that n  s i bi , a= i=0

and we write a =b sn sn−1 , . . . s0 . The number b is called the basis of the representation that is also called b-adic or b-ary. For the decimal representation, we define the first numbers as usual: 0, 1, 2, 3, 4 = 3 + 1, 5 = 4 + 1, 6 = 5 + 1, 7 = 6 + 1, 8 = 7 + 1, 9 = 8 + 1, Z = 9 + 1. Then we have the decimal representation for b = Z and 0 ≤ si < Z. For example, 123 means 1 · Z 2 + 2 · Z 1 + 3 · Z 0 . The decimal representation is omnipresent in any context, where natural numbers have to be written in a precise and standardized way. For b = 2 we have the binary representation where 0 ≤ si ≤ 1. Here 1100111 means 1 · 26 + 1 · 25 + 0 · 24 + 0 · 23 + 1 · 22 + 1 · 21 + 1 · 20 . This representation is chosen for digital technology, for example when representing numbers for amplitudes, times, etc., in sound representation. The hexadecimal representation takes b =Z 16. We then have to rename the numbers up to (decimal) 15, i.e., A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. For example, x =Z 41663 =Hex A2BF = Ab3 + 2b2 + Bb1 + F b0 =2 1010001010111111. The hexadecimal representation is used in the MIDI code for production, management, and storage of musical performance.

9.1 The Infinity of Prime Numbers The Division Theorem has an interesting consequence relating to prime numbers. Definition 26 A natural number p > 1 is prime if p = 1 · p = p · 1 is the only way to write it as a product of two natural numbers.

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It is clear by induction that every natural number > 1 is the product of a finite number of prime numbers. It is a classical result of number theory that goes back to Euclid: Theorem 11 The set of prime numbers is infinite. Proof 6 The proof runs as follows. Suppose there are only finitely many prime ∼ numbers. Then we may enumerate their set P by p : n → P ⊂ N. We consider the number N = i b. The arithmetics developed so far for natural numbers will be extended to integers. –Σ– The extension of the arithmetic of natural numbers to integers, rational, real, and complex numbers follows a general philosophy: To solve a problem in mathematics, you have to take the problematic structures and use them to construct the solution. In short: “The problem is the solution.” Integers are constructed to solve the following problem. We had seen in Sorite 3, statement (xvi), that for a ≤ b, the equation a + x = b has exactly one solution x. But if a > b, no solution is possible in natural numbers. Here is the problem: Such simple equations that may not have solutions in natural numbers. Take two such equations with the same (hypothetical) solution x: a+x = b c+x = d Then, by adding c to the first and a to the second equation, we get c+a+x = c+b a+c+x = a+d The left sides are equal, meaning that c+b = a+d. This leads to an equivalence relation among (the coefficients of) equations, namely that (a, b) ∼ (c, d) iff c + b = a + d. That this relation is an equivalence relation on N × N is an easy exercise. This leads to the definition of integers. Their set is defined by Z = N × N/ ∼ . © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_10

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ˇ “* Example 24 Musical intervals are often not natural numbers. For example, one may want to know the interval between pitch 9 and pitch 7. This leads to the equation 9 + x = 7. One may also want to know the interval between pitch 11 and pitch 9. According to the above equivalence relation we calculate the cross sums 9 + 9 and 7 + 11. Both add to 18, which means that the equations have one and the same non-natural solution x. Integers are invented to deal with exactly this new type of solution. We shall see in a moment, that this solution is denoted by −2, a “negative” interval number. Let us see how we may represent integers by pairs (a, b) of natural numbers. If [a, b] ∈ Z with b ≤ a, then we have a solution x of b + x = a, and we have [a, b] = [x, 0], and no other [x , 0] does the job. If a ≤ b, then we have a solution x of a + x = b, and we have [a, b] = [0, x], and no other [0, x ] does the job. Further, [x, 0] = [0, y] iff x = y = 0. Therefore the integers [a, b] ∈ Z with b ≤ a are represented by natural numbers x, via [x, 0]. This defines injection N  Z : a → [a, 0]. We write −b for the integer [0, b], b = 0, and call such an integer a negative integer, while the images of N are called natural numbers or positive or (when including the zero number) non-negative integers. For a negative integer z = −b, the positive integer b is called its absolute value, which is denoted by |z|. For a non-negative integer z the absolute value is that same number |z| = z. For any integer z = [a, b], we more generally define its negative by −z = [b, a].

10.1 Arithmetic of Integers Addition of integers is defined coordinate-wise by [a, b] + [c, d] = [a + c, b + d]. It is a good exercise to show that this definition does not depend on the representatives (a, b), (c, d) of integers. We then write a − b for a + (−b). Sorite 4 Let Z be provided with the addition + : Z × Z → Z, and let a, b, c be any integers. Then we have these properties. (Associativity) (a + b) + c = a + (b + c) = a + b + c. (Commutativity) a + b = b + a. (Additive neutral element) a + 0 = a. (Additive inverse element) a − a = 0. (Extension of natural arithmetic) If a, b ∈ N, then [a + b, 0] = [a, 0] + [b, 0], i.e., it amounts to the same if we add two natural numbers a, b or the corresponding non-negative integers, also denoted by a, b. (vi) (Solution of equations) The equation a + x = b in the “unknown” x has exactly one integer number solution x, i.e., x = b − a.

(i) (ii) (iii) (iv) (v)

We have a linear ordering on Z by the definition a ≤ b iff b − a ∈ N, see also Figure 10.1.

10.1 Arithmetic of Integers

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Fig. 10.1. The set of integers Z extends the set of natural numbers N and can be represented on a line that extends infinitely to the left and to the right.

It is also possible to extend the multiplication operation defined on N to the integers. The definition is again one by representatives of equivalence classes [a, b]. To understand the definition, we first observe that a class [a, b] is equal to the difference a − b of natural numbers with the above identification. In fact, [a, b] = [a, 0] + [0, b] = a + (−b) = a − b. If we want to extend the arithmetic on the natural numbers, we should try to observe the hoped for and given rules to thereby get the extension. We should have [a, b] · [c, d] = (a − b) · (c − d) = ac + bd − ad − bc = [ac + bd, ad + bc]. This motivates the following definition: Definition 27 Given two integers [a, b], [c, d], their product is defined by [a, b] · [c, d] = [ac + bd, ad + bc]. Check again that this function is well defined. Sorite 5 Let a, b, c be three integers. We have these rules for their multiplication. (i) (Associativity) (a · b) · c = a · (b · c) = a · b · c. (ii) (Commutativity) a · b = b · a. (iii) (Multiplicative neutral element) The element 1 = [1, 0] is neutral for multiplication, a · 1 = a. (iv) (Zero and negative multiplication) a · 0 = 0, a · (−b) = −(a · b). (v) (Distributivity) a · (b + c) = a · b + a · c. (vi) (Integrity) If a, b = 0, then a · b = 0. (vii) (Additive monotony) If a < b, then a + c < b + c. (viii) (Multiplicative monotony) If a < b and 0 < c, then a · c < b · c. (ix) (Extension of natural arithmetic) For two natural numbers a, b, we have [a · b, 0] = [a, 0] · [b, 0]. This allows complete identification of naturals as a subdomain of the integers, if we look at addition and multiplication. (x) (Triangular inequality) We have |a + b| ≤ |a| + |b|. The concept of a prime number is extended to integers by the definition that a ∈ Z is prime iff |a| is so in the set of natural numbers.

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ˇ “* Example 25 Integer arithmetic allows for unrestricted operations on integer representations of musical parameters, such as pitch or onset times (Figure 10.2). A first operation on integers is transposition. Given an integer t ∈ Z, one considers the function of transposition by t, denoted by T t . It is the function T t : Z → Z : a → T t (a) = t + a. In view of property (vi) in Sorite 5, T t is a bijection on Z. A second operation on integers is inversion. Inversion is also a bijection on Z and is denoted by T−t . It is defined by T−t (a) = t − a. Clearly, T−t ◦ T−t = IdZ . It has a fixpoint, i.e., T−t (a) = a iff t = 2a is a multiple of 2, i.e., an even integer. If t is not even, that is, it’s odd, there is no fixpoint. For example, if we want to define an inversion with fixpoint a = 73 (also called inversion at 73) we have to take the inversion T−146 .

Fig. 10.2. Transposition and inversion on integers.

11 Rationals

Summary. In Latin, ratio means rapport, division of two things. Here we introduce rational numbers ab as fractions of two integer numbers a and b. This procedure allows us to solve equations of type a·x = b with a = 0. Our strategy again follows the philosophy that the problem is the solution. –Σ– The philosophy of finding the solution in the problem is also valid for the construction of rational numbers. This time we consider equations of type a·x = b, where a = 0. For integers, this may not have a solution, e.g., 2 · x = 3. Again, we look at two such indexequationequations having the same (hypothetical) solution: a·x = b c·x = d and then multiply the first by c and the second by a to get c·a·x = c·b a·c·x = a·d with identical left side, so c · b = a · d. This defines an equivalence relation among the equations’ coefficients: (b, a) ∼ (d, c) iff c · b = a · d. Check this as an exercise. This defines the set Q of rational numbers by Q = Z × Z∗ / ∼ where Z∗ = Z − {0}. We denote the equivalence class [b, a] by b/a or ab and call b the numerator and a the denominator of the fraction ab . Observe that for any s·b . s = 0, we have ab = s·a The integers can be embedded in the rationals by the injection Z  Q : a →

a . 1

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The linear ordering on integers can be extended to rationals by the following rule: If ab , dc ∈ Q, we can always suppose a = c > 0 by multiplication of numerators and denominators by the other fraction’s denominators, or negative c·b d b d , c = a·d denominators: ab = c·a a·c . Suppose this, then we define a < a (for a > 0) iff b < d.

ˇ “*

Example 26 In music, rationals are very important. Let us look at some basic examples of the use of rationals in the musical domain. For the classical score notation, the horizontal axis represents onset time and durations of notes. This musical time is not the physical dimension, but it is a symbolic time. It is only interpreted in physical units when one adds rules for the shaping of tempo. We come back to the topic of tempo in Chapter 32. For the time being we only want to look at the symbolic time that is denoted on the score. In this environment, onset time is divided into equal portions, called measures. The duration of such measures is indicated at their beginning with a time signature. Typical time signatures are shown in Figure 11.1.

Fig. 11.1. Three typical time signatures.

Time signatures look like rational numbers: They have a numerator and a denominator. For example, the left time signature in Figure 11.1 resembles the rational 44 . The middle time signature resembles rational 34 , while the right signature resembles rational 68 . However, these symbols are not rationals but are representatives for rationals. We know that 34 = 68 mathematically, but their musical meaning is more than this. See again Figure 11.1. You can see that the first measure consists of two half notes, while the second one consists of two quarter notes and two quarter rests. Each of these durations add up to 44 , as shown by the small cross symbols denoting four “beats” in each measure. Look at the other two time signatures in Figure 11.1. The denominators 4 and 8 designate the "beat" durations in their respective measures. Notice the pattern of cross symbols (eighths) above the measures. While the 34 signature is divided into three groups of two cross symbols, the signature 3 4

6 8

shows a division into two three-cross

groups. Simple meters such as divide their beats into two equal parts as shown in Figure 11.1, left and middle examples. In compound meters, as shown in Figure 11.1, right example, the dotted quarter beat is not divided in two, but into three notes. The time signature 68 indicates a total number of six quavers. There are two dotted quarter beats, each one comprising three quavers, for a

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total of six. Figure 11.2 shows three examples of 34 as compared with 68 measures in the literature.

Fig. 11.2. Examples of 6/8 against 3/4 measures: (a) Mi votu e mi rivotu, Sicilian traditional song, (b) America, from West Side Story, by Leonard Bernstein (in the original score, the time signature was a mixed one, showing both signatures at the beginning of this two-measure unit), (c) An der schönen blauen Donau, by Johann Strauss. The sound example is 3_4.

Mathematically, this may be insignificant as long as no other musical parameters to the notes, such as attack or loudness, the two measures sound different. In Figure 11.3, a famous example of a fast-changing time signature construction from Igor Stravinsky’s The Rite of Spring is shown.


     




 

 




     

    



       




                       


        

Fig. 11.3. Complex time signature construction from Stravinsky’s The Rite of Spring. The sound example is Stravinsky.

The solution of an equation of type a · x = b is crucial in music. One is typically given a duration b and wants to play it as a sequence of a durations. Then these durations are the solution x. This is the procedure applied when

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defining tuplets in the score notation. Figure 11.4 shows a 4/4 measure that is 4 4 = 28 . Musical score notation divided into 7 equal durations, each of which is 4·7 uses note durations from the standard repertory of half, quarter, eighth notes, etc. (their durations are powers of 12 ), but indicates by the tuplet number (7 here) how their duration should be interpreted.




 










 






 









     

       

Fig. 11.4. The measure in time signature 4/4 is divided into a septuplet (seven equal durations). The second measure is not complete. The sound example is rhythm_Rationals.

11.1 Arithmetic of Rationals We now ease notation by writing products xy instead of x · y. Addition and multiplication of integers can be extended to rationals as follows: Definition 28 Let

b d a, c

∈ Q. Then their sum is defined by b d bc + ad + = , a c ac

while their product is defined by bd bd = . ac ac √

Exercise 13 Verify that these operations are well defined (independent of the representatives) and that this arithmetic extends the arithmetic of the integers under the above injection Z  Q. Let us denote this by the formula (Z, +, ·)  (Q, +, ·), which will be explained in detail in Chapter 24. We define the absolute value of a rational number ab by   b   = |b|  a  |a|

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ˇ “* Example 27 Addition of rational numbers is a frequent operation in music, especially when adding the duration of tuplets, which can be complex configurations of time. This situation is encountered in many Western compositions, e.g., in the complex scores of Brian Ferneyhough (see Figure 11.5) or in jazz compositions, such as Footprints by saxophonist Wayne Shorter, or, without corresponding score notation, in African traditions of polyrhythms. For exam30 51 ple, it may happen that one has to add 35 to 67 , which yields 35 + 67 = 21 35 + 35 = 35 .

Fig. 11.5. Score example for complex time structures from Brian Ferneyhough’s Third String Quartet.

And here is a summary of important properties of rational arithmetics: Sorite 6 Let

a c e b , d, f

be rational numbers. Then these rules hold.

(Additive associativity) ( ab + dc ) + fe = ab + ( dc + fe ) = ab + dc + fe . (Additive commutativity) ab + dc = dc + ab . (Additive neutral element) ab + 01 = ab . 0 (Additive inverse element) ab + −a b = 1. a c e (Multiplicative associativity) ( b · d ) · f = ab · ( dc · fe ) = ab · dc · fe . (Multiplicative commutativity) ab · dc = dc · ab . (Multiplicative neutral element) ab · 11 = ab . (Multiplicative inverse element) If b = 0, then ab · ab = 11 . (Distributivity) ab · ( dc + fe ) = ab · dc + ab · fe . (Linear ordering) The relation < among rational numbers is a linear ordering. Its restriction to the integers a1 induces the given linear ordering among integers. (xi) (Additive monotony) If ab < dc , then ab + fe < dc + fe .

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

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(xii) (Multiplicative monotony) If ab < dc and 01 < fe , then ab · fe < dc · fe . (xiii) (Archimedean ordering) For any two positive rational numbers ab , dc there is a natural number n such that n1 · ab > dc . (xiv) (Solution of equations) The equation ab · x = dc has a unique solution for a b = 0. (xv) (Triangular inequality) | ab + dc | ≤ | ab | + | dc |.

12 Real Numbers

Summary. We have used the philosophy of the problem being the solution to construct integer and rational numbers when dealing with equations of type a + x = b or a · x = b. But there are many other equations, especially dealing with approximations in music theory, that cannot be solved with Z or Q. In this chapter we apply the above philosophy to find solutions of such problems, namely the real numbers. –Σ– The geometric problem of finding the length l of the diagonal of a square with side length 1 leads us to the Pythagorean equation l2 = 12 + 12 = 2. Suppose that l = pq ∈ Q, and suppose that p, q have no common prime factor. 2

Then we have l2 = pq2 = 2, hence the equation 2q 2 = p2 of integers. But (this is the theorem about uniqueness of prime factorization, to be proved in Chapter 25) the factor 2 on the left side implies that p = 2p . But then 2q 2 = p2 = 4p2 , so q 2 = 2p2 . Therefore, for the same reason, q = 2q  , which contradicts the absence of common prime factors of p, q. Hence the diagonal equation has no rational solution. A fortiori the equation s12 = 2 has no rational solution. Otherwise s6 would solve the diagonal equation. This latter equation is crucial in music theory: s is the frequency ratio between successive semitone steps of the 12-tempered octave tuning. In fact, if the frequency ratio from a pitch x to pitch y is r, and the ratio from y to z is w, then the frequency ratio from x to z is r · w. Therefore, if all semitone steps have equal frequency ratio s, the octave, having frequency ratio 2, and being built from 12 equal semitone steps, must have frequency ratio 2 = s12 . Refer to our Chapter 2 on the history of mathematics in music, where the invention of 12-tempered octave tuning by Zaiyu Zhu is described in Section 2.4. The situation is not hopeless, however, since despite the non-existence of solutions in Q, we can still approximate solutions by rational numbers. Here is the procedure for the diagonal equation l2 = 2. Take the largest integer s0 = 1 such that s20 < 2, and then the smallest integer S0 = 2 such that © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_12

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2 < S02 . Then split this interval [s0 , S0 ] = {x|x ∈ Q, s0 ≤ x ≤ S0 } in the middle, getting (s0 + S0 )/2 = 3/2. Now check whether (3/2)2 < 2. This is wrong, so switch to the new interval [s1 = s0 , S1 = 3/2]. Again, split this interval in its middle 5/4 and check the size of (5/4)2 . It is smaller than 2, so move to the third interval [s2 = 5/4, S2 = S1 ]. This procedure yields a sequence S0 , S1 , . . . Si , Si+1 , . . . of rational numbers that approximate with arbitrary precision the diagonal size, without ever reaching it. There are many such sequences of rationals that remain limited without ever reaching their 1 i+1 ) , i ≥ 0. It approaches the famous “limit,” e.g., the sequence xi = (1 + i+1 non-rational Euler number 2.718281828459 . . ..

ˇ “*

Example 28 The above approximation of a solution of l2 = 2 should be interpreted in musical terms. It is an open question whether one can invent a practical musical realization, if possible with lengths of strings, of this approximation. We have the following calculations: Given two frequencies x, y. These correspond to string lengths lx = 1/x, ly = 1/y. The middle frequency m = x+y 2 corresponds to the length lm =

lx ly (lx + ly )/2

which is not really a simple construction out of lx , ly in terms of mechanical manipulations, except for the denominator.

ˇ “*

Example 29 Another example of real numbers in music is given by glissando. A glissando is a very fast performance of the notes between a starting and an ending point. n the score, usually only the first and last notes are indicated, together with a connecting line. While a glissando on piano implies discrete frequencies, a glissando on a violin, for example, is “continuous” because on strings continuous frequency values can be performed. The movement of a glissando is thought to glide through all real numbers between initial and final pitch. The idea of a continuity is delicate, however, since the rational numbers are also “dense”—between any two rational numbers there is an infinity of rational numbers1 . We come back to these now mysterious concepts of density and continuity when discussing questions of tuning systems in Chapter 28 and continuity in Chapter 30. We now see that we have identified the problematic objects. They are sequences (xi )I∈N of rational numbers that are in some sense limited. The philosophy now applies: We shall use such sequences to construct the set R of real numbers. Definition 29 A Cauchy sequence of rational numbers is a sequence (ai )i∈N ∈ QN with the following property: For every positive natural number L, there is 1

But also, between any two rational numbers there is an infinity of non-rational real numbers.

12 Real Numbers

101

a natural number N such that whenever m, n > N , we have |an − am | < 1/L. The set of Cauchy sequences is denoted by C. √

Example 7 Constant sequences of rational numbers are Cauchy. The se1 )i is Cauchy. If (ai )i is Cauchy, then so is (a+ai )i for every rational quence ( i+1 number a.

ˇ “*

Example 30 The Fibonacci numbers are a fixpoint sequence F ixrF ibonacci = (1, 1, 2, 3, 5, 8, 13, 21, . . .)

of a recursive function and were defined in Musical Example 20. The fractions F (i) = F ixrF ibonacci ,i+1 /F ixrF ibonacci ,i of rational numbers are a Cauchy sequence. They are used to define a real number g = 1.6180339887 . . . (real numbers are to be defined below), known as √the golden ratio. It can also be represented as an explicit real number g = 1+2 5 (the existence of nth roots is discussed in Chapter 13). This number is the solution of the geometric problem of constructing a rectangle with sides a, a + b such that a a+b = , b a see Figure 12.1. This means that the ratio of the longer side a + b to the shorter a is the same as the ratio of the shorter a to the remainder b = (a + b) − a. The ratio turns out to be g. This problem of ratios was first discussed by the mathematician and cosmologist Johannes Kepler in 1597. Apart from being an artistic principle of constructing aesthetically pleasing ratios, the golden ratio is present in many biological and physical contexts. The golden ratio has also been used in compositions of Karlheinz Stockhausen and Gérard Grisey. See also [64] for more information about Fibonacci numbers and the golden ratio.

Fig. 12.1. The golden ratio is the solution of the geometric problem of constructing a rectangle with sides a, a + b such that ab = a+b . a

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Definition 30 A sequence (ai )i∈N ∈ QN is said to converge to a rational number a iff for every positive natural number L, there is a natural number N such that whenever n > N , we have |an − a| < 1/L. Lemma 7 If a sequence (ai )i ∈ QN converges to a rational number a, then it is Cauchy. The proof is an easy exercise. We would like to “forget” about the sequences that converge to zero. These are called zero sequences, and their set is denoted by O. We now want to define arithmetic operations on C and then use them to define real numbers. Definition 31 Let (ai )i , (bi )i ∈ QN . We set (ai )i + (bi )i = (ai + bi )i (ai )i · (bi )i = (ai bi )i Proposition 9 (Zero sequences are an ideal) (i) If (ai )i , (bi )i ∈ C, then (ai )i + (bi )i and (ai )i · (bi )i are in C. (ii) If (ai )i , (bi )i ∈ O, then (ai )i + (bi )i ∈ O. (iii) If (ai )i ∈ C and (bi )i ∈ O, then (ai )i · (bi )i ∈ O. The fact that O is closed under addition, and that any product of a zero sequence with a Cauchy sequence is a zero sequence are important properties that we shall discuss in Chapter 24, and which are the reason O is called an ideal in C. Lemma 8 On C, the relation (ai )i R(bi )i iff (ai )i − (bi )i ∈ O is an equivalence relation. Proof 7 Reflexivity and symmetry are trivial. Transitivity results as follows. Let (ai )i R(bi )i and (bi )i R(ci )i . Then we have |ai − ci | = |ai − bi + bi − ci | ≤ |ai −bi |+|bi −ci | ≤ 1/2L+1/2L = 1/L for i > N such that both |ai −bi | < 1/2L, and |bi − ci | < 1/2L. Definition 32 The set of real numbers is defined using the above equivalence relation R: R = C/R. And here is a more concrete description of the equivalence classes that define real numbers: Lemma 9 If [(ai )i ] ∈ R, then we have [(ai )i ] = (ai )i + O = {(ai )i + (zi )i |(zi )i ∈ O}. This representation is called the O-coset of (ai )i .

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We have a canonical injection Q  R that sends a rational number a to the constant Cauchy sequence coset [(a)i ] = (a)i + O = {(a + zi )i |(zi )i ∈ O}. And we can now also extend addition and multiplication to real numbers in a very straightforward way: If [(ai )i ], [(bi )i ] ∈ R, we set [(ai )i ] + [(bi )i ] = [(ai + bi )i ] [(ai )i ] · [(bi )i ] = [(ai bi )i ], which means that we use the arithmetic on Cauchy sequences and simply “project” it to the O-cosets. This is a standard procedure in algebra, which we shall study in Chapter 24. √

Exercise 14 Give a proof of fact that these arithmetic operations are well defined. The arithmetic properties of these operations on R are: Sorite 7 Let x, y, z be real numbers. (i) (Additive associativity) (x + y) + z = x + (y + z) = x + y + z. (ii) (Additive commutativity) x + y = y + x. (iii) (Additive neutral element) The rational zero 0 is also neutral on the reals, x + 0 = x. (iv) (Additive inverse element) x + (−x) = 0. (v) (Multiplicative associativity) (x · y) · z = x · (y · z) = x · y · z. (vi) (Multiplicative commutativity) x · y = y · x. (vii) (Multiplicative neutral element) The rational unity 1 is also neutral on the reals, x · 1 = x. (viii) (Multiplicative inverse element) If x = 0, then there is exactly one multiplicative inverse x−1 , i.e., x · x−1 = 1. More precisely, there exists in this case a Cauchy sequence (ai )i representing x and such that ai = 0 for all i, and we may represent x−1 by the Cauchy sequence (a−1 i )i . (ix) (Distributivity) x · (y + z) = x · y + x · z. (x) If a, b, c are real numbers such that a = 0, then the equation ax + b = c has exactly one solution x. This means that we have “saved” the algebraic properties of Q to R. But we wanted more than that. Let us first look for the linear ordering structure on R. Definition 33 A real number x = (ai )i + O is called positive iff there is a positive rational number 0 < ε such that ε < ai for all i but a finite set of indexes. This property is well defined. We set x < y for two real numbers x, y iff y − x is positive. In particular, x is positive iff 0 < x.

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Proposition 10 The relation < on R from Definition 33 defines a linear ordering. The set R is the disjoint union of the subset R+ of positive real numbers, the subset R− = −R+ = {−x|x ∈ R+ } of negative real numbers, and the singleton set {0}. We have R+ + R+ = {x + y|x, y ∈ R+ } = R+ . R+ · R+ = {x · y|x, y ∈ R+ } = R+ . R− + R− = {x + y|x, y ∈ R− } = R− . R− · R− = {−x · −y|x, y ∈ R+ } = R+ . R+ + R− = {x − y|x, y ∈ R+ } = R. R+ · R− = {−x · y|x, y ∈ R+ } = R− . (Monotony of addition) If x, y, z are real numbers with x < y, then x + z < y + z. (viii) (Monotony of multiplication) If x, y, z are real numbers with x < y and 0 < z, then xz < yz. (ix) (Archimedean property) If x, y are positive real numbers, there is a natural number N such that y < N x. (x) (Density of rationals) If 0 < ε is a positive real number, then there is a rational number ρ with 0 < ρ < ε. (i) (ii) (iii) (iv) (v) (vi) (vii)

Definition 34 The absolute value |a| of a real number a is a if it is nonnegative, and −a if a is negative. Proposition 11 (Triangular Inequality) If a, b are two real numbers, then we have the triangular inequality: |a + b| ≤ |a| + |b|. We now have a completely general convergence criterion on R. But first, we have to define convergence on R, using the concept of convergence of rational sequences. Definition 35 A sequence (ai )i of real numbers is said to converge to a real number a iff for every real ε > 0, there is an index N such that n > N implies |an − a| < ε. Clearly, convergence can only take place for one a, and therefore we denote convergence by limi→∞ ai = a. The sequence (ai )i is Cauchy iff for every real number ε > 0, there is a natural number N such that n, m > N implies that |an − am | < ε. Theorem 12 (Convergence on R) A sequence (ai )i of real numbers converges iff it is Cauchy. This result leads to a huge number of existence theorems of special numbers. We just mention one particularly important situation.

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105

Corollary 2 (Existence of Suprema) If A is a bounded, non-empty set, i.e., there is an upper bound b ∈ R such that b > a for all a ∈ A (in short: b > A), then there is a uniquely determined supremum or least upper bound s = sup(A), i.e., an upper bound s ≥ A such that for all t < s, there is a ∈ A with a > t. √

Exercise 15 Give a proof of this corollary.

13 Roots, Logarithms, and Normal Forms

Summary. Corollary 2 in Chapter 12 is crucial for the construction of some important structures for real numbers, such as general roots and logarithms. These are introduced in this chapter. We also discuss musical applications to pitch theory. –Σ–

13.1 Roots, and Logarithms Theorem 13 (Existence of nth Roots) Let a ≥ 0 be a real number and n a positive natural number. Then there exists a unique non-negative real√number x such that xn = a. We call x the nth root of a and denote it by x = n a or by a1/n . Proof 8 The proof uses Corollary 2. Consider the set A = {q|q ∈ R AND q n < a}. It is limited from above since we can show that (a + 1)n > a. For the supremum sup(A), which exists by Corollary 2, it is easily seen that sup(A)n = a. √

Exercise 16 Show that for real numbers a, b ≥ 0, we have

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_13

√ n

ab =

√ √ n a n b.

107

108

13 Roots, Logarithms, and Normal Forms

ˇ “* Example 31 The existence of nth roots is the basis of many tuning systems in music. We have already discussed the case of the 12-tempered tuning, where the √ octave frequency ratio 2 is divided into 12 equal frequency ratios of size 12 2 ≈ 1.059463094359295 . . . For microtonal tunings—for example, quarter√ tone or 24-tempered tuning, one needs the quarter-tone frequency ratio 24 2 ≈ 1.029302236643492 . . . The quarter-tone piano of Czech composer and theorist

Fig. 13.1. Jin Pang and his erhu.

Fig. 13.2. Alois Hába and his quarter-tone piano.

Fig. 13.3. Reproduction of the archicembalo described by Nicola Vicentino in 1555. This harpsichord had 36 pitches per octave.

Fig. 13.4. Clavemusicum Omnitonum Modulis Diatonicis Cromaticis et Enarmonicis, a harpsichord by Vito Trasuntino of Venice. It has 31 pitches per octave.

Alois Hába (Figure 13.2), the archicembalo (Figures 13.3, 13.4) or the Chinese erhu (Figure 13.1) string instrument have realized such microtonal tunings.

13.1 Roots, and Logarithms

109

Hába has also written interesting string quartets for quarter-tone, fifth-tone, and sixth-tone temperaments. Many composers from different cultures have written and played compositions for microtonal tunings. Definition 36 We define rational exponents xq , q ∈ Q, for 0 < x as follows: Set x0 = 1, then for negative integers −n, x−n = 1/xn , then for q = n/m ∈ Q, xn/m = (x1/m )n . This definition is independent of the representation q = n/m. √

Exercise 17 Show that for two rational numbers p, q, and positive real numbers x, y, we have xp+q = xp xq , xpq = (xp )q , and (xy)p = xp y p . To define real exponents al = x for x > 0, a > 0, consider the set A = {q|q ∈ Q AND aq < x}. This set is bounded from above. Its supremum is defined to be the logarithm of x for basis a, denoted by loga (x), see Figure 13.5. It is the supremum of all

Fig. 13.5. The logarithm function for a = 10.

rational exponents q that yield a power aq < x, so the logarithm is a kind of real exponent: aloga (x) = x. It defines a function loga : R+ → R, and by its very definition, it has the characteristic property that for two positive numbers x, y, we have loga (xy) = loga (x) + loga (y).

ˇ “*

Example 32 The selection of admissible musical pitches (also known as tunings) is a major topic in the construction of musical instruments and in music theory. For string instruments, any conceivable pitch can be played within

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13 Roots, Logarithms, and Normal Forms

the range of the instrument, but for keyboards, only a discrete subset of pitches is available (see also Figure 30.1). In music theory, the totality of possible pitches is not conceived.

Fig. 13.6. A time-periodic pressure variation, here a sinusoidal function, is responsible for our perception of pitch.

But what is pitch? In physics, sounds with a determined pitch are generated by a variation of the air pressure p(t) (in pascals, where one pascal (Pa) is the force of one Newton per square meter N/m2 ) as a function of time t (in seconds (sec), say) that shows periodicity, i.e., it repeats its shape after a time period P . Figure 13.6 shows a sinusoidal function of pressure variation around the average air pressure 101325 P a. The frequency of a pressure function is defined to be f = 1/P if P is the time period, and the frequency unit is Hertz, Hz = 1/sec. For example, the chamber a in music is frequently (but not always, some regions have slightly different standards) associated with 440 Hz. However, humans don’t perceive frequency as such. It is the logarithm P itch(f ) = log(f ) that our brain perceives as pitch1 . This law is called the Weber-Fechner law. For example, if we are given a pitch P itch(fc ) = log(fc ), say of middle c on a piano, then the octave c of this pitch has the double frequency 2fc . This translates to the logarithmic equation P itch(2fc ) = log(2) + log(fc ). In other words, going up one octave means adding the constant log(2) to the given pitch. This is the reason why the distance between keys an octave apart 1

In psychoacoustics, the pitch number is defined by a slightly different formula, namely P itch(f ) = log1200(2) log10 (f ) + v. The factor log1200(2) is chosen such that the 10 10 octave is divided into 1200 units. In fact, log1200(2) log10 (2) = 1200. The pitch unit 10 that is defined by this formula is called Cent (Ct). So the octave is divided into 12 times 100 Cents, which means that each semitone is divided into 100 Cents—hence the name “Cent”.

13.2 Adic Representations

111

on the piano is constant. If the keyboard had to represent frequency differences, octaves would be separated more and more as the keys go to the right. From c to c we have 2fc − fc = fc , but for c we have 4fc − 2fc = 2fc , double the difference of the previous octave. Musical tunings are defined by mathematical formulas that specify admissible pitches. √ For example, 12-tempered tuning selects frequencies f of the shape f = f0 · ( 12 2)p , p ∈ Z, f0 a basic frequency, with corresponding pitches p log(2). Just tuning is defined by f = f0 · 2o 3q 5t , where P itch(f ) = log(f0 ) + 12 o, q, t ∈ Z, such that P itch(f ) = log(f0 ) + o log(2) + q log(3) + t log(5). This construction stems from three basic traditional musical intervals, namely octave with frequency ratio 2 : 1, just fifth with 3 : 2, and just third with 5 : 4. One may also make these interval ratios evident in the above formula, since P itch(f ) = log(f0 )+o log(2)+q log(3)+t log(5) = log(f0 )+(o+q +2t) log(2)+ q log(3/2) + t log(5/4). Recall here our discussion of the Pythagorean theory in section 2.1 that was based on the tetractys. For just tuning, one would have to add a fifth row with five points. A generalization of both, tempered and just tunings is given by the formula f = f0 · 2o 3q 5t , o, q, t ∈ Q p , p ∈ Z, while just which includes 12-tempered tuning for q = t = 0 and o = 12 tuning takes integer exponents only, and Pythagorean tuning is just tuning with t = 0. More general tunings involving higher prime numbers are also proposed in music theory [113]. We shall discuss just tuning from a geometric point of view in Chapter 28.

13.2 Adic Representations The problem of real numbers is that despite their elegant construction, they are difficult for humans to handle and impossible for computers, since infinite sequences are not representable by finite machines. This is all the more complicated since most real numbers have no simple rule that would describe their defining Cauchy sequences. We come back to this observation at the end of this chapter. A construction that helps humans describe and calculate real numbers is the adic representation, which generalizes the adic representation of natural numbers described in Chapter 9. Recall that for a natural basis 1 < b and a = 0, we could exhibit asequence a0 , a1 , . . . an , an = 0, of natural numbers n 0 ≤ ai < b such that a = i=0 ai bi . We now generalize this construction to a b-adic representation of real numbers, taking, for an integer n, an infinite sequence (ai )i∈n] with 0 ≤ ai < b, n] = {i|i ∈ Z AND i ≤ n}. We then look at partial sums   = ai bi = an bn +an−1 bn−1 +. . . a0 b0 +a−1 b−1 +. . . an−j bn−j . j

i=n,n−1,n−2,...n−j

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Lemma 10 The sequence ( known, it is also denoted by



j )j

converges to a real number

 i∈n]

ai bi . If b is

an an−1 . . . a0 .a−1 . . . for n ≥ 0 or

0.0 . . . an an−1 . . .

with an at the nth position to the right of the dot if n < 0. Proof 9 The proof essentially consists of an estimation of the speed with which the sum converges. The point can be made for b = 2 andassuming that all coefficients are ai = 1. Then we have to consider the sums i=0,1,...j 2−j . But  this is the so-called classical geometric series: sj (x) = i=0,1,...j xj for x = 1/2. j+1

One easily verifies that sj (x)x − xj+1 + 1 = sj (x), so sj (x) = 1−x 1−x . Taking x = 1/2, we get sj (1/2) = 2(1 − (1/2)j+1 ), and this clearly converges to 2. The number zero is denoted by 0 or 0.0. If aj = 0 eventually, we also write an . . . a0 .a−1 . . . a−m and, if the number is negative, we prepend a − and write −an an−1 . . . a0 .a−1 . . . Theorem 14 Every real number can be represented in this b-adic form for any given basis b > 1. The rational numbers are precisely of the following type: There is a sequence am , am−1 , . . . am−k such that their b-adic representation is as follows: an . . . a0 .a−1 . . . am+1 am am−1 . . . am−k which means that the overlined sequence acts as a period that is repeated ad infinitum to the right. For example, if we have the period 2, 3 for the decimal representation, we get for example such a number 1.23232323 . . . This adic representation is nearly unique, but for every basis b, there is a situation where some rational numbers have two representations. This happens each time where we have a period b − 1 of length one, for example in the decimal representation 0.1239. This number is the same real number as the number 0.124. Let us see why, and look at the simplest example of a decimal representation. Take x = 0.999999 . . . with period 9 of length one. This number x is in fact equal to 1.0. Let us see their differences when we consider the defining Cauchy sequences. For 1.0, its sequence is constant (1)i . For 0.999999 . . . we have the sequence 0.9, 0.99, 0.999, . . . The differences of the members of these sequences are 0.1, 0.01, 0.001, . . . which is a zero sequence, hence our claim.

14 Complex Numbers

Summary. Square roots of negative real numbers are not defined yet. We introduce complex numbers to solve this problem. Essentially, we introduce an imaginary number i, the square root of −1, and thereby add a new dimension to the real numbers. –Σ– We can now solve equations of type ax + b = c for all a = 0, and we can find points of convergence of all Cauchy sequences in R, but we cannot solve all equations yet. It can be shown that one can also solve any equation of type x3 + bx2 + cx + d = 0, but equations of type x2 + bx + c = 0 cannot be solved in general. For example, the simple equation x2 + 1 = 0 has no solution in R since x2 ≥ 0 for all real number x. The problem now is types of equations with higher powers of the unknown x. We shall see later in Chapter 24 that in fact, here again, the problem is the solution. But this requires more structures than we have yet. Therefore, we present a solution with less theory. The method we use now goes back to the German mathematician Carl Friedrich Gauss (1777-1855). He invented the valid theory of complex numbers. Mathematicians had worked with solutions of equations such as x2 + 2 = 0, but nobody figured out how to conceive such strange numbers that would solve those equations. This is one reason they are called “complex numbers”. Here is Gauss’ construction: The set of complex numbers C is identical to the cartesian product R × R. A complex number is a pair z = (x, y) of real numbers, where x is called the real part x = Re(z) and y is called the imaginary part y = Im(z) of z. The interesting new structure on C is its arithmetic, the addition and multiplication of complex numbers. Let z = (x, y), w = (u, v) be two complex numbers. We set z + w = (x, y) + (u, v) = (x + u, y + v), z · w = (x, y) · (u, v) = (xu − yv, xv + yu). Here is the sorite for this arithmetic structure: © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_14

113

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14 Complex Numbers

Sorite 8 Let x, y, z be complex numbers, and denote 0 = (0, 0), 1 = (1, 0). Then (i) (Additive associativity) We have (x + y) + z = x + (y + z) and denote this number by x + y + z. (ii) (Multiplicative associativity) We have (x · y) · z = x · (y · z) and denote this number by x · y · z, or xyz, if no confusion is likely. (iii) (Commutativity) We have x + y = y + x and x · y = y · x. (iv) (Distributivity) We have x · (y + z) = x · y + x · z. (v) (Additive and multiplicative neutral elements) We have 0 + x = x and 1 · x = x. (vi) (Solution of equations) If a = 0, then every equation a · x = b has a unique solution; in particular, the solution of a · x = 1, the multiplicative inverse of a, is denoted by a−1 . The solution of a + x = 0, the additive inverse (or negative) of a, is denoted by −a. We shall see below how to calculate the inverse explicitly.

Fig. 14.1. The complex numbers are points in the Gauss plane R2 , together with arithmetic operations. Every complex number is determined by its real and imaginary components.

The geometric view of Gauss is this: We have an injection R → C that sends a real number a to the complex number (a, 0). Similar to the embedding Q  R discussed above, all arithmetic operations, addition and multiplication, “commute” with this embedding, i.e., (a + b, 0) = (a, 0) + (b, 0), (ab, 0) = (a, 0) · (b, 0). We therefore identify real numbers a with their image (a, 0) in C. With this convention, denote the complex number (0, 1) by i, and call it the imaginary unit. It is easy to see that i2 = −1. This means that in C, the equation x2 +1 = 0 now has a solution, x = i. Complex numbers of the shape (0, b) are called imaginary. Clearly, x is uniquely determined by its real and imaginary parts, in fact:

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115

x = (Re(x), Im(x)). We then have this crucial result, which justifies the geometric point of view: Proposition 12 For any complex number x, we have a unique representation x = Re(x) + i · Im(x), as a sum of a real number Re(x) and an imaginary number i · Im(x). √

Exercise 18 Using the representation in Proposition 12, show that we have these arithmetical rules: 1. (a + i · b) + (c + i · d) = (a + c) + i · (b + d), 2. (a + i · b) · (c + i · d) = (ac − bd) + i · (ad + bc). The complex numbers have a rich inner structure that is related to conjugation. Definition 37 The conjugation is a map C → C : x → x ¯ defined by x ¯ = Re(x) − i · Im(x), i.e., Re(¯ x) = Re(x), Im(¯ x) = −Im(x). √ ¯, which is The norm of a complex number x is defined by |x| = x · x defined, since x · x ¯ = Re(x)2 + Im(x)2 ≥ 0. Observe that the norm of a complex number x = a + i.b is the Euclidean length of the vector (a, b) ∈ R2 known from high school! Sorite 9 Let x, y ∈ C. Then x=x ¯ iff x ∈ R, x = −x iff x is imaginary. |x| = 0 iff x = 0. x x−¯ x Re(x) = x+¯ 2 , Im(x) = 2i . ¯. If x = 0, then the multiplicative inverse of x is x−1 = |x|−2 · x ¯ = x; in particular, conjugation is a bijection. x x+y =x ¯ + y¯. x·y =x ¯ · y¯. If x is real, then |x| in the sense of real numbers coincides with |x| in the sense of complex numbers, which justifies the common notation. (ix) |x · y| = |x| · |y|. (x) (Triangle inequality) |x + y| ≤ |x| + |y|.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

√

Exercise 19 Calculate the inverse z −1 of z = 3.5 + i ·

√

5.

Complex numbers are omnipresent in physics. Roger Penrose [93] even claims that nature is built upon C rather than R. Recently, we have also been able to model musical processes using complex numbers [68]. Let us describe this approach here without delving into technical details.

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ˇ “* Example 33 It is a deep philosophical problem to conceive an ontology that comprises the cartesian res cogitans (the thinking thing) and res extensa (the extended thing), meaning the mental and physical ontologies. This famous cartesian duality can be solved in principle using complex numbers. One considers the cartesian product R3 × C. The three-dimensional component R3 carries the spatial coordinates, while the complex factor C is split into the real R and the imaginary iR. This defines two subspaces. The “physical” space R3 × R with the real axis R for physical time, and the “mental” space R3 × iR with the “imaginary” time axis iR. The first subspace represents the cartesian res extensa while the second space represents the cartesian res cogitans, see also Figure 14.2. This ontological model can be applied to music, where the score is

Fig. 14.2. The five dimensional space-time with complex time. The “physical” space R3 × R with the real axis R for physical time, and the “mental” space R3 × iR with the “imaginary” time axis iR.

positioned in the mental component, and its physical performance lives in the physical component. This implies that our mental activity while thinking about the score or creating it as a set of symbols occurs in imaginary time, while performance has to switch time to its real component. In our model [68], we have developed a theory of transition from imaginary to real time, using ideas from physical string theory. In this model, not only are there imaginary and physical

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states, but we also designed an entire family of intermediate states that share real and imaginary time. We shall come back to this model in Chapter 33 where gestural aspects of performance are discussed.

ˇ “*

Example 34 In sound technology, complex numbers are indispensable. When describing sounds that have a determined frequency, there is a classical mathematical theory that meets the needs for a complete description, namely the formalism discovered by Joseph Fourier around 1800. His theory allows for a decomposition of a sound function as a sum of sinusoidal functions, called partials or overtones. To perform calculations with Fourier’s theory, one works with complex numbers. Fourier’s theory has also been applied to create fast algorithms for the calculation of numerical data associated with partials. The most famous is called Fast Fourier Transform (FFT). It is the basis of the global Internet-based sound technology. Fast transmission of sound data would be impossible without FFT.

Part IV

Graphs and Nerves

15 Directed and Undirected Graphs

Summary. Up to now, we have been able to construct all basic number domains N, Z, Q, R, C. But we have not considered geometric objects. This chapter begins to fill that gap. It introduces the most elementary geometric objects: graphs—systems of points and arrows connected by directed or undirected lines. We shall conclude part IV with the introduction of higher-dimension graphical objects that relate to coverings of sets by a system of subsets. –Σ– In music theory, the systematic use of graphs was introduced by one of the authors (Guerino Mazzola) in 1980, see [72], who used category theory where arrows are the natural language. His idea of replacing sets of notes with directed graphs was motivated by the need for a method to define chords in 12-tempered tuning without reference to overtone arguments, which had never been a good logic for the justification of harmonic arguments. Let us give a single example of the graph-theoretical method for the construction of chords. Consider the major triad c, e, g that we place in the set P12 = {0, 1, 2, 3 . . . 11} of pitch classes. We apply the function F = T 7 3 : P12 → P12 : x → 3x + 7 to

Fig. 15.1. The major triad as a kind of “orbit” of c under one single function T 7 3.

c ∼ 0 and get F (0) = 7, the fifth 7 ∼ g. Applying F to g we get F (7) = 28, which generates the pitch class 4, namely, the third e. Applying F to the third generates nothing new, as F (4) = 3 · 4 + 7 = 19, generating again the fifth 7. This allows us to view the major triad as a kind of “orbit” of c under one single function F . © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_15

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The second, simultaneously invented introduction of graphs to music theory stems from David Lewin [60]. His idea was to move away from what he called the “cartesian” paradigm in music theory, meaning that the passive observer of musical objects “down there” should be replaced by the gesturally interactive “dancer” within music. Figure 15.2 shows a typical graph, where a chord is the set {f, c, g} of vertices of the graph, and the relations among the chord’s notes (pitch classes) are given by either transposition (F is transposed by T 2 to G) or inversion (f is inverted to c under I5, which is T−5 in our terminology). Such graphs are also called Klumpenhouwer networks in honor of Lewin’s student Henry Klumpenhouwer.

Fig. 15.2. In network theory, as shown from this original graphic, the triad {f, c, g} is interpreted using transposition or inversion relations (I5 and I7) among its elements.

ˇ “* Example 35 There are many musical situations where arrows that connect musical objects are adequate. The nature of such arrows can vary considerably, and we shall see some general examples when we discuss category theory in Chapter 29. A simple illustration of the graphical approach to music is shown in the following example. If we consider the black key f = g on a keyboard, we may view it as either a sharpened version f of the white key f , or as the flattened version g of the white g key. It is not an independent key; it is thought of as a key that results from two possible movements. The graphical   representation f - f = g  g represents this idea. 15.1 Directed Graphs Definition 38 A directed graph, or digraph for short, is a function Γ : A → V 2 from a set A of so-called arrows to the cartesian square of a set V of socalled vertices. The projection t = pr1 ◦ Γ : A → V is called the digraph’s tail function, while h = pr2 ◦ Γ : A → V is called the digraph’s head function. If a a ∈ A is an arrow, we represent it by t(a) - h(a). √

Example 8 Here is a digraph with one point t and one arrow T , a loop, the so-called final digraph 1 = t b T . The following digraph has two vertices u

and two parallel arrows u, v that connect them • v

%9 • . The following digraph

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is important in graph theory: P F

9f j N

*t Q

Q

T

For every natural number n, we have the chain digraph [n]. It has V = n + 1 = {0, 1, 2 . . . n} as vertex set and the set A = {(i, i + 1)|i = 0, 1, . . . n} as arrow set: (0,1) (1,2) (n−1,n) - n. [n] = 0 - 1 - 2 . . . n − 1 For n = 0 we have the trivial digraph with one vertex 0 and no arrow. The number n is called the length of [n]. Similar to sets and functions, digraphs must also be related to each other by “digraph functions.” Here is their definition: Definition 39 If Γ : A → V 2 , Δ : B → W 2 are two digraphs, a morphism f : Γ → Δ is a pair f = (u, v), u : A → B, v : V → W such that Δ ◦ u = v 2 ◦ Γ , i.e., the following square commutes: Γ

A −−−−→ ⏐ ⏐ u

V2 ⏐ ⏐ 2 v

Δ

B −−−−→ W 2 Similar to sets and functions, there is an identity morphism IdΓ = (IdA , IdV ) for every digraph Γ : A → V 2 , and if f = (u, v) : Γ → Δ, g = (w, z) : Δ → Σ are two morphisms, their composition g ◦ f = (w ◦ u, z ◦ v) is a morphism, and composition is associative. The identity morphisms are also neutral under composition similar to set identity functions. Isomorphisms are morphisms that have an inverse, which means that both parts, the arrow and the vertex functions, are bijections. √

Example 9 An important example of morphisms are the morphisms p : [n] → Γ whose domain is the chain of length n. Such a morphism is called a path of length n in Γ . The set of paths in Γ is denoted by P ath(Γ ). It follows that if p : [n] → Γ, q : [m] → Γ are two paths in Γ such that p(n) = q(0) then we can concatenate them and get a path q ◦ p : [n + m] → Γ . Concatenation of paths is associative. And the “lazy paths” [0] → Γ are neutral with their concatenations, whenever these are defined.

ˇ “*

Example 36 Melodies within compositions are typical examples of paths in music. Consider a composition K as shown at the top of Figure 15.3. Take the digraph Γ (K) defined to have the notes of K as the vertex set and the subset A ⊂ K 2 of all pairs of notes (n, m) such that Onset(n) < Onset(m).

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Fig. 15.3. Three melodic paths in the digraph Γ (K) of composition K, Song of Yi II—A Se by Mingzhu Song. The sound example is Yi.

This is sketched with some of the arrows in the middle of Figure 15.3. Now, we look at paths [n] → Γ (K). These are by construction connected sequences of notes that follow each other by increasing onset times. This is what could be defined as melodies in K. We have selected three such melodies, f1 , f2 , f3 , which are defined on the chain digraphs [7], [13], [6], respectively (Figure 15.3).

15.2 Undirected Graphs Undirected graphs, or simply “graphs,” are similar to digraphs except that their arrows have no direction. Here is the precise definition: For a set V of vertices, denote by 2 V the set {{x, y}|x, y ∈ V } of one or two element subsets of V . These are interpreted as undirected arrow connections: Definition 40 A graph is a function Γ : A → 2 V from a set A of edges into the set 2 V of unordered pairs {x, y} of vertices x, y. √

Example 10 The following general method produces many examples of graphs. For a set V of vertices, take the set function 2 V 2 : V 2 → 2 V : (x, y) → {x, y}. Then we have for each digraph Γ : A → V 2 an associated graph |Γ | : A → 2V = 2V 2 ◦ Γ .

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Definition 41 If Γ : A → 2 V, Δ : B → 2 W are two graphs, a morphism f : Γ → Δ is a pair f = (u, v), u : A → B, v : V → W such that Δ◦u = 2 v ◦Γ , i.e., the following square commutes: Γ

A −−−−→ 2 V ⏐ ⏐ ⏐2 ⏐ u v Δ

B −−−−→ 2 W Clearly, the above function 2 V 2 commutes with the functions between vertices, thus for v : V → W , we have a commutative diagram 2

V2

V 2 −−−−→ 2 V ⏐ ⏐ ⏐2 ⏐ v v2  2

W2

W 2 −−−−→ 2 W Combining this diagram with the commutative diagram (39), we see that every morphism f = (u, v) : Γ → Δ of digraphs induces a morphism |f | = (u, v) : |Γ | → |Δ|, and the composition of morphisms commutes with the composition of their images. Therefore the assignment Γ → |Γ | is defined not only on digraphs, but also on their morphisms, and in such a way that | | commutes with morphisms. This type of assignment is very important in modern mathematics (and in music!) and is called functorial: It maps one type of structure (here: digraphs plus their morphisms) to another type of structure (here: graphs plus their morphisms) in a compatible way. √

Example 11 If we look at the directed chains [n], we get their undirected images |[n]| =: |n|. Then, walks of length n are the morphisms |n| → Δ of graphs. Denote by W alk(Δ) the set of walks in graph Δ. Walks can be composed much like paths, and we have an obvious function P athW alk(Γ ) : P ath(Γ ) → W alk(|Γ |) for any digraph Γ , which again commutes with the composition of paths and also maps lazy paths to lazy walks. This is again such a functorial assignment. We can even go one step further: If f : Γ → Δ is a morphism of digraphs, we have, by obvious composition of digraph morphisms, a map P ath(f ) : P ath(Γ ) → P ath(Δ), and mutatis mutandis for graphs. This yields a corresponding commutative diagram P athW alk(Γ )

P ath(Γ ) −−−−−−−−−→ W alk(|Γ |) ⏐ ⏐ ⏐W alk(|f |) ⏐ P ath(f )  P athW alk(Δ)

P ath(Δ) −−−−−−−−−→ W alk(|Δ|) This last commutative diagram shows how “natural” the assignment Γ → |Γ | is, and this is the reason why this diagram is called a natural transformation. This concept will be explained in Chapter 29.

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15.3 Cycles We have seen that we can compose paths or walks. Now we can address cycles, special paths or walks that terminate on the same vertex whence they started. Cycles of length one are called loops. Graphs are said to be connected iff any two vertices can be the extremal values of a walk. Digraphs Γ are called connected iff their undirected images |Γ | are so. Two important types of cycles are Euler and Hamilton cycles. Definition 42 A cycle in a graph is called a Euler cycle iff it contains all vertices and each edge is traversed only once. A cycle in a graph is called a Hamilton cycle iff every vertex appears only once, except the first vertex, which reappears as the last by definition of a cycle. Proposition 13 A graph has a Euler cycle iff it is connected and every vertex has a positive even number of edges that contain it.

ˇ “*

Example 37 The harmonic sequence I − IV − V − V I − II − V − I can be seen as a Euler cycle as shown in Figure 15.4.

Fig. 15.4. The Euler cycle of the harmonic sequence I − IV − V − V I − II − V − I.

ˇ “*

Example 38 Dodecaphonic series can be interpreted as Hamilton cycles. We work in the complete graph K12 that has 12 vertices of P12 and all possible unordered pairs {x, y}, x = y as edges, see the left graph in Figure 15.5. In this representation, the set P12 is shown as a circular arrangement of twelve points similar to the twelve hours on a clock. Later, in Section 19.2, when dealing with group theory, we shall understand why this is a good representation. In section ∼ 6.3, a series was interpreted as a permutation s : P12 → P12 . We now interpret this as a walk that starts at s(0), goes to s(1), etc., and when arrived at s(11) closes to s(0). This defines a Hamilton cycle. This one for Webern’s op. 17.2 (shown in Figure 4.8) is drawn to the right in Figure 15.5. Connected graphs without cycles are called trees. A spanning tree of a graph is a subgraph that is a tree and contains all vertices. Every graph has a spanning tree.

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Fig. 15.5. The complete graph K12 of P12 (left) and a Hamilton cycle (right) associated with Webern’s series of op. 17.2.

16 Nerves

Summary. In biology, nerves connect different parts of a body. In music, we also can construct “nerves,” which are structures that connect different parts of a composition, making communication between such parts possible—similar to biology. –Σ– Often, we encounter the situation of a set X that is covered by a collection of subsets Vi ⊂ V , similar to a geographic atlas of charts that cover a

given region. Mathematically, to be a covering of a set V means that V = i Vi . We then would like to understand how these charts intersect and represent this information in a geometric way. This leads to the concept of the nerve of the covering. But let us first recall that a graph follows a similar construction: We are given the set V of vertices and then consider the set 2 V consisting by definition of all one- or two-element subsets of V , which are small charts. The vertices x ∈ V may then be recovered as intersections {x} = {x, y1 } ∩ {x, y2 } of two such charts. In other words, the idea of a graph is a special case of the idea of a nerve. For the next definition, we need a generalization of the coproduct to a number of cofactors. Recall the definition of a coproduct from Section 5.3. If  (Xn )0≤n≤m isa finite family of sets, the coproduct 0≤n≤m Xn is defined by  recursion via 0≤n≤m Xn = ( 0≤n≤m−1 Xn )  Xm . Definition 43 Given a set X, a covering of X is a set C ⊂ 2X of non-empty

subsets such that C = X. The nerve of C is the  subset N (C) ⊂ 2C consisting of all finite non-empty subsets s ⊂ C such that s = ∅. If card(s) = n + 1, we say that s is an n-simplex.The subset of n-simplices of N (C) is denoted by Nn (C), therefore N (C) = n≥0 Nn (C). In particular, N0 (C) = C if we identify 0-simplices with their single elements. If s ∈ N (C), then any nonempty subset t ⊂ s is in Nn (C). It is called a face of s.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_16

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In a geometric representation of simplices of a nerve, one represents 0simplices as points, 1-simplices as lines between their two 0-simplex points, 2simplices as triangles spanned by their three 0-simplex vertices, and 3-simplices as tetrahedra spanned by their four 0-simplex points.

ˇ “*

Example 39 A classical example of a nerve in music is given by the covering C (3) ⊂ 2C of the diatonic scale X, here X = C-major = {c, d, e, f, g, a, b}, by the seven standard triadic degrees I, II, . . . V II, as shown in Figure 16.1. The seven note names of C-major are shown as full points. This nerve is called the harmonic band of the given scale [72].

Fig. 16.1. The covering of the C-major scale by the seven standard degree chords.

The nerve N (C (3) ) has the seven degrees as 0-simplices, and the 14 lines for all pairs of degrees that intersect, e.g., I ∩ III = ∅, as 1-simplices. It has 7 filled triangles for triples of degrees that intersect, e.g., I ∩ III ∩ V = ∅, as 2-simplices. The overall geometry of this nerve is shown in Figure 16.2. The geometry of the harmonic band is the reason for the failure of Hugo Riemann’s function theory program. Refer to Musical Example 9, where we have explained Riemann’s ideas. Recall that Riemann wanted to define tonality functions X − T onality : Ch → T DS, starting from three values X − T onality(IX ) = T, X − T onality(VX ) = D, X − T onality(IVX ) = S. He then imposed conditions of function values for the remaining degrees IIX , IIIX , V IX , V IIX . These conditions were in fact geometric if one works with the harmonic band. Riemann’s first condition was that successive degrees on the band’s boundary, i.e., I → V → II → V I → III → V II → IV → I, should have different function values, this succession

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Fig. 16.2. The nerve N (C (3) ) of a diatonic scale is a Moebius band. Its lack of orientation is a reason for problems in Riemann harmony.

being the “fifths sequence.” Denote by δX the successor of X on the boundary. This would be satisfied since the fifth sequence IV → I → V has different values S, T, D. The second condition was to require that parallel degrees should have same values. Parallel here means that standing on a degree Y on the Moebius band and having the band’s surface to your right, the parallel πY would be the degree to your right in the direction of the boundary path. For example, standing on V , you look in direction of successor degree δV = I and you have degree III = πV . This means that πV is the third member of the 2-simplex spanned by V, δV, πV . So harmonic parallelism is really a geometric concept when we work with the Moebius band. The problem with Riemann’s approach comes into play when he asks that X − T onality(πY ) = X − T onality(Y ). In fact, we have these equations: D = X − T onality(V ) = X − T onality(πV ) = X − T onality(III) = X − T onality(πIII) = X − T onality(I) = T, which contradicts the first condition of different values for successive degrees on the band’s boundary. This contradiction is due to the lack of orientation on a Moebius band. If you stand on one side of the band and walk on the band’s surface, you end up standing upside down, see Figure 16.3. √

Example 12 For X = n + 1 = {0, 1, . . . n}, the covering C1 (n) = {{i, i + 1}|i = 0, . . . n − 1} defines a nerve N (C1 (n)), where N0 (C1 (n)) is the edge ∼ set of the chain graph |n|, and N1 (C1 (n)) → {{1}, {2}, . . . {n − 1}} the vertex singleton set of |n|, except the first and the last vertices 0, n. The bijection is defined by taking the intersection of successive members {i, i+1}∩{i+1, i+2} = {i + 1}.

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Fig. 16.3. The harmonic band has no orientation: If you walk around you end up upside down.

More generally, one may take the covering Ck (n) of n + 1 that has the charts i, k := {i, i + 1, i + 2, . . . i + k} for i = 0, . . . n − k. We now have not only 1-simplices as above, but also Nj (C(n)) = ∅ up to j = k, as {i, i + 1, . . . i + k} ∩ {i + 1, i + 2, . . . i + 1 + k} ∩ . . . {i + k, i + 1 + k + 1, . . . i + 2k} = {i + k}, see Figure 16.4 for the nerve N (C3 (16)).

Fig. 16.4. The nerve of the covering C3 (16) is visualized; it is a chain of tetrahedra (3-simplices) that are connected on one side with each other.

ˇ “*

Example 40 Compare the nerve C3 (16) to the nerve of the harmonic band. Can you see similarities? Can you guess how a harmonic band could be defined so it would resemble the nerve C3 (n)? Try first try to think about a harmonic band that resembles C1 (n). Similar to morphisms between graphs, we have morphisms between nerves: Definition 44 If C, D are two coverings, a simplicial morphism f : N (C) → N (D) is a set map f : C → D such that f (s) ∈ N (D) for each simplex s ∈ N (C). Clearly, the identity IdC is a simplicial morphism N (C) → N (C),

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and the composition g ◦ f of two simplicial morphisms f : N (C) → N (D), g : N (D) → N (E) is simplicial. Composition is associative. √

Example 13 Suppose we have coverings C, D of sets X, Y respectively, by non-empty subsets. Suppose we are given a set map f : C → D and a set map φ : X → Y such  that φ(S)  ⊂ f (S) for each S ∈ C. Then for every simplex s ∈ N (C), ∅ = φ( s) ⊂ S∈s f (s), therefore f (s) is a simplex, and f defines a simplicial morphism f : N (C) → N (D). The map f can always be found if we have φ, so only φ is the tricky point in this construction. That is, we need to know that for every S ∈ C, there exists at least one T ∈ D such that φ(S) ⊂ T .

The set N1 (C) of 1-simplices of N (C), together with the set C of 0simplices, defines a graph N1 (C) → 2 C : {S, T } → {S, T }, which we denote by N1 (C), and whose edges are the 1-simplices. We therefore may define walks |n| → N1 (C) in a nerve, also called nervous walks in C. The set of nervous walks in C is denoted by W alk(C). And for a simplicial morphism f : N (C) → N (D), we have the associated functorial map W alk(f ) : W alk(C) → W alk(D). More generally, we may consider simplicial morphisms defined on the nerve N (Ck (n)). These morphisms map edges not only to 1-simplices but also higherdimensional simplices in N (Ck (n)).

16.1 A Nervous Sonata Construction In this section, we want to show how the geometry of nerves of coverings can be used to compose motivic structures in a sonata. Our example is the sonata Allegro movement op. 3 by one of the authors (Mazzola), published and recorded under the title L’essence du bleu [76]. We only describe the motivic construction of a specific part of the composition and don’t discuss the harmonic and rhythmic aspects. This sonata was composed in the spirit of Pierre Boulez’s “creative analysis” [20], which means that the composition was analyzed and this analysis was used to create a new composition by changing some analytical parameters. In our analysis of Beethoven’s “Hammerklavier” Sonata, op. 106, we exhibited the central role of the diminished seventh chord C −7 = {c , e, g, b}, see Figure 16.5. Apart from determining the sonata’s harmonic modulations (changes of tonalities, see Chapter 23 for details), this chord also determines Beethoven’s motivic work. As the chord C −7 is reproduced after transposition T 3 of its pitch classes, it is reasonable to consider an elementary motivic “zigzag” of period 3. This is shown in Figure 16.5 to the left, where a chromatic ascending and descending melodic movement is visible. In op. 106, this zigzag is a dominating motivic structure [72]. In our creative part of the analysis, we replaced the minor third transposition T 3 by a major third transposition T 4 , since 12 = 3 · 4 is the decomposition

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Fig. 16.5. The motivic zigzags in Beethoven’s op. 106 and Mazzola’s op. 3.

of 12 into different prime number powers. The corresponding chord is the augmented triad1 C + = {c , f, a} that is reproduced after a transposition T 4 of pitch classes. And the corresponding motivic zigzag of period 4 is shown below the chord C + to the right in Figure 16.5. Inspired by the harmonic band of triads, the idea was to construct a covering of the unit of zigzag, shown on top of Figure 16.6, that would have a nerve in the shape of a Moebius band, too. Figure 16.6 shows a solution that consists of nine three-element motives.

ˇ “* Exercise 5 Check that the nine three-element motives really define a Moebius-shaped nerve. Using this scheme of motives, the construction of a concrete melodic structure in this sonata runs as follows, see Figure 16.7. We enumerate the nine three-element motives on the Moebius band according to their path of neighboring motives, yielding the sequence 1, 2, 3, . . . 8, 9 of motive numbers. We then select four consecutive motives A = (2, 3, 5, 6) as well as their mirror image A = (2, 1, 9, 7). Then, the selections A and A are moved down clockwise and counterclockwise to get the selections B, C, D, E, F and B  , C  , D , E  , F  . We then have two groups A, B, C, D, E, F and A , B  , C  , D , E  , F  of six groups of four motives each. These motives are distributed in the pitch and onset time plane as shown in Figure 16.8 for the group A, B, C, D, E, F , yielding Dr (r for right hand). The geometric position is given within the original zigzag that is slightly rotated to generate increasing pitches with time. The same geometric positioning 1

For music theorists: The sharpened note names do not mean that we think in alterations, as explained in Musical Example 35, but simply denote the position of the note on the set P12 .

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Fig. 16.6. The covering of the motivic unit (top) by nine three-element motives has a Moebius band nerve.

Fig. 16.7. The construction of a concrete melodic structure in this sonata.

is applied for the second group A , B  , C  , D , E  , F  of three-element motives, yielding Dl (l for left hand). See Figure 16.9 for this configuration. This produces the score part of measures 33-38 in the sonata, see Figure 16.10. We come back to this construction in Chapter 23 where the modulation theory is applied to this score part.

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Fig. 16.8. Distribution of the three-element motives in the pitch-onset plane. Group A, B, C, D, E, F yields Dr (r for right hand), while Group A , B  , C  , D , E  , F  yields Dl (l for left hand).

Fig. 16.9. The distribution Dr , Dl .

16.1.1 Infinity of Nervous Interpretations Let us terminate this chapter with a remark on the interpretative power of nerves in music analysis. It may seem that everything is finite since we are dealing with a finite number of notes that may be covered by a finite atlas of charts. But this is erroneous, as may become evident by the following thoughts.

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Fig. 16.10. The score part of measures 33-38 corresponding to the distribution Dr , Dl . The sound example is sonata.

A covering C of a set X of notes is a subset of 2X . Its nerve N (C) is X a subset of 22 , its simplices are finite sets of charts from C. There is no reason, why we would not be interested in the nerve of the nerve, N (N (C)), whose simplices would be the finite sets of simplices in N (C) having non-empty 2X

intersection. We then have N (N (C)) ⊂ 22 . This may be of interest when looking for connections between simplices to understand the transition from one simplex to one of its neighbors. This makes evident that there is no reason to stop stepping from one powerset 2X to its X

...2X

powerset 22 , and so on to the multiply repeated powerset of powersets 22 . This perspective, which can be taken for both, analysis and composition, opens up an infinity of structures in higher powersets of a given finite set X. In a more philosophical understanding, this makes plausible the potential infinity of understanding and construction of the basic finite set of notes in a musical composition. 16.1.2 Nerves and Musical Complexity Nerves are a precise and powerful tool to discuss the difficult concept of musical complexity in analysis and composition. The question is here what makes a composition complex. There are two extremal positions when considering a score: One can say that this object is just a set of notes. This would reduce the score to an atomic perspective. The corresponding covering would be the one which has exactly one note in each chart. The nerve of this covering would be trivial, consisting only of zero simplices. And all these charts would be isomorphic by shifting around one note to produce all the others. This would amount to simply counting notes, a totally destructive classification: to reduce a composition to the cardinality of its note set.

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The other extremal position would be to cover the composition by one single chart that comprises all the notes. This big chart would then be a big set of points in the score’s parameter space. Even though the nerve of this covering would also be trivial, consisting of one single zero simplex, the classification of this chart would be extremely difficult. In mathematical music theory [75, Appendix C.3.6] it is shown that, for example, there are 2 230 741 522 540 743 033 415 296 821 609 381 912 ∼ 2.2336 isomorphism classes of 72-element compositions. This makes understanding impossible: how could one understand a class in this virtually infinite system? Complexity is better understood if one chooses a system of charts that are not too big—chords, small motives, short rhythmic units, for example—and then looks at nerves of such a covering which may include many intersecting charts (giving rise to higher-dimensional simplices, such as we have seen for the harmonic Moebius band). Beethoven’s compositions are excellent examples for this type of complexity: His constructions are “locally simple”, but generate complex nerves and nerves of nerves, and so on. The subject of musical complexity is far from being fully understood, but the tool of nerves of coverings is a first step towards a mathematical theory of musical complexity.

16.1 A Nervous Sonata Construction

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Fig. 16.11. A harmonic band. Copy it to a separate paper, cut it out there, fold and glue it using the small purple tab.

Part V

Monoids and Groups

17 Monoids

Summary. Monoids are the simplest type of algebraic structure, and for this reason they are omnipresent in mathematics. This situation is parallel to the hierarchy of numbers. The monoids will be extended structurally (not as sets!) to groups, rings, and modules later. –Σ– Definition 45 A monoid is a pair (M, ∗), where M is a set, and ∗ : M ×M → M a function, called composition, such that (i) ∗ is associative. (ii) There exists a neutral element e, such that e ∗ m = m ∗ e = m for all m ∈ M . It is uniquely determined by this property. (iii) (M, ∗) is called commutative iff m ∗ n = n ∗ m for all m, n ∈ M . √

Example 14 Here is a first set of examples:

1. The number domains N, Z, Q, R, C with the multiplication of numbers and the unit e = 1 are commutative monoids. 2. The unit circle {z| |z| = 1} ⊂ C, together with multiplication of complex numbers is a commutative monoid. 3. For any set X, the set End(X) := Set(X, X), together with the composition ∗ = ◦ of functions and the identity e = IdX is a (in general non-commutative) monoid. 4. The subset Sym(X) ⊂ End(X) of permutations of X, i.e., of bijections, is a monoid. 5. For a digraph Γ , the set End(Γ ) of digraph morphisms f : Γ → Γ with the composition ∗ = ◦ and with e = IdΓ is a monoid. 6. For a digraph Γ and a vertex x of Γ , the set of cycles in x, together with the composition of paths and the lazy path at x as neutral element, is a monoid Cyc(Γ, x). As a special case, given a set A (the “alphabet”), we have the word monoid W ord(A) = Cyc(A), where A is the digraph with one © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_17

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single vertex and A the set of arrows. Here, Cyc(A) consists of sequences of arrows (all being loops!), i.e., words built from letters in A.

ˇ “*

Example 41 In music theory, scales play an important role. In the pitchclass set,P12 , the D-major scale is represented by d, e, f , g, a, b, c , d, when starting from d. This information is often encoded by examining the intervals (in multiples of semitone steps) between consecutive pitch classes. In our example, this would be the sequence (2, 2, 1, 2, 2, 2, 1). This information is then encoded as a word in the monoid W ord(A), where A = {a, b} is the alphabet whose letters encode the two intervals, a stands for 1, and b stands for 2. This encoding represents the scale by the word bbabbba. If we had taken the melodic D-minor scale d, e, f, g, a, b, c , d, the word representation would be babbbba, exchanging the second and third letters in the major scale word. Of course, this representation is not reliable since it does not define the pitch class where the intervals start. We shall come back to the mathematical nature of this abstraction in Chapter 19. In recent times, this formalism of word monodies has been a strong research direction in mathematical music theory, see the corresponding papers in [5]. The implicit idea of such a scale word is that the sum of the intervals corresponding to the letters a, b is 12, the octave interval. But the formalism is indeed more general. One may define any word w ∈ W ord(A) for any alphabet A with the interpretation of letters as intervals being given by a set function int : A → P12 . Then the scale scale(w) would be the sequence of pitch classes, starting from 0, say, and defined by the succession of intervals associated to letters by the function int. It is understood that after the last interval of w, the word starts over. For example, the above major scale of word w = bbabbba yields the multiple concatenation of intervals, for example the triple one www = bbabbba ∗ bbabbba ∗ bbabbba = bbabbbabbabbbabbabbba. For a non-octave scale word w = b, the word wwwwww yields the scale c, d, e, f , g , a , c, the first Messiaen scale, also known as the whole-tone scale. For the non-octave scale word w = ba, on gets wwww = babababa, which corresponds to the scale c, d, d , f, f , g , a, b, c, which is known as the second Messiaen scale. For w = aab, the iterated word www = aabaabaab yields the scale c, c , d, e, f, f , g , a, a , c, the third Messiaen scale. But one may also get non-octave compatible words such as w = bba, whose iteration eventually fills every pitch class. However, one may also work in the pitch set Z and then generate a selection of pitches that don’t repeat themselves after octaves. In our example, w = bba, one would get this scale: c, d, e, f, g, a, a , c, d, d , f, g, g , . . ..

ˇ “*

Example 42 Another use of the word monoid W ord(A) in music is the formal representation of harmonic syntax, in the Riemann formalism, say. Here we have an alphabet (we use only a simple model of harmonic functions here)

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A = {TX , DX , SX , tX , dX , sX |X ∈ P12 } where TX , DX , SX represent the tonic, dominant, and subdominant functions of major tonality X, while tX , dX , sX represent the tonic, dominant, and subdominant functions of minor tonality X. Then an expression of such a harmonic syntax might look like T C SC D C T C T C D F T F , where we see after a cadence Cad = TC SC DC TC the harmonic trace M od = TC DF TF of a modulation from C-major to F -major. This word is the product Cad ∗ M od of the cadence and modulation subwords.

ˇ “*

Example 43 In the theory of rhythm of music and language, word monoids are also the standard representation. For example, in poetry, a rhyme scheme, such as ABAB, represents the occurrence of a (end)rhyme word in a succession of a poem’s lines. Within a line of a poem, the distribution of short (symbolized by “letter” ∪) and long (symbolized by “letter” —) syllables defines a metrical line. Typical “elementary” words of meters are the dactyl —∪∪ or the spondee — —. Here is a classical hexameter, a product of dactyls and spondees, from Virgil’s Aeneid: Arma vi(∗)rumque ca(∗)no, Tro(∗)iae qui(∗) primus ab(∗) oris (—∪ ∪ ∗ —∪ ∪ ∗ — —∗ — —∗ —∪ ∪ ∗ — —) These structures have been used for the rhythm of musical compositions with lyrics. A wonderful example being Franz Schubert’s composition op. 72 for Leopold Stolberg’s poem Lied of dem Wasser zu singen. See [75, Section 11.6.2] for a thorough discussion of the poetic rhythm of musical three-note motives in Schubert’s composition. In music theory, certain large forms, such as the song form ABA or the sonata form ABAD represent the succession of parts and their iteration. For the sonata form, it is A for the exposition, B for the development, again A for the recapitulation, and D for the coda. Definition 46 Given two monoids (M, ∗M ), (N, ∗N ), a morphism f : M → N is a set function f : M → N such that f (m ∗M n) = f (m) ∗N f (n) for all m, n ∈ M and f (eM ) = eN . The set of monoid morphism f : M → N is denoted by Mon(M, N ). Monoid morphisms f : M → N, g : N → L can be composed as set functions and define monoid morphisms, and the identity IdM is always in End(M ) := Mon(M, M ). A monoid isomorphism is a morphism that is invertible, and this is the case iff it is a bijection of sets. Proposition 14 (Universal Property of Word Monoids) Let A be an “alphabet” set and N a monoid. Then there is a bijection ∼

Mon(W ord(A), N ) → Set(A, N ). The bijection is defined taking the restriction of f : W ord(A) → N to the subset A of loops.

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Exercise 20 Give a proof of this proposition.

√

Exercise 21 If N = N, with addition of natural numbers, what is the meaning of the morphism W ord(A) → N (N with addition) that is defined on A by f (a) = 1 for all a ∈ A? Definition 47 For a monoid (M, ∗), a submonoid is a subset N ⊂ M such that m ∗ n ∈ N for all m, n ∈ N and e ∈ N .

The above numbers N ⊂ Z ⊂ Q ⊂ R ⊂ C are submonoids for both addition and multiplication of their larger successor sets.  If (Ni )i is a nonempty family of submonoids Ni ⊂ N , their intersection i Ni is a submonoid. Definition 48 Let (M, ∗) be a monoid and let S be any subset of N . Then the intersection  N S := N ⊂M submonoid with S⊂N

is the smallest submonoid of M that contains S. It is called the submonoid generated by S. It is also identified as the submonoid {s1 ∗ s2 ∗ . . . sk |si ∈ S for all i} of all products of elements of S, including the empty product that is by definition the neutral element e. √

Example 15 The word monoid W ord(A) is generated by A, which means W ord(A) = A.

18 Groups

Summary. Monoids that have only invertible elements are called groups. Groups are the most important single structure in algebra and have enormous applications in physics as well. –Σ– Definition 49 A monoid (G, ∗) is a group iff every element g ∈ G is invertible, which by definition means that there is h ∈ G such that g ∗ h = h ∗ g = e. This inverse h is uniquely determined since for any inverse h , we have h = h ∗ e = h ∗ (g ∗ h) = (h ∗ g) ∗ h = e ∗ h = h. We denote it by h = g −1 . A commutative group is also called abelian. For two groups G, H, a group homomorphism is a monoid morphism f : G → H. The set of these homomorphisms is denoted by Grp(G, H). Group homomorphisms can be composed, and the identity IdG is a group homomorphism. A subgroup H ⊂ G is a submonoid that contains all inverses of its elements. √

Example 16 The group of symmetries of a square (Figure 18.1). There are eight symmetries i, r1 , r2 , r3 , h, v, d1 , d2 . √

Exercise 22 In the group from Example 16, write down the multiplication table (x∗y)x,y . In each of the 8 rows/columns there is a x/y, and in the crossing point of these rows and columns, write the product x ∗ y. √

Example 17 Here are some examples of groups.

1. For a monoid M , the subset M ∗ of invertible elements g is a group. This means that we consider the elements g such that there is h ∈ M with h ∗ g = g ∗ h = e. The product g1 ∗ g2 of two such invertible elements g1 , g2 has the inverse g2−1 ∗ g1−1 . 2. The multiplicative monoids N, Z, Q, R, C have the subgroups of invertible elements N∗ = {1}, Z∗ = {1, −1}, Q∗ = Q−{0}, R∗ = R−{0}, C∗ = C−{0}.

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Fig. 18.1. The symmetries of the square build a group under composition.

3. The additive monoids N, Z, Q, R, C have the subgroups of invertible elements N∗ = {0}, the others are already groups. 4.  Let G be a group and (Hi )i any family of subgroups. Then the intersection i Hi is a subgroup of G. 5. If (G, ∗) is a group, the opposite group Gopp is (G, ∗ ) where g ∗ h = h ∗ g. The neutral element of Gopp is the same as the neutral element of G, while g ∗ (h ∗ k) = (k ∗ h) ∗ g = k ∗ (h ∗ g) = (g ∗ h) ∗ k and the inverses of elements in Gopp coincide with the inverses of G. In Section 6.3 was introduced the set Sn of permutations of n, together with the composition π ◦ ψ of permutations π, ψ ∈ Sn . We also exhibited the inverse π −1 of any permutation, and the existence of the neutral permutation Idn . This is a core group in mathematics, and also in music.

ˇ “* Example 44 In Section 6.3, a dodecaphonic series was identified with a permutation in S12 . We shall see below (Proposition 15) that the cardinality of S12 , the number of all dodecaphonic series, is 12! = 479,001,600. Definition 50 Let G be a group (we suppose that the multiplication ∗ is given) and let S be any subset of G. Then the intersection  S := H H⊂G subgroup with S⊂H

is the smallest subgroup of G that contains S. It is called the subgroup generated by S. It is also identified as the subgroup {t1 ∗ t2 ∗ . . . tk |ti = s±1 i , si ∈ S for all i} of all products of elements of S, or of their inverses, including the empty product that is by definition the neutral element e.

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Exercise 23 Find a minimal set of generators for the square group in Example 17.

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Example 45 Let us consider the cartesian product space OP = R × R of onsets and pitches, for which we allow real number values, just to view the situation in a geometric way. Within the group Sym(OP ) of bijections on OP , there are some bijections that are classical in music: retrograde R, indexinversioninversion I, and retrograde inversion RI. The operation R is defined by R(x, y) = (−x, y), inversion is I(x, y) = (x, −y), and retrograde inversion is RI(x, y) = (−x, −y), see Figure 18.2. (We should not care about getting negative pitch values here, we can always re-calibrate pitch to get positive values by shifting the zero down.)

Fig. 18.2. The Klein 4-group of retrograde R, inversion I, and retrograde inversion IR = RI.

Retrograde R and inversion I are reflections at the vertical axis and the horizontal axis, respectively. But RI is not a reflection, it is a 180o rotation around the origin (0, 0) of the plane P O. In music theory, RI is not understood as a rotation, but as the composition R ◦ I = I ◦ R of retrograde and inversion. In terms of group theory, we have a small group R, I that is generated by R and I, with four elements IdOP , R, I, RI = IR. You can easily check that this is in fact a group. It is called the Klein 4-group K4 . This group is generated by retrograde and inversion, and the product RI is not understood geometrically as a rotation, but as a composition of two musically understandable operations. It has two non-trivial subgroups R = {IdOP , R} and I = {IdOP , I}. The understanding of RI as a rotation has, however, been realized in a very practical way by Ludwig van Beethoven. At his time there were public piano competitions. Several pianists would perform whatever they liked and then be judged by the audience. In such a competition, Beethoven was fighting against the quite famous pianist Daniel Steibelt. After Steibelt had played his

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score, Beethoven took it and rotated it by 1800 , thereby creating a RI version of Steibelt’s composition. Beethoven then played this rotated score and defeated Steibelt, who ran away in bitter anger. This fact can be generalized and gives rise to a general concatenation principle in music theory [75, Section 8.3], namely that all groups that are important to music are in fact so because they admit sets of generators that are musically understandable. One of the most important groups is the group of permutations of a set X, Sym(X). In particular, if X = {1, 2, . . .}, the group Sym(X) is called the symmetric group of rank n and is denoted by Sn . The former symbol Sn denoted the group Sym(n), but by abuse of language, we use the same symbol ∼ for Sym({1, 2, . . .}). Clearly, if card(X) = n, then Sym(X) → Sn . We shall discuss this group in Chapter 20. Definition 51 A group G is called finite if G is a finite set. If the cardinality of G is n (in symbols card(G) = n), we say that it is a group of order n, and we write ord(G) = n. Clearly, any two groups of order one are isomorphic to the trivial group 1 = {e}. Proposition 15 The symmetric group Sn has order ord(Sn ) = n! Proof 10 The proof goes by induction on n. For n = 1, this is clear. Suppose x the subset p in we know that ord(Sn ) = n! and denote by Sn+1  of permutations x . We Sn+1 such that p(1) = x ∈ {1, 2, . . . n+1}. Then Sn+1 = x=1,2,...n+1 Sn+1 x ) = n! holds. In fact, denote by (x, n + 1) the permutation claim that card(Sn+1 that exchanges x and n + 1 and leaves fixed all other numbers. Then clearly n+1 n+1 ∼ x (x, n + 1) = Sn+1 . Therefore we have to prove Sn+1 → Sn . But the set Sn+1 n+1 Sn+1 consists of all permutations of numbers 1, 2, . . . n+1 that leave n+1 fixed. This is obviously in bijection with Sn . Hence, card(Sn+1 ) = (n + 1)card(Sn ) = (n + 1)! proves the proposition.

19 Group Actions, Subgroups, Quotients, and Products

Summary. In musical creativity, actions are very important. They always deal with two components: the agent who acts in a determined way, and the object on which this action is being performed. Of course, not just any action can be performed on any object, so one has to specify agent/action as well as domain of objects that are suitable for a determined action. In this chapter we develop the formalism of group actions and provide introductory examples. –Σ– Figure 19.1 shows actions taken while building musical instruments or composing a score. Of course, these intuitive situations are far from precise mathematical formalism.

Fig. 19.1. Left: Building instruments is an action that combines parts to produce a whole. But this is not always possible. Right: The musical composition is also an action that (usually) generates a score. The neutral action could be the creation of an empty score (recall John Cage’s empty composition in Figure 3.3), while analysis could be seen as the inverse action.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_19

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19.1 Actions Definition 52 A group homomorphism f : G → Sym(X) for a set X is called an action of G on X. Equivalently, a group action is a function G × X → X : (g, x) → g · x such that (i) e · x = x for all x ∈ X and (ii) (gh) · x = g · (h · x) for all x ∈ X. The correspondence being g · x = f (g)(x). Definition 53 Given a group action G × X → X, the relation on X, defined by x ∼ y if there exists a g ∈ G such that g · x = y, is an equivalence relation. The equivalence classes for this relation are called the orbits of the given action. We also denote [x] by G · x. An action is said to be transitive if X = G · x, i.e., if there is only one orbit. It is said to be simply transitive if the application G → X : g → g · x is a bijection. It is straightforward that the bijection does not depend on the chosen x, i.e., that a bijection for x implies a bijection for any other y ∈ G, and that simple transitivity implies transitivity.

ˇ “*

Example 46 In his seminal book [60, Definition 2.3.1], David Lewin introduces a core concept of his transformational theory, the Generalized Interval System. Here is his original text: DEFINITION: A Generalized Interval System (GIS) is an ordered triple (S, IV LS, int), where S, the space of the GIS, is a family of elements, IV LS, the group of intervals for the GIS, is a mathematical group, and int is a function mapping S × S into IV LS, all subject to the two conditions (A) and (B) following. (A): For all r, s, and t in S, int(r, s)int(s, t) = int(r, t). (B): For every s in S and every i in IV LS, there is a unique t in S which lies in the interval i from s, that is a unique t which satisfies the equation int(s, t) = i. We want to show that this concept is that of a simply transitive action of a group. We have the group IV LS opp (see Example 17) that acts on the space S in the following way. Call · the composition in IV LS and · the composition of the opposite group. For any ordered pair (i, s) ∈ IV LS×S we have the action i∗s = t for that unique element t ∈ S such that in Lewin’s approach i = int(s, t). We can also state this by the equation int(s, t) ∗ s = t. Let us show that this is indeed an action. The equation in Lewin’s point (A) shows that the neutral element e ∈ IV LS acts with e ∗ s = s. In fact, int(s, s)int(s, s) = int(s, s). Multiplying this equation with int(s, s)−1 yields int(s, s) = e. Moreover, if i = int(t, r), j = int(s, t) are two group elements, and if j ∗ s = int(s, t) ∗ s, i ∗ (j ∗ s) = i ∗ (int(s, t) ∗ s) = i ∗ t = int(t, r) ∗ t = r, while (i · j) ∗ s = (int(t, r) · int(s, t)) ∗ s = (int(s, t) · int(t, r)) ∗ s = int(s, r) ∗ s = r. The action

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is simply transitive since the map IV LS opp → S : i → i ∗ s = t is a bijection for every s by the equation int(s, t) ∗ s = t for i = int(s, t). Conversely, we get Lewin’s GIS for a simply transitive action G × S → S ∼ since for every s ∈ S, there is a bijection G → S : g → g ∗ s, and the conditions (A) and (B) are verified for the opposite group Gopp . √

Example 18 Let X = Z, the set of integers. Consider the group T ⊂ Sym(Z) of transpositions on Z. Its elements are the transpositions T t , t ∈ Z, and the action is T t (z) = t + z. Also consider the larger group T I ⊂ Sym(Z), whose elements are the transpositions T+t = T t and inversions T−t with T−t (z) = t − z, so its elements are T±t with T±t (z) = t ± z. The multiplication is T+t ◦ T+s = T+t+s , T+t ◦ T−s = T−t+s , T−t ◦ T+s = T−t−s , T−t ◦ T−s = T+t−s . These groups act on 2Z , too, and also on the subset F in(Z) of finite subsets of Z. Neither action is transitive.

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Example 47 Group actions are very frequent in music theory. Take G = (Z, +), the additive group of integers and S = Z the set of pitches. Then G acts on S by transposition, i.e., g ∗ s = T g (s) = g + s. This action is obviously simply transitive. If we take the subset F in(Z) ⊂ 2Z of finite subsets of Z, which can be interpreted as chords, we have the action Z × F in(Z) → F in(Z) : (g, c) → g ∗ c = {T g (x)|x ∈ c}. This action is not simply transitive because it is not transitive, although it is free, that is, g ∗ c = h ∗ c iff g = h, which means that the map G → S : g → g ∗ s is injective. The orbits of chords under this action are called transposition classes of chords. For example, the chords in the orbit [{0, 3, 6}] are called diminished triads, while the chords in the orbit [{0, 4, 7}] are called major triads.

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Example 48 We may also consider a larger group T I that acts on the pitch set Z and consists of the transpositions T+t = T t and inversions T−t of Z. This group acts transitively on Z, but its action is not free because for every s ∈ Z, we have T−2s (s) = 2s − s = s, together with T+0 (s) = s. This group acts on F in(Z) in the same way (Z, +) acted by transpositions. For a given chord c, the subgroup F ix(c) of transpositions and inversions g such that g ∗ s = s is called the fixpoint group of s. It describes the inner symmetries of chord c. The diminished triad c = {0, 3, 6} has F ix(c) = T−6 , while for c = {0, 4, 7}, the major triad, the fixpoint group is trivial. The T I-orbit [{0, 4, 7}] is the disjoint union of the T -orbit of {0, 4, 7} and the T -orbit of {0, 3, 7}, major and minor triads. The most important primary application of group actions are actions of subgroups H ⊂ G by cosets. These actions are defined by right multiplications within the given group, i.e., G × H → G : (g, h) → gh. The orbits of such an action are called left cosets of H. They are the sets gH = {gh|h ∈ H}. In particular, two left cosets gH, g  H are equal iff g −1 g  ∈ H. The set of left cosets is denoted by G/H. If we choose one representative g for each left coset

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 gH, we have G = representative g gH. Moreover the maps H → gH : h → gh are bijections. Therefore we have a bijection ∼

G → G/H × H. One may also define right cosets Hg that correspond on an action from the left (corresponding to a function H × G → G : (h, g) → hg). The orbits of this action are the right  cosets Hg. They also define a partition by disjoint right cosets, i.e., G = representative g Hg. The set of these equivalence classes is denoted by H\G, and we have a bijection ∼

G → H\G × H. ∼

Left and right cosets correspond to each other by the bijection ?−1 : G → G : g → g −1 . This implies that left cosets are mapped one-to-one to right cosets. ∼ Therefore G/H → H\G. The shared cardinality of G/H and H\G is called the index of H in G and is denoted by (G : H). We therefore have ord(G) = (G : H) × ord(H). If G is finite, we have the famous Lagrange equation ord(G) = (G : H)ord(H) of natural numbers. In particular, the order of a subgroup H ⊂ G always divides the order of the group G. For example, if G = S6 , we have ord(S6 ) = 6!, which implies that there is no subgroup H whose order does not divide 6!—e.g., there is no subgroup of order 7.

ˇ “*

Exercise 6 Recall the Klein 4-group discussed in Musical Example 45. Are there subgroups of this group of order 3? Interpreting a dodecaphonic series as a permutation in S12 , can its order be 12, 13, or 23? If x ∈ G is a group element, its order is by definition the order of the group it generates, i.e., ord(x) = ord(x). For a finite group G, the order of an element must be a divisor of the group’s order: ord(x)|ord(G) by Lagrange’s equation. How do we find this order for x? One looks at all the powers e, x, x2 , x3 , . . . of x. Since G is finite, they cannot all be different from each other. Let xm be the first power of x that equals a preceding power, xm = xn , n < m. Then, after dividing by xn , we see that e = xm−n . Since m was the first power of x that equals a preceding power, this means that n = 0, and the group x consists of the different powers e, x, . . . xm−1 . For example, we have x−1 = xm−1 . In particular, m = ord(x).

19.2 Subgroups and Quotients Subgroups where left and right cosets coincide play a central role in group theory: They are the kernels of group homomorphisms, and here are the precise concepts.

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Definition 54 If f : G → K is a group homomorphism, then the subset {g|g ∈ G, f (g) = eK } is a subgroup, the kernel of f , denoted by Ker(f ). √

Exercise 24 Prove that a group homomorphism f is injective iff Ker(f ) is trivial.

Definition 55 A subgroup H ⊂ G of group G is called normal iff gH = Hg for all g ∈ G, i.e., all left and right cosets coincide. For example, all subgroups of commutative groups are normal. If H ⊂ G is normal, we may construct a group from G/H by the multiplication gH · kH = gkH. The neutral element is H. The multiplication is well defined since gH = g  H, kH = k  H implies gkH = gHk = g  Hk = g  kH = g  k  H. It is evident that multiplication is associative, has the neutral element H, and has inverse g −1 H of gH. This group is called the quotient group of G modulo H, denoted by G/H. We have a canonical surjection G → G/H. Proposition 16 Let H be a subgroup of group G. Then the following two properties are equivalent—(i) iff (ii): (i) There is a group homomorphism f : G → K such that H = Ker(f ). (ii) H is normal. The group homomorphism associated with a normal subgroup H is the canonical morphism G → G/H. Proof 11 If H is the kernel H = Ker(f ) then gHg −1 ⊂ H for all g ∈ G, and also g −1 Hg ⊂ H, hence gH = Hg. Conversely, if H is normal, we take the canonical homomorphism f : H → G/H, whose kernel is H.

Fig. 19.2. The cyclic groups Z3 , Z6 , Z9 , Z12 viewed as subgroups of the unitary group U of complex numbers.

√

Example 19 The simplest example of a quotient group is given by taking for every positive n ∈ N the subgroups n = nZ ⊂ Z generated by n. It

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19 Group Actions, Subgroups, Quotients, and Products

consists of all integer multiples zn of n. The quotient group Zn = Z/nZ is called the cyclic group of order n. It has, in fact, n elements. This follows from the division theorem. If a ∈ Z, we have a = bn + r, 0 ≤ r < n, so we only have the cosets r + nZ for 0 ≤ r < n, and r is uniquely determined by the division theorem, so we have exactly n cosets. If for two integers a, b, their cosets are equal in Zn , we also write a = b(mod n) (in words: “a is equal to b modulo n.”). All these cyclic groups can be realized as subgroups of the multiplicative group U = {z|z ∈ C AND |z| = 1} (the unitary group of complex numbers of norm one), see Figure 19.2. Addition of pitch classes corresponds to addition of angles on the unit circle. Negative pitch classes −x are obtained from x by reflection at the vertical middle axis through 0, 6. In music, cyclic groups of higher order are used in the context of microtonal compositions. For example, quarter-tone music works in Z24 , while sixth-tone music works in Z18 . But in music theory, sometimes the diatonic scale 0, 2, 4, 5, 7, 9, 11 is also modeled in Z7 as if the tonal distances were equal.

Fig. 19.3. The quotient group Z12 , where the coset representatives 0, 1, 2 . . . 11 are shown together with their standard interpretation as pitch classes of pitches in Z.

ˇ “*

Example 49 The case n = 12 is central for the quotient group Z12 . See Figure 19.3, where the coset representatives 0, 1, 2 . . . 11 are shown together with their standard interpretation as pitch classes of pitches in Z modulo the octave of 12 semitones. Theorem 15 If f : G → K is a group homomorphism, then we have the following commutative diagram, which represents f by its kernel and image:

19.2 Subgroups and Quotients

G

f

157

- K 6

f

proj ∼f

→

--

? ? G/H

inj

6 Im(f )

where proj is the canonical surjection, inj is the embedding of Im(f ), and f is the isomorphism of groups that sends the coset gKer(f ) to f (g). The proof is straightforward, and the only point is the definition of f . This function is well defined since gKer(f ) = g  Ker(f ) means g −1 g  ∈ Ker(f ), so f (g) = f (g  ). √

Example 20 The subgroup T ⊂ T I is normal, the quotient group is ∼ T I/T → {±1} = Z∗ . 19.2.1 Classification of Chords of Pitch Classes We have seen that for chords ch ∈ F in(Z), the action of the group of transpositions T or the group T I of transpositions and inversions defines orbits that are musically significant, for example defining a major triad as an element of the T -orbit [{0, 4, 7}]. We can transfer this type of group action to pitch classes if we deem irrelevant multiple appearances of a pitch name or the absolute position in Z. We therefore consider the set of chords of pitch classes that is defined by P CChords = 2Z12 . We include the empty or one-element chords for completeness, although they are not considered chords in common terminology. On P CChords = 2Z12 , we have the action of group T IZ12 of permutations of Z12 that acts as before: T±t (x) = t±x, where t ∈ Z12 . The formal rules on this group are as explained for T I in Example 20. We also have the normal subgroup T Z12 ⊂ T IZ12 of transpositions T+t . The action of T IZ12 on P CChords follows the same scheme as described in in Example 20—a chord is transformed by f ∈ T IZ12 via the transformation of all of its elements by f . See Figure 19.4 for the action of an inversion on a chord. The action of T Z12 and of T IZ12 define two classifications of chords: T Z12 -orbits are called transposition classes, while T IZ12 -orbits are called transposition-inversion classes. These classes are redundant, however, since we also have the action of the two-element group Z2 on P CChords by complementation, i.e., −ch = 12 − ch is compatible with the T IZ12 -action. It is therefore sufficient to classify chords of cardinality ≤ 6, and those with cardinality > 6 can be recovered from the classification of their complements. The complete list of representatives of such classes is folklore in mathematical music theory for half a century. We refer to a complete list in [75, Appendix L.1]. This list is far from random. The representatives of chord classes are chosen according to the following criteria: We take the lexicographic ordering of

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19 Group Actions, Subgroups, Quotients, and Products

Fig. 19.4. The inversion T−5 applied to chord ch = {c, e, g, b}.

chords, when we represent chords as functions ch : Z12 → 2 as explained in Section 5.5. To begin with, the cardinality of chords is taken as the ordering principle. We start with cardinality 0, represented by its complement, the full Z12 . Then follow with chords of one element (single notes), then chords of two elements (unordered intervals), then triads, etc., until chords of six elements. Within a given cardinality, we take from each T IZ12 -orbit the first chord according to the lexicographic ordering of chords as functions ch : Z12 → 2 qua sequences (ch0 , ch1 , . . . ch11 ), where chi = 0 if ch(i) = 0 and chi = 1 if ch(i) = 1. In the list [75, Appendix L.1], the value 0 is represented by a • while the other value is represented by a ◦. In that list, we have also displayed classes of chords under a larger group (class numbers XX and XX.1 being in the same bigger class. We shall come back to this option in Section 21.2.1). In the American tradition, the classification of chords follows not the above lexicographic ordering, but the ordering of “most packed to the left” as described in Section 15, yielding the so-called prime form of chords.

ˇ “*

Exercise 7 Calculate the prime form of chord ch = {c, e, g , b}.

19.3 Products Cartesian products of groups are defined as follows. Definition 56 For two groups G, H, their cartesian product G × H is the settheoretical cartesian product of the underlying sets G, H, and the multiplication goes by factors: (g, h) ∗ (k, l) = (g ∗G k, h ∗H l). Inverses and neutral elements are given factor-wise, too.

19.3 Products

159

We have two embedding group homomorphisms G  G × H : g → (g, eH ), H  G×H : h → (eG , h). We have the two projection group homomorphisms pr1 : G × H → G, pr2 : G × H → H with Ker(pr1 ) = H, Ker(pr2 ) = G, using the identification of G, H with their embeddings.

ˇ “*

Example 50 In music theory, cartesian products of groups are important. We shall see in Chapter 21 that the product Z3 × Z4 , together with its two projections pr3 : Z3 ×Z4 → Z3 and pr4 : Z3 ×Z4 → Z4 is the single most important mathematical structure needed to understand the structure of intervals, and in particular core properties of counterpoint theory. The universal property of cartesian products of sets holds for groups mutatis mutandis: Theorem 16 (Universal Property of the Cartesian Product of Groups) For any three groups G, H, K, the function ∼

Grp(K, G × H) → Grp(K, G) × Grp(K, H) : f → (prG ◦ f, prH ◦ f ) is a bijection.

ˇ “* Example 51 In Chapter 21, we shall construct an isomorphism of groups ∼ f : Z12 → Z3 × Z4 . This will be achieved using the universal property ∼

Grp(Z12 , Z3 × Z4 ) → Grp(Z12 , Z3 ) × Grp(Z12 , Z4 ) of the cartesian product Z3 × Z4 , meaning that we construct two homomorphisms f3 : Z12 → Z3 and f4 : Z12 → Z4 . We present a small list of a number of important chords in Z12 , the class numbers relate to the complete list in [75, Appendix L.1].

160

19 Group Actions, Subgroups, Quotients, and Products Class Representative No.

Name Symbol

Visualization in Z12

Co diminished triad 15

• ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ ◦◦

C+ augmented triad 16

• ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦◦

C major triad 10.1

• ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ ◦◦

Table 19.1. Some important chords I.

19.3 Products Class Representative No.

Name Symbol

Visualization in Z12

Cm minor triad 10.1

• ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦◦

C M 7 or C maj7 major 7th 28.1

• ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ ◦•

C7 dominant 7th 29

• ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ •◦

Table 19.2. Some important chords II.

161

162

19 Group Actions, Subgroups, Quotients, and Products Class Representative No.

Name Symbol

Visualization in Z12

CmM 7 minor major 7th 30

• ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦•

Cm7 minor 7th 22.1

• ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ •◦

C o7 dimin. 7th 37

• ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦◦

Table 19.3. Some important chords III.

20 Permutation Groups

Summary. This chapter deals with the study of the permutation groups Sn . We also give examples of compositional methods using permutation groups. –Σ– Let us now investigate minimal sets of generators of Sn , and recall that ord(Sn ) = n! There are different ways to represent permutations p ∈ Sn . A simple basic representation is to write the full table of ordered pairs (i, pi = p(i)), i = 1, . . . n. This is usually done by a 2 × n-matrix, a rectangular system with two rows and n columns:   1 2 ... n p1 p 2 . . . p n A more economical way is to use cycles. A cycle is an ordered k-tuple C = (c1 , c2 , . . . ck ) of pairwise different numbers 1 ≤ ci ≤ n. This represents a permutation that sends ci to ci+1 for i = 1, . . . k − 1 and then ck to c1 . The number k is the length l(C) of C. The k-element set c1 , c2 , . . . ck of C is denoted by |C|, so card(|C|) = l(C). Cycles C = (c1 , c2 ) of length 2 are called transpositions. We have already encountered transpositions in the calculation of ord(Sn ). Cycles are standard generators of Sn . Theorem 17 The permutation group Sn is generated by the n − 1 transpositions (1, n), (2, n), . . . (k, n), . . . (n − 1, n). Proof 12 The proof goes by induction on n, n = 1, 2 being trivial. Suppose the theorem holds for n. Then if p ∈ Sn+1 fixes 1, by induction it is a product of transposition (2, n + 1), . . . (n, n + 1). If p(1) = k > 1, take q = (1, n + 1)(k, n + 1)p. Then q(1) = 1. Therefore q is a product of transposition (2, n + 1), . . . (n, n + 1), and we have the representation p = (k, n + 1)(1, n + 1)q as desired.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_20

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20 Permutation Groups

ˇ “* Example 52 We consider a well-known melody M , shown as “Original” on the top left side of Figure 20.1. The notes of this melody are numbered with 1, 2, . . . 10 according to increasing onsets. The retrograde π∗M of this sequences corresponds to the permutation   1 2 3 4 5 6 7 8 9 10 π= 10 9 8 7 6 5 4 3 2 1 By Theorem 17, π can be written as a product of transpositions (i, 10), i > 1. In our case we have π = (1, 10)(2, 10)(9, 10)(2, 10)(3, 10)(8, 10)(3, 10) · (4, 10)(7, 10)(4, 10)(5, 10)(6, 10)(5, 10). This mean that the retrograde π ∗ M can be obtained by successive transpositions: M, (5, 10) ∗ M, (6, 10)(5, 10) ∗ M, . . . π ∗ M . These 13 intermediate compositions are played one after another, starting from M (“Original”) and ending with the “Retrograde” on the top right of Figure 20.1. This enables the listener to understand the difficult retrograde transformation as a succession where only two notes are exchanged and always with respect to the last note, so the beginnings of the successive versions are kept as stable as possible. In our score in Figure 20.1, we have doubled the durations of the Original and the Retrograde to make the start and end easier to be recognized. And here is the theorem about the cycle representation of a permutation: C 1 ◦ C 2 ◦ . . . Cr Theorem 18 Every permutation p ∈ Sn is the product p =  of cycles Ci , such that |Ci | ∩ |Cj | = ∅ for i = j, and n = i |Ci |. The set {|Ci ||i = 1, . . . k} is uniquely determined by p. Proof 13 The proof is straightforward, we just present the basic idea. By definition of permutations, the group p acts on the set {1, 2, . . . n}. This action generates a partition {1, 2, . . . n} = i p · ci by orbits p · ci . The powers of p act on the orbit elements ci , and the sequence of these actions pk (ci ) defines the orbit of ci in the form of a cycle Ci . √

Exercise 25 Write the permutation  π=

1 2 3 4 5 6 7 8 9 10 11 12 12 11 10 9 8 7 6 5 4 3 2 1



as a product of disjoint cycles. The symmetric group Sn has the alternating subgroup An that consists of all permutations that can be written as a product of an even number of transpositions (i, j). It is in bijection with the set Bn of all permutations that

20 Permutation Groups

165

Fig. 20.1. The retrograde (Retrograde) of a simple melody (Original) is achieved by a series of transpositions of two notes. New exchanged notes are indicated by a *. The sound example is permutation.

can be written as a product of an odd number of transpositions, a bijection ∼ being defined by a multiplication An → Bn : p → (12)p. It can be shown that An ∩ Bn = ∅. Since transpositions generate Sn , we have Sn = An  Bn , i.e., An covers half of the n! permutations, ord(An ) = n!/2. This implies that An

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20 Permutation Groups

is a normal subgroup of Sn since there can only be one coset of An , namely Bn , i.e., left and right cosets coincide (in fact with Bn ). We have the quotient ∼ group Sn /An → {±1} = Z∗ . The number ±1 associated with a permutation p under this quotient group is called the sign sig(p) of p. Even/odd permutations are those with a positive/negative sign.

20.1 Two Composition Methods Using Permutations Using permutations to compose music is not an new concept. Renaissance Man Athanasius Kircher designed a composition machine based on a matrix, as described in Section 2.6, Figure 2.20. More ideas came to fruition in the 18th century, the most famous being Wolfgang Amadeus Mozart’s Musikalisches Würfelspiel, or musical dice game. Exploration of compositional methods continues today, including efforts by one of the authors of this book, Maria Mannone. Using Rubik’s cubeT M , Mannone has designed a composition device called Cubharmonic. We describe the dice game and cube device in the following sections.

Fig. 20.2. Mozart’s first dice number table. Rows correspond to dice numbers, columns to measure numbers. The coefficients within the table are measure numbers Mozart defined.

20.1.1 Mozart’s Musical Dice Game Mozart’s musical dice game is entitled “Walzer und Schleifer mit zwei Würfeln zu componieren ohne Musikalisch zu seyn, noch von der Composition etwas zu verstehen.” (To compose a waltz with two dices without being musical and

20.1 Two Composition Methods Using Permutations

167

knowing anything about composition.) The idea is that the user can define a sequence of eight measures I, II, III, . . . V III of music by throwing two dice and then using a table (Figure 20.2) to look up a corresponding musical measure. The table has eleven rows that correspond to the result of adding the two dice rolls (2 = 1 + 1 on top to 12 = 6 + 6 at the bottom). For measure I, the user finds the dice result under column I. For example, we get 119 if the dice roll yields 9, corresponding to measure 119 on the list of 176 possible musical measures Mozart provided. The first thirty measures (all in waltz time signature 3/8) are shown in Figure 20.3. This system allows creation of 118 = 214,358,881 possible compositions of eight measures each. 20.1.2 Mannone’s Cubharmonic A well-known game that uses the concept of group is the Rubik’s CubeT M . It is the world’s best-sold toy. Invented by Hungarian professor and architect Ernö Rubik, the cube was first shown to his students as a teaching tool. Years later, under the initial name of Magic Cube (perhaps in analogy with magic square of letters and numbers), the cube was produced for sale [97]. The classic version is a 3 × 3 × 3 cube, where each face has a different color. Sides can be moved. Among all possible combinations, there is only one solution that completes the six faces with correct colors. Solving techniques are studied by mathematicians. There are other versions of the cube, for example 2×2×2 and 4×4×4 (Pocket and Rubik’s RevengeT M ). The latter was invented by Péter Sebestény [70]. One of the authors (Maria Mannone) used a 4 × 4 × 4 cube to create a harmonic game, the Cubharmonic, shown in Figure 20.4. The idea is to experiment with creating new harmonic sequences. On each face of the cube we can write a harmonic sequence of a four-part harmony. For example, the cadence I − IV − V − I in C-major can be written as c g e c

c a f f

b g d g

c g e c

We can write these letters on one face of the cube. If we write a different harmonic sequence on each face, by rotating and twisting sides we get a huge number of different harmonic combinations. The number of combinations is greater than for the classic 4 × 4 × 4 cube. The reason is that little squares of the same color, having been marked with different names of notes, are not undistinguishable. The scope of the original game was to recover the initial configuration of one color on each face, which means from chaos to cosmos. Here the main interest is to mix sequences, from cosmos to chaos. We can exchange chords, and we can also modify horizontally voices in the same harmonic sequence.

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20 Permutation Groups

Fig. 20.3. The first thirty measures, in waltz time, out of a total of 176 on Mozart’s list.

The cube of Figure 20.4 presents a choice of six different harmonic sequences, shown in Figure 20.5. Three different results, obtained via simple permutations, are given in Figure 20.6.

20.1 Two Composition Methods Using Permutations

169

Fig. 20.4. Mannone’s Cubharmonic.


  

 


      

 


  




 



    

 

 





  

    

 

 

 







  




   





 

 





  

  

   



Fig. 20.5. A choice of six different cadential sequences (one for each side of the cube) as a starting position. The sound example is cubharmonic_1.


  




 

  

  

      

  

  

  

     

  

Fig. 20.6. Three different results obtained via simple permutations, i.e., scrambling the cube. The sound example is examples_cubharmonic.

21 The Third Torus and Counterpoint

Summary. This chapter deals with the mathematics of the third torus group Z3 × Z4 and its symmetries and then applies these structures to music theory, in particular to counterpoint. –Σ–

21.1 The Third Torus The third torus is a group that has a rich structure despite its simple mathematical shape. It is also a central object of mathematical music theory, which will be discussed in Section 21.2. The third torus is the finite abelian group T3×4 = Z3 × Z4 . Let us look at the finite cyclic groups Zn that define the torus’ background structure. If we consider elements z ∈ Zn , they are cosets z = x + nZ, and we denote these by xn = x + nZ. The group structure on Zn yields (x + y)n = xn + yn . We also write x ≡ y(mod n) and say that x is congruent to y modulo n for xn = yn . If n|m, we have a canonical surjective group homomorphism Zm → Zn : xm → xn . It is well defined since mZ ⊂ nZ, therefore xm = ym implies xn = yn . In particular, we have canonical surjections pr3 : Z12 → Z3 and pr4 : Z12 → Z4 . The kernels are the four-element subgroup Ker(pr3 ) = 312  and the three∼ ∼ element subgroup Ker(pr4 ) = 412 . I.e., Z12 /3 → Z3 and Z12 /4 → Z4 . √

Exercise 26 Verify these facts.

Using the universal property of cartesian products of groups, this data can be combined to define a group homomorphism (pr3 , pr4 ) : Z12 → Z3 × Z4 . This homomorphism is inserted in our commutative diagram

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_21

171

172

21 The Third Torus and Counterpoint (pr3 ,pr4 )

Z12

(p r

3 ,p

proj

- Z3 × Z4 6

r4 )

inj

? ? ∼ (pr3 ,pr4 ) 6 Z12 /Ker((pr3 , pr4 ))→ Im((pr3 , pr4 ))

The kernel Ker((pr3 , pr4 )) is the intersection Ker(pr3 ) ∩ Ker(pr4 ) of the two kernels Ker(pr3 ), Ker(pr4 ), but this is trivial. Therefore, in our diagram, the left projection is an isomorphism, and so is the homomorphism onto the image Im((pr3 , pr4 )). This image of Z12 has 12 elements, and Z3 × Z4 also has 12 elements, so the image is all of Z3 × Z4 , which means that (pr3 , pr4 ) is an isomorphism of groups. It is advantageous for music theory to compose this

Fig. 21.1. The third torus. We see four vertical circles—copies of Z3 —attached to the four pitch-class points 0, 3, 6, 9.

isomorphism with the automorphism ∼

Id × (−1) : Z3 × Z4 → Z3 × Z4 : (x, y) → (x, −y). Then, the composed isomorphism ∼

t : Id × (−1) ◦ (pr3 , pr4 ) : Z12 → T3×4 : x12 → (x3 , −x4 ) has the property that t(412 ) = (13 , 04 ) t(312 ) = (03 , 14 ) t(712 ) = (13 , 14 ) t(512 ) = (−13 , −14 ) t(112 ) = (−13 , 14 ) t(1112 ) = (13 , −14 )

21.1 The Third Torus

173

The inverse t−1 is given by t−1 (x3 , y4 ) = 4x3 + 3y4 which means that the two components on T3×4 represent the multiples of 4 and 3, respectively, in Z12 . As these quantities represent major and minor thirds in the pitch-class set Z12 , the name third torus is explained. Figure 21.1 visualizes the third torus. 21.1.1 Geometry on T3×4 On the third torus T3×4 we can define a metrical distance function by d(z, w) being the minimal number of minor or major third steps to reach w from z. The third steps are addition or subtraction of (13 , 04 ) or (03 , 14 ). This metrical geometry on T3×4 is important because it is invariant under all symmetries of T3×4 . Let us now explain what symmetries of T3×4 are. On Z we had considered action of the group T I whose elements are the function t : z → t ± z. This means that we considered the invertible elements ±1 of T±1 ∗ Z and used the multiplication with such elements, together with a shifting by t (z) = t + (±1)z. t: T±1 We want to carry over this idea to Z12 . The problem is that so far we have no multiplication of elements of Z12 . This is what we shall introduce now. Later, in Chapter 24, we will see that the present procedure is a very simple special case of a general method to introduce multiplication in quotient groups. Multiplication on Zn is defined by xn · yn = (x · y)n . If this works, the arithmetic on Z can be carried over “literally” to Zn as addition and multiplication are both defined by going back to Z, doing it there, and then taking cosets; also refer to Musical Example 19. Take two different representatives of our cosets, x = x + nz, y  = y + nw instead of x, y. Then x · y  = (x + nz) · (y + nw) = xy + n(xw + zy + nwz) = xy(mod n), and we are done. For Z12 this multiplicative structure defines four elements that are invertible, namely Z∗12 = {112 , 512 , 712 , 1112 }. We have 1212 = 5212 = 7212 = 11212 = 112 . All other elements are not invertible, for example 312 412 = 012 . With this multiplicative structure we now generalize the above construction of T I for Z. Rewrite T I = T Z∗ . We now define T Z∗12 as the group of permutations of Z12 defined by the elements T t .s, t ∈ Z12 , s ∈ Z∗12 defined by T t .s(z) = t + sz for z ∈ Z12 . This is a group under usual composition of functions. In fact T t .s ◦ T u .v = T t+su .sv, and (T t .s)−1 = T −st .s, while T 0 .112 is neutral. Symmetries on Z12 are by definition the permutations in the group T Z∗12 . The next step consists of proving that all symmetries are in fact geometrically reasonable if we transport them to the third torus using the isomorphism t. Here is the geometric interpretation, which we specify for generators of the symmetry group T Z∗12 . We usually omit the factor 1 if the symmetry is T t .1, see Figure 21.2.

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21 The Third Torus and Counterpoint

Fig. 21.2. The symmetries of the torus all conserve the third distances.

The transposition T 3 translates to a rotation of 90o around a vertical middle axis of the torus. The transposition T 4 translates to a 120o tilting movement of the torus. An inversion T 0 .11 becomes a 180o rotation of the torus around the horizontal axis through 0 and 6. The symmetry T 0 .5 becomes a reflection of the torus at the horizontal plane through the middle of the torus. The symmetry T 0 .7 becomes a reflection of the torus at the vertical plane through 0 and 6. This means that all symmetries are combinations of such classical symmetries (such as reflection and rotation) of the geometric object the torus is defining. This means: Theorem 19 The group of symmetries T Z∗12 defines on the torus T3×4 a group of symmetries that conserve metrical distances, i.e., all symmetries are isometries on the third torus. This is a remarkable result since on Z12 , multiplication with 5 or 7 does not conserve distances in the circle representation of Z12 .

21.2 Music Theory Let us apply the above results to some questions in music theory. 21.2.1 Chord Classification We have seen that chord classes under the action of T Z∗ are standard in music theory. But we may as well look at classification of chords by orbits of the larger

21.2 Music Theory

175

group T Z∗12 . This is the classification that has been achieved in [75, Appendix L.1]. There, two T Z∗ classes XX and XX.1 are representatives of the same T Z∗12 class. 21.2.2 Key Signatures The first application relates to a simple observation. If we look at the C-major scale C = {c, d, e, f, g, a, b} ⊂ Z12 , it consists of white keys only. If we move from the C-major scale to the F -major scale T 5 C = {c, d, e, f, g, a, b} = F , we get one black key b and lose one white key b. This is why the key signature of F -major has one . If we continue in the same way, moving to T 5 F = B = {c, d, e, f, g, a, b}, we get a scale with two black keys b, e, losing the corresponding white keys b, e. The key signature now has two . This phenomenon holds on: Each time we move a scale X to T 5 X, we add one  to a white note. This is true until we have reached G-major with the scale {c, d, e, f, g, a, b}, six  signs in the key signature. Why does the system break down here? We could argue that you always move one fourth up, and because this movement is always the same, the change of key signature must also always be the same, adding each time one . The argument is wrong, however, because the black keys do not move along with the fourth transposition, meaning the situation is not universal.

ˇ “*

Exercise 8 Try to find the number of key alterations ( or ) for the transpositions T 5 X starting from a scale X0 of seven white keys if the piano has the following white keys: {c, d, e, f, g, a, b}. You will see that the above rule does not hold. The explanation comes from a different representation of the diatonic major scale C. We use the symmetry T 0 .5 of Z12 and see that the scale C is mapped to 5C = {7, 8, 9, 10, 11, 0, 1}, see Figure 21.3. This image is an uninterrupted sequence of fourths. In this representation, we also see the distinguished roles of f and b. They mark the boundaries of the fourth sequence of the scale. They are known in music theory as the leading notes. We shall come back to this observation in Chapter 23. What is important here is that the transposition by 5, one fourth, moves the sequence of fifths to a new sequence of fifths by one unit clockwise. One black key (b) is added while one white key (b) is lost. This transposition by one fourth now clearly always added one more black key, namely the succession e, a, d, g, c = b to the transposed scales and simultaneously removes the keys e, a, d, g, c. Therefore the 5-symmetry explains this key signature phenomenon for the major scales. 21.2.3 Counterpoint In counterpoint, one starts with the construction of a composition from two voices: cantus firmus (CF) and discantus (D). The rules that determine admis-

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21 The Third Torus and Counterpoint

Fig. 21.3. The representation of the C scale in Z12 after multiplication by 5. The key T 5 C = F is the rotation of 5C by one unit in clockwise direction.

sible constructions began developing in the 9th century in Europe and were quite stabilized in the 16th century, typically as used by the great composer Giovanni Pietro Aloisio Sante da Palestrina. These rules (or what they thought were such rules) were written down in the small book Gradus ad parnassum by Johann Joseph Fux in 1725 [42]. Despite the simplicity of Fux’s rules, there are a number of unsolved problems with this theory. It commences with the first species counterpoint. It is the basis of all subsequent situations, namely second species, third species, fourth species, and fifth species (florid counterpoint). We focus on the first species here. It is defined by the CF melody and defines rules to add a second voice (D), where for each note of CF one defines one note of D, of same duration and onset. See Figure 21.4 for an example. The example shows the two voices and a central feature: CF and D notes of same onset define consonant intervals. These intervals (differences of pitch classes) must be of six types (if we work in Z12 ), prime (0=12), minor third (3), major third (4), fifth (7), minor sixth (8), and major sixth (9). The other six interval numbers—minor second (1), major second (2), fourth (5), tritone (6), minor seventh (10), and major seventh (11)—are dissonant. In our example, we see that the consonances 0, 4, 4, 9, 9, 0 are played. This definition of consonances is the first big problem in counterpoint: The selection of these intervals is not justified by acoustical arguments. The critical interval is the dissonant fourth. In fact, recall that in the Pythagorean tuning tradition, the fifth frequency ratio 3/2 was consonant, but also the fourth ratio 4/3. Because the common basis of music theory is the just tuning system, this constitutes a serious problem, which has been recognized by leading scholars such as Carl Dahlhaus (see [79, Chapitre 13] for a discussion). He argues that

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Fig. 21.4. Different motions in first species counterpoint.

the fourth being dissonant must be justified by a not-yet-discovered rule of polyphonic texture. The big question that opens up in view of these problems is how the distribution K = {0, 3, 4, 7, 8, 9}, D = {1, 2, 5, 6, 10, 11} of intervals in the consonant and dissonant halves can be constructed without a invalid reference to the acoustical background. Our solution results from the following observation. There is a unique symmetry AC = T 2 .5 of Z12 that transforms K into D. More precisely, AC(0) = 2, AC(3) = 5, AC(4) = 10, AC(7) = 1, AC(8) = 6, AC(9) = 11 and vice versa, AC(D) = K since AC 2 = IdZ12 . This is the autocomplementarity function. There are five other interval dichotomies (X, Y ) that have unique autocomplementarity functions, we call them strong dichotomies. Here is the total list, the labels are from the classification of chords in [75, Appendix L.1]: #64 : ({2, 4, 5, 7, 9, 11}, {0, 1, 3, 6, 8, 10}), AC = T 5 .11 #68 : ({0, 1, 2, 3, 5, 8}, {4, 6, 7, 9, 10, 11}), AC = T 6 .5 #71 : ({0, 1, 2, 3, 6, 7}, {4, 5, 8, 9, 10, 11}), AC = T 11 .11 #75 : ({0, 1, 2, 4, 5, 8}, {3, 6, 7, 9, 10, 11}), AC = T 11 .11 #78 : ({0, 1, 2, 4, 6, 10}, {3, 5, 7, 8, 9, 11}), AC = T 9 .11 #82 : (K, D), AC = T 2 .5 Of course, for any symmetry f ∈ T Z∗12 , the dichotomies f (X), f (Y ) are also of this type if (X, Y ) are so. But there are no other such dichotomies. It is remarkable that the dichotomy #64 has as first half {2, 4, 5, 7, 9, 11} the set of proper intervals of a major scale when counted from its tonic. It is therefore called the major dichotomy. The question now is to exhibit a property that distinguishes the consonance-dissonance dichotomy (K, D) (#82) from the others. The answer is found when we consider these dichotomies within the third torus. We want to learn how the interval numbers are distributed on the torus. To this end we define two numbers, diameter and span.

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Definition 57 For a strong dichotomy (X, Y ) with autocomplementarity p, we call its diameter the number 1  δ(X) = d(u, v), 2 u,v∈X

where d(u, v) is the distance between u and v as defined at the beginning of Section 21.1.1. The span of (X, Y ) is defined by  d(u, p(u)). σ(X) = u∈X

It is obvious by the invariance of distance on the torus that diameter and span are invariant within a class of dichotomies that is generated by the symmetries f ∈ T Z∗12 . Figure 21.5 shows these numbers (a), and the configuration for the Fux dichotomy (K, D) (b).

σ(X) 82

16

68

10

24

a)

75

71, 78

64

25

28

29

4

δ(X)

K

8

7

0

11

3 b)

1 10 6

2

5 9

D

Fig. 21.5. The diameters and spans of the six strong dichotomies are shown (a), and the geometric configuration for Fux dichotomy (K, D) is shown in (b).

From this result we learn that the Fux dichotomy (K, D)—#82 in Figure 21.5 a)—has the smallest diameter and the largest span. In other words,

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its consonant elements are separated in an optimal way from its dissonant elements. This distinguishes this dichotomy from the other five possibilities. It is interesting that the major dichotomy #64 plays a polar role, its members are “mixed” in an optimal way. The role of strong dichotomies is far from investigated in music theory. Let us just give one more remarkable example. The Russian composers Alexander Scriabin is known for his quite intriguing approach to harmony (among other far out approaches). His most famous discovery is the so-called mystic chord. Figure 21.6 shows the chord in its common representation (left) and in its representation on Z12 (right). The chord has six notes, and we recognize immediately that it is isomorphic to the dichotomy of class #78. The chord also is covered by the four prominent triads: major, minor, augmented, and diminished (middle of Figure 21.6).

Fig. 21.6. Scriabin’s mystic chord (left) is one half of the strong dichotomy #78 (right). Four prominent triads—major, minor, augmented, and diminished—cover the chord.

ˇ “* Exercise 9 Calculate the nerve of the covering of the mystic chord by major, minor, augmented, and diminished triads. Is it a tetrahedron? This arsenal of six dichotomies can be used to create six “counterpoint worlds.” This means that the classical Fuxian counterpoint is but one of six possibilities to define a counterpoint theory. More precisely, it is possible to define a model of counterpoint for each strong dichotomy, where the classical rules of counterpoint can be described based upon the geometry of these dichotomies. In particular, the important rule of forbidden parallels of fifths, i.e. the succession of two intervals of size 7, in Fuxian counterpoint can be deduced from the geometry of #82. This theory does not refer to any psychological arguments, which are standard for the explanation of forbidden parallels of fifths. See [8] for a thorough exposition of this mathematical theory of counterpoint, including composition software for such “exotic” counterpoints. This theory is not only developed for all six counterpoint worlds on Z12 but it works for microtonal pitch classes in Z2n for all n > 2, too. This is a strong argument for mathematical methods in music theory. They enable the creation of new theories of composition that extend given theories for future musical creativity.

22 Coltrane’s Giant Steps

Summary. This chapter deals with an analysis of John Coltrane’s famous composition Giant Steps from 1959, released in 1960 on the synonymous LP. –Σ– Figure 22.1 shows the original LP cover of Giant Steps. This composition is

Fig. 22.1. John Coltrane’s composition Giant Steps from 1959.

known for its fast harmonic changes. Several pianists, such as Tommy Flanagan and Cedar Walton, experienced difficulties in performing it. © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_22

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There is also Coltrane’s pianist McCoy Tyner’s live solo performance, which, despite the technically perfect rendition, sounds somehow too controlled. The composition is presented in a score form that is standard in jazz, called the lead sheet. Such a score contains only the essential melodic movements, together with a sequence of chord symbols, written in more or less standard notation. Figure 22.2 shows the lead sheet of this composition. One immediately recognizes the fast chord changes (this is the official term for the harmonic sequence of a lead sheet).

Fig. 22.2. The lead sheet of John Coltrane’s composition Giant Steps.

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In the first two measures, every melodic note has another chord, and not just slightly different ones: Coltrane jumps around in the entire harmonic spectrum. We want to not only present an analysis, but also sketch how the analytical part can be used to generate a new composition that incorporates the analytical insights. This new composition by one of the authors (Mazzola) is entitled Giant’s Steps and has been performed live and also recorded for a music DVD [44] in fall 2014 at the Airegin Club in Yokohama, Japan. This procedure of analyzing and then recreating a composition has been described by composer and theorist Pierre Boulez as analyse créatrice (creative analysis) in [22], see also [20]. It is also discussed in [79, Chapter 7] and [84, Chapter 25].

22.1 The Analysis Jazz harmony is usually based on tetrads, seventh chords, not triads. As you can see from the lead sheet (Figure 22.2), all chords are given the “7” for this reason. There are three types of seventh chords here: mostly major seventh chords X maj7 (XM A 7 in the lead sheet), some seventh chords X 7 (X7 in the lead sheet), and some minor seventh chords Xm7 . See our tables at the end of Chapter 19 for these chord names and symbols. In our analysis, we only consider X maj7 chords and replace the rare cases of X 7 chords by the same chord with major seventh. The minor chords are taken as such. We work in pitch class-space Z12 here. In this space, seventh chords may have inversion symmetries. Both X maj7 and Xm7 (XM I 7 in the lead sheet) are symmetric around their center. For example, Am7 is symmetric around d, by the symmetry of TT−4 , and Gmaj7 is symmetric around the middle of c, c by the symmetry T−1 . We can see this from the tables at the end of Chapter 19. This implies that any transposition or inversion of these two types of chords produces chords of the same type. The first astonishing fact about Coltrane’s chord changes in Giant Steps is that he defines large portions of chords that relate to one another by one single symmetry, the unique symmetry Id = T−4 of the C-major scale. One could therefore say that Giant Steps is a big architecture around C-major. Here are the details. The first symmetric group of chords comprises nine consecutive chords, starting from the second chord in measure one, as shown in Figure 22.3. More precisely, the chords are inverted by Id and also retrograded. The symmetry is a harmonic retrograde inversion. The symmetry is not only applied only once: The left half and the right half are both also symmetric with the same symmetry. Moreover, the middle chord Am7 is symmetric to itself by Id , as observed above. So not only is the architecture of chords highly symmetric, but the symmetry is realized within the central minor chord as well. The second symmetric group comprises the next eleven chords, as shown in Figure 22.4. The symmetry is the same, a harmonic retrograde inversion

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Fig. 22.3. These nine consecutive chords are arranged in a perfect symmetry around d.

Fig. 22.4. These eleven consecutive chords are arranged in a perfect symmetry around d.

for the inversion Id . Again, the symmetry is expressed not only by the inner symmetry of the central minor chord Am7 , it is also visible for the first and last two chords of this group.

Fig. 22.5. The two melodies, the initial and the second on measure five, are arpeggios of chords in those positions, and the notes following these melodies are Id -symmetric to notes within these melodies.

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Moreover, the lead sheet’s first and last chord, B maj7 and F maj7 (because we’re treating F 7 as F maj7 ), are symmetric under Id . Coltrane’s deeply harmonic style of thought1 is visible in his melodic approach, see Figure 22.5. The initial melody is an arpeggio of Gmaj7 , the first chord in the second measure. The subsequent b is the Id symmetric note to the initial f . The second melodic part of measure five is an arpeggio of Emaj7 , the first chord of measure six. Again, the subsequent note f is Id -symmetric to the second melodic note b. And these two symmetric note groups are also symmetric with each other (b, f ) → (f , b)!

Fig. 22.6. The totality of chords. We have connected any two chords that are Id symmetric with each other. The two groups of nine and eleven chords, respectively, are shown. We recognize that there are nine chord names, all except the three names C, E, G.

The totality of chords is shown in Figure 22.6. We have connected any two chords that are Id -symmetric with each other, and the two groups of nine and eleven chords, respectively, are shown. We recognize that there are nine chord names, all except C, E, G . This is exactly the third Messiaen scale, complement of the augmented triad C + . It also has the inner symmetry Id . Some published analyses of Giant Steps have, in fact, stressed the role of the augmented triad in this composition. Let us see how the sequences of our nineand eleven-chord groups unfold on this Messiaen scale. Figure 22.7 shows the chord paths for the nine-element group (left) and the eleven-element group 1

See also our analysis of Coltrane’s A Love Supreme and Ascension in [82, Chapters 2.2 and 5.2].

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(right). Whereas the left path is not very regular, the right one is a perfectly

Fig. 22.7. The chord paths for the nine-element group (left) and the eleven-element group (right).

symmetric trajectory. It also fulfills the frequent changes from II to V to I, the so-called 251 sequence, e.g., in the sequence a → d → g in G-major. That Coltrane chooses such symmetric paths is not typical of him. For example, in the first movement of A Love Supreme, the sequence of the composition’s main motive f − a − f − b does not follow a recognizable harmonic scheme, see [82, Chapter 2.2]. The total chord change system is reproduced in Figure 22.8. To the left on the vertical axis we see the chords in their relations under the Id symmetry and fifth transposition T 7 . Rectangular regions show the chord groups that are related by these relations. We recognize a highly organized system. It has two parts, the first one to the left, which “rotates” in the circle of the four positions Gmaj7 , Bmaj7 , Emaj7 , Dmaj7 , centered by the Am7 chord. The second half then moves down to F maj7 and its Id -symmetric and fifth-related partner B maj7 . In the second half, after this movement, the original group around Am7 is recovered, and followed again by the downward movement to F maj7 and B maj7 .

22.2 The Composition Given these analytical results, we have recreated the composition in the vein of Boulez’s creative analysis. The overall insight was that there are two levels in Coltrane’s harmonic construction: the one to the left in Figure 22.8, rotating in that circle of four positions, and the one to the right, “descending” to the symmetric pair F maj7 and B maj7 . In Coltrane’s construction, these two levels are taken in alterations: circle goes to the symmetric pair goes to circle goes to

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Fig. 22.8. The total chord change system parametrized by Id symmetry and fifth transposition T 7 chord relations.

the symmetric pair. We have therefore added a dynamic down and up trajectory as shown in Figure 22.9. The given harmonic units, such as the circle, have been reshaped by motivic structures that are arpeggios of harmonies given by Coltrane’s architecture. Figure 22.10 shows such constructions for the circle part. The circle is shown to the bottom right together with four arpeggios and motives that represent the harmonic positions as well as the symmetry Id . The harmonic positions are made evident by circle of fifths representations of G-major and D-major tonalities. This enables us to represent Coltrane’s short spots in a more explicit melodic rendition. The overall structure of our Giant’s Steps composition alternates between a relatively fast upper part around the circle of four positions shown in Figure 22.10, on the one hand, and around the symmetric pair F maj7 and B maj7 on the other. This lower part is associated with a more bluesy mood and interestingly is easily associated with Coltrane’s A Love Supreme main motive. Our score of Giant’s Steps as it is used in performances is shown in Figure 22.11. It shows the two-level scheme in terms of a sequence of chords and motivic cells.

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Fig. 22.9. The dynamics between the upper “speed” level and the lower “blues” level. The trajectory alternates between them and shows a circular up-down path.

Fig. 22.10. Motivic structures that are arpeggios of harmonic structures given by Coltrane’s architecture.

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Fig. 22.11. The two-level scheme in terms of a sequence of chords and motivic cells.

23 Modulation Theory

Summary. Modulation is a central theme in tonal musical composition. It means the transition from one tonality to another. Of course such a situation is not omnipresent since either the very concept of a tonality is not given (in the composer’s or in the theorist’s mind), or when present, there is no real theory of how to modulate. This chapter dös not intend to present the one and only modulation theory, but is written to prove that precise mathematical conceptualization and the application of mathematical methods can lead to explicit and efficient models of modulation. –Σ– Our model was originally designed to understand Arnold Schönberg’s tonal modulation theory written in 1911 [103]. The theory was first published in 1985 in [72]. The result of our model is that it provides us with all of Schönberg’s results wherever he deals with specific modulations. For example, he deals with the modulation from C-major to F -major, but he does not discuss a (direct) modulation from C-major to D-major. However, our model deals with all transitions between two major tonalities. This model also allows us to understand some complex modulations in that most difficult sonata of tonal music, Beethoven’s “Hammerklavier” Sonata op. 106. Most of Beethoven’s modulations could not be understood before our model was applied. For a thorough reference to this mathematical theory of modulation, including the discussion modulations in the classical literature, we refer to [75, Chapters 27, 28]. We should add that our harmonic (and motivic) analysis of Beethoven’s op. 106 was also applied to the construction of a new sonata, the Sonata op. 3 L’essence du bleu already discussed in Section 16.1. This approach is understood to be a kind of “experimental music theory,” akin to experimental physics, where the presence of a law in the physical reality is tested by an adequate aggregate of physical objects. In music, the experiment would be the construction of a musical object, a composition, to test the efficiency or aesthetic validity of a theoretically designed method. © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_23

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Unfortunately, the Schenkerian heritage in American music theory has created a poor approach to modulation, mainly, mainly because this was not Heinrich Schenker’s dreams. The consequence thereof is that this approach is incapable of describing and explaining modulatory processes, let alone those complex constructions by great composers such as Mozart and Beethoven. It is not sufficient to reduce a modulation to the simple statement “I hear a dominant;” such a statement is as poor as stating that one understands nuclear fusion because one feels that “it’s very hot out there.” In this chapter, we first develop a precise conceptualization regarding modulations. We shall then present a mathematically precise model of such a conceptualization. We shall apply this model to calculate components of modulations that were given by Schönberg. We also will present the nerves of modulatory structures, together with their role in voice leading. We end this chapter with a short set of examples of modulations in Beethoven’s op. 106.

23.1 The Concept of a Tonal Modulation Schönberg describes a modulation from an old to a new tonality as a tripartite process where the first part consists in a “neutralization” of the old tonality, the second part brings what Schönberg calls the “fundamental” degrees of the new tonality, and the third part presents a cadence in the new tonality, see Figure 23.1.

Fig. 23.1. Schönberg’s tripartite modulation, comprising the old tonality neutralization step, then the fundamental degrees in the new tonality, and ending up with a cadence in the new tonality.

This scheme is logical: We check out from the old hotel, and then we move to the new one, and then we check in there. In fact, Schönberg stresses that a modulation is more or less an involved trajectory. Here is his original text (in an English translation): There is, for example, a very popular harmony treatise in which modulations are almost exclusively made using the dominant seventh or diminished seventh chord. And the author merely demonstrates that after each major or minor triad, any of those two chords can be played,

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and thereby go to any tonality. If I wanted that, I could have finished even earlier. In fact, I am able to show (using “gauged” examples from literature) that you may use any triad after any other triad. So if that reaches every tonality and thereby modulation has been realized, the procedure would be even simpler. But a traveler, recounting his journey, would not describe it as the crow flies. The shortest path is the worst. The bird’s perspective is the perspective of a bird’s brain. If everything is blurred, everything is possible. Differences disappear. And it is then irrelevant whether I have made a modulation with a dominant or diminished seventh chord. The essential of a modulation is not the target, but the trajectory. In Schönberg’s theory [103], we need to specify the following concepts: 1. 2. 3. 4. 5.

What is a tonality? What is a degree of a tonality? What is a cadence? Which is the modulation mechanism? How do these structures determine the fundamental degrees of a modulation?

These questions are never dealt with explicitly in common music theory, but they are very important. For example, if the first question is not answered, then a modulation theory should be able to deal with modulations from standard major tonalities to exotic pentatonic or Turkish microtonal tonalities. We don’t know of a single serious modulation theory that would deal with such cases. With these caveats in mind, we shall now define these concepts for our specific present context. This endeavor is a useful exercise in reliable conceptualization. Here are the answers to the above questions. 1. Tonality We take the set Dia of twelve diatonic scales T i C, i = 0, 1, 2, . . . 11 with C = {0, 2, 4, 5, 7, 9, 11}. For each scale X we take the covering of X by the standard triadic degrees IX , IIX , . . . V IIX , and denote this data by X (3) . The set Dia(3) = {X (3) |X = T i C, i = 0, 1, 2, . . . 11} is the collection of tonalities. We present them by their nerves N (X (3) ), which are their harmonic bands as discussed in Musical Example 39 in Chapter 16. Observe that we tacitly identify these tonalities with the tonalities T i C (3) whose tonic is i, although no tonic is selected. In music theory, this choice of the tonic i is one of seven possible choices. This is called the ionian mode. There is one mode for every choice of a pitch class in T i C. Here are the names of the modes, which we explain for the C scale:

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tonic c : ionian tonic d : dorian tonic e : phrygian tonic f : lydian tonic g : mixolydian tonic a : aeolian tonic b : locrian This means that in our modulation theory, we may also change the mode without changing the theoretical model, since the tonic is just an additional specification without deeper consequences. Figure 23.2 shows the system Dia(3) of all tonalities, which we also call triadic interpretations of the major scales.

Fig. 23.2. The twelve tonalities of our modulation model.

We should add that this modulation theory relates to tonalities in 12tempered tuning. There is also a theory for the just tuning system that works as well, but the mathematical content is somewhat different, see [75, Section 27.1.6]. 2. Degree A degree in tonality X (3) is one of the seven triadic degrees IX , IIX . . . V IIX .

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3. Cadence Let T ria be the subset of 2Z12 consisting of all major, minor, or diminished triads. There are five minimal subsets of degrees of a tonality X (3) such that only this tonality contains these degrees. The types are as follows: type 1 : {IIX , IIIX } type 2 : {IIX , VX } type 3 : {IIIX , IVX } type 4 : {IVX , VX } type 5 : {V IIX }

We recognize the widely used cadence of type 4 in classical music: It is the cadence that is realized in the cadential sequence I − IV − V − I. The degree I is only used to define the tonic while degrees IV, V are sufficient to determine the scale uniquely. Type 2 is frequently used in jazz—recall the typical 251 movement, corresponding to II, V, I. There is a simple explanation for these five cadences. When we represent a major scale as a sequence of pitch classes in fourth or fifth distance, the cadences are those collections of pitch classes that contain the leading tones and one tone in the interior of the scale. For C, these are f, b, which is already contained in cadence of type 5, degree V II, whereas another pitch class, for V II pitch class d, determines which half of the seven-element sequence of fourths or fifths is specified. The C scale is specified by d, whereas cadence of type 5 for G contains a instead of d. A cadence is a map cadi : Dia(3) → 2T ria , i = 1, 2, 3, 4, 5 that maps a tonality into one of the five cadence types. For example, cadi (X (3) ) = {IIIX , IVX }. 4. Mechanism We are given two different tonalities X (3) , Y (3) . We model the modulation mechanism using an idea from particle physics. In physics, there are four basic forces: electromagnetic, strong, weak, and gravitational force. Every force is supposed to materialize in bosons or force quanta. Electromagnetism interacts via photons, weak force interacts via W + , W − , and Z bosons, strong force interacts via gluons, and gravitation interacts via gravitons. In modulation theory we interpret the forces that “transform” X (3) into Y (3) as symmetries g ∈ T Z12 that map the scale X into the scale Y . Observe that such a symmetry automatically maps the degrees of X (3) into the degrees of Y (3) . This yields our concept: Definition 58 A modulation m : X (3) → Y (3) is an ordered pair m = (cadi , g) of a cadence cadi and a symmetry g such that g(X (3) ) = Y (3) .

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The symmetry g of modulation m is called the modulator and is the mechanism of modulation. There are always two symmetries g with g(X (3) ) = Y (3) . In fact, one such g is the transposition from X to Y , and the other is the inversion g(X) = Y . This follows from the fact that every major scale X has an inversion as an inner symmetry, e.g., Id for C. See Figure 23.3, where we have shown the inner symmetry A of X together with the transposition T t from X to Y .

Fig. 23.3. The inner symmetry A of X together with the transposition T t from X to Y .

5. Fundamental degrees To get the fundamental degrees, we need the model of a modulation quantum (in analogy to the bosons in physics), because ultimately we look for some fundamental degrees and not only symmetries. Here is the concept of a modulation quantum. Definition 59 Given a pair X (3) , Y (3) of different tonalities and a modulation m = (cadi , g) : X (3) → Y (3) , a modulation quantum for m is a subset Q ⊂ Z12 such that a) g ∈ Sym(Q), where Sym(Q) is the subgroup of T Z12 that leaves Q invariant. b) All triads in cadi (Y (3) ) are subsets of Q. c) The intersection T Z12 ∩ Sym(Y ∩ Q) is trivial and Y ∩ Q is covered by degrees of Y (3) , i.e., it is the union of certain degrees of Y (3) . d) The quantum Q is a minimal set with properties a) and b). A modulation that admits a modulation quantum is called a quantized modulation. The fundamental degrees of a quantized modulation are by definition the degrees of Y (3) that cover Y ∩ Q. The hope now is that we

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find enough quantized modulations and that their fundamental degrees are those Schönberg found in his harmony.

23.2 The Modulation Theorem The modulation theorem provides us with the information about the existence of quantized modulations for the set Dia(3) of tonalities. The theorem is valid for much-more general sets of tonalities in 12-tempered and just tuning, see [75, Chapter 27], but for our modest needs, the case Dia(3) is sufficient. Theorem 20 For the system Dia(3) of triadic interpretations of diatonic major scales in 12-tempered tuning, there is a quantized modulation m = (cadi , g) : X (3) → Y (3) for each ordered pair X (3) , Y (3) of tonalities. It has a total of 26 such modulations from a fixed tonality X (3) . The list of all quantized modulations is given in Section 23.5 at the end of this chapter. In the wider context of all scale types with seven notes that have triadic interpretations with an inner symmetry and have quantized modulations for all pairs of tonalities, the diatonic major scale case Dia(3) has the smallest number of quantized modulations starting at a fixed tonality [75, Chapter 27, Theorem 30]. This means that the behavior of a type of tonality under modulation also qualifies it: The diatonic major tonalities are the most economic candidates. If we compare the table of modulations in Section 23.5 to Schönberg’s lists of fundamental degrees, they coincide for every case where Schönberg has discussed direct modulations.

Fig. 23.4. The nerves of a modulation C (3) → E(3) . On top is the nerve Q(3) of the modulation quantum.

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23 Modulation Theory

23.3 Nerves for Modulation Since all modulations in our model involve triadic interpretations, we can look for the nerves of our objects. The nerves of diatonic major scales are the harmonic bands, and we now add the nerves Q(3) of the modulation quanta, which refers to the triads from both tonalities that cover the quanta Q. Figure 23.4 shows the situation for modulation C (3) → E(3) . The nerve N (Q(3) ) is shown on top. It connects the two harmonic bands and is covered by eight triads: IIC , IVC , VC , V IIC from C and IIE , IIIE , VE , V IIE from E. It shares two 2-simplices (triangles) with each harmonic band and contains a 3simplex spanned by IIC , V IIC , VE , V IIE as well as two 5-simplices, spanned by IIC , VC , V IIC , IIIE , VE , V IIE and IIC , IVC , V IIC , IIE , VE , V IIE . As these are five-dimensional, we cannot represent them (like with tetrahedra) in three-space. The horizontal beams symbolize them instead. For a complete list of modulation nerves, see [75, Figure 27.3]. This geometric representation of a modulation not only is nice to look at, but it also carries important musical information. In fact, when we think about the succession of degrees within a modulatory process, we move from vertices v (degrees!) in the harmonic band of the old tonality to vertices u in the new tonality. It is advantageous to look for walks on the connecting lines between v and u, meaning that one moves from a vertex to another on a 1-simplex, which guarantees common notes and therefore a connection that is preferred by voice-leading arguments. Also, moving along lines that are embedded in higher simplex configurations is preferred, because then the common notes are even higher in number.

Fig. 23.5. A very short modulation B → G between distant tonalities in measures 238-239 of Beethoven’s “Hammerklavier” Sonata op. 106.

23.4 Modulations in Beethoven’s op. 106

199

23.4 Modulations in Beethoven’s op. 106 Beethoven’s “Hammerklavier” Sonata op. 106 is a challenge for modulation theory. We find some modulations that are executed very fast, with a minimum of notes, although they connect tonalities that are far from each other in terms of fourth distance. For example, in the Allegro movement, there is a modulation B → G between distant tonalities in measures 238-239. But there are other modulations that have a huge anatomy that is difficult to understand without a deeper understanding of the modulatory process.

Fig. 23.6. A complex modulation G → E.

Let us look at an example of such a complex modulation. Consider the modulation G → E in measures 124-129 of the Allegro movement, see Figure 23.6. All modulations in the Allegro movement have been analyzed and completely understood using our modulation theory, see [75, Section 28.2]. Here we want to give just one example, namely the modulation G → E in measures 124-129 of the Allegro movement. This modulation is bipartite (first part: measures 124-127, second part: measures 128-129). Before we encounter the fundamental degrees V II − V − V II in E in part two, according to our

200

23 Modulation Theory

modulation table in Section 23.5, we hear note g as an octave interval: pedal and stationary voice in the first part. The pitches of the first part, when transposed into the octave spanned by the two g notes, show a regular melodic structure, see Figure 23.6, bottom. This structure has two parts: the first in measures 124-125, and the second in measures 126-127. They are related to each other by the inversion at d, which is the same as the inversion at g in pitch classes. This first part of the modulation makes evident the inversion Id before we see the fundamental degrees in the second part of the modulation. But why this preliminary inversion? It is the modulator for the modulation in our model, Id (G(3) ) = E(3) . This strategy is a beautiful compositional realization of what our model specifies. The model does predict fundamental degrees, and it does so on the basis of modulation forces that are provided by modulator symmetries. Beethoven not only writes down the fundamental degrees, but also makes evident the modulator in the first part of the modulation. Our interpretation in this analysis does not assume that Beethoven has performed his construction using the ideas of our model. But he might have done so instinctively; one cannot know such hidden layers of creativity. This situation is parallel to what happens in physics. We discover physical laws, but we cannot know whether a divine creator (if this is the underlying cosmological hypothesis) has constructed the universe according to these laws, which are our way to understand nature. Nevertheless, the laws hold, and so does our modulation module for the critical system of modulations in Beethoven’s composition. Concluding this chapter, we should add that our model also holds for other compositions by Beethoven, for example for modulations in the Cavatina movement of String Quartet op. 130.

23.5 Quanta and Fundamental Degrees

201

23.5 Quanta and and Fundamental Degrees for the Modulations Between Diatonic Major Scales (Dia(3) ) The translation p indicates the relation Y (3) = T p X (3) . Transl. p

Cadence Quantum

1

{II, V }

Modulator Fund. Degrees

• ◦ • • ◦ • • • • • •• e5 11 5

1

{II, III} • ◦ • • ◦ • • • • • •• e 11

2

{V II}

2

{II, V }

◦ • • ◦ • • ◦ • ◦ ◦ ◦• e6 11

{II, III, V, V II} {II, III, V, V II} {II, IV, V II}

6

{II, IV, V, V II}

6

{II, IV, V, V II}

◦ • • ◦ • • ◦ • ◦ • ◦• e 11

2

{IV, V } ◦ • • ◦ • • ◦ • ◦ • ◦• e 11

3

{II, V }

• ◦ • ◦ ◦ • ◦ • • • •• e7 11

{II, III, V, V II}

3

{II, III} • ◦ • ◦ ◦ • ◦ • • • •• e7 11

{II, III, V, V II}

4

{V II}

4 4 5 6 6 6 6

◦ ◦ • • ◦ • • ◦ ◦ • ◦• e8 11

{II, III, V, V II}

8

{V, V II}

9

{II, IV, V II}

6

{II, III, V, V II}

10

{II, IV, V, V II}

6

{II, IV, V, V II}

10

{II, III, V, V II}

11

{IV, V } ◦ • • • • • • • ◦ • ◦• e 11 {II, III} • • • • ◦ • • • • • ◦• e 11 {V II}

◦ ◦ • ◦ • • ◦ • ◦ ◦ •• e 11

{II, III} ◦ • • • • • ◦ • • • •• e

{IV, V } ◦ • • • • • • • • • ◦• e 11 {IV, V } • • • • ◦ • • • • • ◦• e

{II, IV, V, V II}

8

{II, III} • • • • ◦ • ◦ • • • •• e 11

7

{V II}

• ◦ • ◦ ◦ • • ◦ ◦ • ◦• e 11

{III, V, V II}

8

{V II}

◦ • • ◦ ◦ • ◦ • ◦ ◦ •• e0 11

{II, V II}

0

8

{IV, V } ◦ • • • • • ◦ • • • •• e 11

{II, IV, V, V II}

8

{II, III} • • • • ◦ • ◦ • ◦ • •• e0 11

{II, III, V, V II}

9

{II, V }

9 10 10 10 11 11

◦ ◦ • ◦ • • • • • • ◦• e1 11

{II, IV, V, V II}

2

{III, V, V II}

2

{II, III, V, V II}

2

{II, III, V, V II}

3

{II, IV, V, V II}

3

{II, IV, V, V II}

{IV, V } ◦ ◦ • ◦ • • • • • • ◦• e 11 {V II} {II, V }

• ◦ • • ◦ • ◦ ◦ ◦ • ◦• e 11 • ◦ • • ◦ • ◦ • ◦ • ◦• e 11

{II, III} • ◦ • • ◦ • ◦ • ◦ • ◦• e 11 {II, V }

{II, IV, V, V II}

1

◦ • • ◦ • • • • • • •• e 11

{IV, V } ◦ • • ◦ • • • • • • •• e 11

202

23 Modulation Theory C -> F

8 X & 4 XX 8 X ?4 8 4

IC

VIIF IIF

A

B

C -> As

X ? X A

XX

?

XX

XX

X X

XX

IC

VC

IIC

C -> A

X ? X

IC A

XX

X X

VC

# XX

# XX

XX

IVH

IIH VIIH

IVH

IC VC IIEs VEs VEs IEs VIIEs VIIEs B

A

C

b XXX b b XXX bb XXX D

bX b XX XX b X D

VH

XX XX

X

X X X X

IH

IC VI C A

X XX

XX

X X X #X

IID IC VI C VC VII D A

B

XX # XX D

# XX X X

VE VIIE

# XX D

XX

# XX

IVE

VE

X XD

X #X

X #X

B

IE

C

C -> G

XX X

X X

X XX X D

XX

VD ID

X X

IC

VC

C

C

B

A

C -> E

C

X XX # XX XX XX D XX X C

b X X b XX D

bX X X

C

XX # # XX

C -> D

B

b b XX b X XXX D X X

X XX

C -> Ges b X XX b XX D X XX XX b XX bX X X X bX X X bX b X b X b X D X X XX X

X #X X X # X # XX

# XX XX XX # XXX D XX XXX IVA V I VIIA A A

XX XX X X

C

B

B

IVA

D

1

C -> Es

VGes I II VI VII VIIGes Ges IC IIIDes Des IIDesVDesIDes IC C C VIIDes IVC VIC IIGes IIGes VIIGes A

X #X

X bX

B

XX b XX b XXX D XX b XX b b XX X XX b XX X bX bX bX D X C

X b XX D X X

VIIB VII I B B VB

IIIB

A

b XX bX

X X

& XX

X XX

C -> Des

A 10

XX b XX X

IC V C

C

B

C -> H

&

IF IVF VF IF

VIIAs IV V As As IAs IIAs

IC IVC

7

XX X X XX XXX D X X XX XX X X X

bX X X X X D X X

X & XX XXX

4

C -> B

A

X XX # XX X D X X XX # XX XX XX X D X

IIIG B

IIG VG IG C

Fig. 23.7. Examples of modulations C (3) → Y (3) according to our model. The three Schönberg steps are notated with A, B, C. The modulation to B (3) is notated by the German letter H for B, while the symbol B stands for English B. The sound example is modulation.

Part VI

Rings and Modules

24 Rings and Fields

Summary. Rings are the basic structures for algebra. We already have many examples of rings: the integers, real and complex numbers, and the structure of addition and multiplication that was defined on Zn in the chapter about the third torus and its geometry. –Σ– Definition 60 A ring is a triple (R, +, ∗), where (i) (R, +) is a commutative group with additively written operation and neutral element 0R , (ii) (R, ∗) is a monoid with multiplicatively written operation and neutral element 1R , and (iii) the two operations are connected by the two distributive laws x ∗ (y + z) = x ∗ y + x ∗ z (y + z) ∗ x = y ∗ x + z ∗ x for all x, y, z ∈ R. The ring is commutative if (R, ∗) is so. A subring S ⊂ R is an additive subgroup that is also a multiplicative submonoid. √

Example 21 The chain Z ⊂ Q ⊂ R ⊂ C represents successively increasing subrings. The ring Zn defined in Section 21.1.1 is a ring. All these rings are commutative, but we shall see important examples of non-commutative rings. For any family of rings (Rn , +n , ∗n )n , the cartesian product n Rn is a ring whose operations +, ∗ are simply the operations +n , ∗n on each coordinate ring Rn .

ˇ “* Example 53 The rings Z3 , Z4 , Z12 , as well as the cartesian product ring Z3 × Z4 defined in Section 21.1.1, are rings that play a major role in music theory. √

Exercise 27 Prove that in any ring, r ∗ 0R = 0R .

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_24

205

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24 Rings and Fields

24.1 Monoid Algebras and Polynomials An important non-commutative ring is deduced from any commutative ring R, together with a monoid M . The ring is called monoid algebra over R and M and is denoted by RM . The underlying set of RM  is the set {f |f : M → R such that f (m) = 0 for a finite number of monoid elements}. Addition of two such functions f, g is defined in a straightforward manner: (f + g)(m) = f (m) + g(m), which is clearly a function of the required type. Multiplication of f, g is defined by  (f ∗ g)(m) = f (n) ∗ g(l), n,l,n∗l=m

again the required type. The 0R M  element here is the zero function f (m) = 0 for all m ∈ M . The 1R M  element is the function f (eM ) = 1R and f (m) = 0R for all m = eM . There is a natural commutative subring of RM , namely the set of functions fr (eM ) = r and fr (m) = 0R for m = eM . We denote these functions fr by r if no confusion of notation results. Therefore we may identify R with the subring of these special functions within RM . The monoid M is also embedded in RM  by fm (m) = 1R and fm (n) = 0R for n = m. Check that the monoid multiplication carries over to the multiplication among these special functions, and we may identify M with this multiplicative submonoid of RM , see the following diagram. R

- RM  6

M A core example of such an algebra is provided when taking the word monoid M = P ath(A) of an alphabet A, denoted by RA. For our purposes, the most important case is the simple one-letter alphabet A = {X}, where X is usually called the indeterminate of this algebra. The path monoid consists of the natural powers X n , n ∈ N of X. This algebra is called polynomial algebra and is denoted by R[X]. The algebra’s elements are represented as follows: For an element a ∈ R and a natural number n, we can consider the function aX n defined by aX n (X n ) = a and aX n (X m ) = 0 for m = n. Then if the natural powers of X, where the function values f (X n ) = an might not vanish, are ≤ N , we can write  an X n . f= n≤N

24.1 Monoid Algebras and Polynomials

207

And this representation is unique. Such functions are called polynomials in the indeterminate X and coefficients in R. In particular, the 1 element is 1 = 1R X 0 , which we identify with eP ath(A) and the zero element is 0 = 0R eP ath(A) = 0R 1. The ring elements an in a polynomial are called its coefficients of a determined power of the indeterminate. The addition of two polynomials is as follows:    an X n + bn X n = (an + bn )X n n≤N

n≤N

n≤N

whereas the product of two polynomials is     an X n ) ∗ ( bn X n ) = (an bl )X (n+l) = (an bl )X m . ( n≤N

n≤N

n,l≤N

n+l=m

It’s clear that the polynomial algebra R[X] is commutative. √

Exercise 28 Verify all these ring properties for the polynomial algebra R[X]. Similar to monoid morphisms and group homomorphisms, there are also ring homomorphisms. Definition 61 If (R, +R , ∗R ), (S, +S , ∗S ) are two rings, a ring homomorphism is a map f : R → S that is a group homomorphism for addition and a monoid morphism for multiplication. The set of ring homomorphisms f : R → S is denoted by Rings(R, S). Clearly, the composition g ◦ f : R → T of two ring homomorphisms g : R → S, g : S → T is a ring homomorphism, and the identity IdR : R → R is a ring homomorphism. √

Example 22 The embeddings Z  Q  R  C are ring homomorphisms. The projection Z → Zn , as well as the projections Z12 → Z3 , Z12 → Z4 discussed in Section 21.1.1, are ring homomorphisms. Another example is the embedding R  RM  of monoid algebras. Also, if n Rn is a cartesian prod uct of rings Rn , the projection prm : n Rn → Rm to the mth component rm of a sequence (rn ) is a ring homomorphism.

ˇ “*

Example 54 The map (pr3 , pr4 ) : Z12 → Z3 × Z4 : x12 → (x3 , x4 ) defined in Section 21.1 is a ring homomorphism, but the map (also defined there) Id × (−1) × (pr3 , pr4 ) is not a ring homomorphism because the 112 does not map to the unit (13 , 14 ). Proposition 17 Let R be a commutative ring, A an alphabet, and S a ring. Then for any given ring homomorphism f : R → S, we have a bijection ∼

Ringsf (RA, S) → Set(A, S) : g → g|A

208

24 Rings and Fields

where Ringsf (RA, S) denotes the set of ring homomorphisms that restrict to f when restricted to R. In particular, if A = {X}, the polynomial ring homomorphisms g that extend a given ring homomorphism f : R → S are in bijection with the elements a ∈ S by the association g → g(X). Such homomorphisms are called polynomial functions. For example, if f : R → C is the canonical embedding of the reals in the complex numbers, the polynomial functions g : R[X]  → C nare given by the value x = f (X), and the functions map a polynomial n an X to the complex  number n an xn .

ˇ “*

Example 55 Polynomial functions play a crucial role in the theory of musical performance. We shall deal with that theory in Chapter 32. But let us preview the role of polynomials in this theory. Often, musicians have to perform changes of musical parameters that are not explicitly notated. A basic example is glissando, where the score notation shows only the initial pitch p1 and the final pitch p2 of glissando. The musician then has to move from the beginning to the end in a continuous curve of intermediate pitches. Whenever this movement has to be defined precisely, be it for a software that implements glissandi or for theoretical reasons, such a glissando curve must be defined in explicit terms. Then we have to define a function gliss : [a, b] → R from a time interval [a, b] to the real-number-valued pitch domain R. Typically such a function gliss is defined by a polynomial function of shape P [X] = rX 3 + sX 2 + tX + d. But we also want that P (a) = p1 , P (b) = p2 and that the slope of the function is horizontal in the two limit times. Slopes will be discussed in Chapter 31. Such a function is shown in Figure 24.1. For other parameters, such as time changes with tempo, ritardando or accelerando, and dynamics changes such as crescendo or diminuendo, similar polynomials functions are used, but see Chapter 32 for details. Much like homomorphisms of groups, ring homomorphisms have kernels, too, and we have a similar result describing images of ring homomorphisms in terms of quotient rings. This result is also valid for non-commutative rings, but we don’t need it, and it looks somewhat more complicated. Definition 62 If f : R → S is a ring homomorphism, its kernel Ker(f ) is defined to be the group-theoretical kernel of f .

ˇ “* Example 56 The kernel of the ring homomorphism Z12 → Z3 : x12 → x3 is the subgroup 4Z12 . It is the diminished seventh chord C o7 in the pitch-class set Z12 . The kernel of the ring homomorphism Z12 → Z4 : x12 → x4 is the subgroup 3Z12 . It its the augmented triad C + in the pitch-class set Z12 . Proposition 18 Let R be a commutative ring. Then a subgroup I ⊂ R is the kernel I = Ker(f ) of a ring homomorphism f : R → S iff

24.1 Monoid Algebras and Polynomials

209

Fig. 24.1. Glissando by a polynomial function P [X] = rX 3 + sX 2 + tX + d, starting at pitch 60 (middle c) at time 0, and ending at time 2 on pitch 67 (fifth g above middle pitch). Slopes are horizontal at beginning and ending times.

rI ⊂ I for all r ∈ R. In this case the subgroup is called an ideal in R. Proof 14 The proof is a construction that we need to spell out. First, suppose that I = Ker(f ), f : R → S. Then for i ∈ I and r ∈ R, we have f (ri) = f (r)f (i) = f (r)0S = 0S , and thus the property rI ⊂ I. Conversely, suppose this property holds for I. Then, as R is a commutative additive group, we may apply the theorem in Section 19.2 with its commutative diagram of groups: G

f

- K 6

f

proj

inj ∼f

→

--

? ? G/H

6 Im(f )

to our situation for G = R, K = S, and H = I, getting a group diagram R

f

- S 6

f

proj ∼f

→

--

? ? R/I

inj

6 Im(f )

210

24 Rings and Fields

The missing part here is that R/I is only a group, not a ring. The ring structure on R/I is defined as follows: For r + I, s + I in R/I, we define r + I ∗ s + I = rs + I. This multiplication is well defined since for other representatives r + i, s + j, we have (r + i)(s + j) + I = rs + rj + si + ij + I, and because rj, si, ij ∈ I by the property of the ideal I, the product is well defined. Once this holds, the ring axioms for R/I are easily verified, and we also know that the maps in our commutative diagram are all ring homomorphisms. We call the quotient group R/I with this ring structure the quotient ring of R modulo the ideal I. √

Example 23 For any subgroup nZ ⊂ Z, n ∈ N, we have the quotient ring Zn := Z/nZ that was defined in Section 21.1.1. This example is a special case of a principal ideal, which by definition is an ideal of shape I = rR, which one denotes by (r). The other example that also played a major role in our earlier theory is the ideal O ⊂ C of zero sequences in the ring C of Cauchy sequences used to define real numbers as a quotient ring R = C/O in Chapter 12. We now come back to the general philosophy of solving problems by transforming the problem into its solution. Here, we want to apply this philosophy to the construction of complex numbers from real numbers. Recall that the problem of real numbers was that polynomial equations, in particular X 2 + 1 = 0, have no solutions in general. So now the problem is the ring R[X] of polynomials with real coefficients. We want to reconstruct C from this ring. To this end, we consider the homomorphism   an X n → an in f : R[X] → C : n

n

defined by sending X to the imaginary unit i, and by the identity on the real coefficients. We now that f is surjective because the images of aX + b are all imaginary numbers ai + b. The kernel of f contains the principal ideal (X 2 + 1) since f (X 2 + 1) = i2 + 1 = 0. We now show that Ker(f ) = (X 2 + 1). This will imply that R[X]/(X 2 + ∼ 1) → C, i.e., the complex numbers can be constructed from the problem set R[X] and a quotient construction! The claimed equation follows from the fact that any polynomial P ∈ R[X] can be written in the form P = H(X 2 + 1) + aX + b. We shall prove this fact in the next section. If we use this formula, then P ∈ Ker(f ) implies 0 = f (P ) = f (H)f (X 2 + 1) + ai + b = ai + b, and so a = b = 0, i.e., P = H(X 2 + 1) ∈ (X 2 + 1), and we are done.

24.2 Fields

211

24.2 Fields Definition 63 A non-zero commutative ring R is called a field iff every nonzero element x ∈ R is invertible, i.e., R∗ = R − {0}. In particular, in a field R the product xy of non-zero elements x, y is non-zero. If it were zero, we could multiply it with the inverse x−1 and get 0 = x−1 xy = y, a contradiction. √

Example 24 The rings Q, R, C are fields. If p is a prime number, the quotient ring Zp is a field.  Definition 64 If P = n an X n ∈ R[X] is a non-zero polynomial, the highest natural number N such that aN = 0 is called the degree of P and denoted by deg(P ). Proposition 19 If R is a field, then for any two non-zero polynomials P, Q ∈ R[X], we have deg(P Q) = deg(P ) + deg(Q). In fact, if P = aN X N + aN −1 X N −1 + . . . a0 , Q = bM X M + bM −1 X M −1 + . . . b0 with aN , bM = 0, i.e., deg(P ) = N, deg(M ) = M , then the highest power of X with non-zero coefficient in P Q is N + M since P Q = aN bM X N +M + (aN bM −1 + aN −1 bM )X N +M −1 + . . . a0 b0 , and aN bM = 0 for the field R. Theorem 21 (Division Theorem) If P and Q are non-zero polynomials in the polynomial algebra K[X] over a field K, then there are uniquely determined polynomials H, R such that Q = HP +R with either R = 0 or deg(R) < deg(P ). The proof is easy and works by induction on deg(P ), we omit it. This theorem implies our above claim that any polynomial P ∈ R[X] can be written in the form P = H(X 2 + 1) + aX + b. Corollary 3 The ring of integers Z and the polynomial algebra K[X] for a field K are principal ideal rings, i.e., every ideal is principal. In the case of Z, take a minimal positive number n ∈ I in an ideal I (the case I = (0) is trivial). Any x ∈ I can be written as x = an + b, 0 ≤ b < n. But then, b = x − an is also in I, and so b = 0 by the choice of n. In the case of K[X], we take a polynomial P with minimal degree in the ideal I. Any element x ∈ I can rewritten as x = HP + R, where either R = 0 or deg(R) < deg(P ). The second case means that x − HP = R is in I, but this contradicts the choice of P , so only the first case is possible, and we are done. Proposition 20 A proper ideal I in a commutative ring R is maximal (i.e., there is no strictly larger proper ideal in R) iff R/I is a field.

212

24 Rings and Fields

Proof 15 Any proper ideal J with I ⊂ J ⊂ R is mapped to an ideal J/I ⊂ R/I. If R/I is a field, it contains no proper ideal except the zero ideal (0). Therefore J/I = (0), i.e., J = I. Conversely, if I is maximal, then I and any element x ∈ R − I must generate the ideal of the entire ring R. This means that 1R = yx + j, j ∈ I. But then 1R/I = (x + I)(y + I), so in R/I, every non-zero element is invertible. √

Example 25 The last proposition shows that the ideal (X 2 +1) is maximal ∼ in R[X] because C → R[X]/(X 2 + 1) is a field.

25 Primes

Summary. Prime numbers play a crucial role in music theory, and in particular in the theory of tuning. In this chapter, we prove uniqueness of prime decomposition for the integers and polynomial rings. –Σ– In a commutative ring R, if x = yz we write also y|x and say that y is a divisor of x. Definition 65 An element p ∈ R in a commutative ring R = 0 is prime if it is not invertible and if any decomposition p = qr implies that either q or r is invertible. For polynomial algebras, prime polynomials are also called irreducible. Proposition 21 If R = Z or R = K[X], and K is a field, then every noninvertible element x = 0 has a factorization x = p1 p2 . . . pk as a product of primes pi . The proof in the case Z has already been indicated in Section 9.1. It goes by induction on |z|: If z is not prime, we can write it as a product z = xy with 1 < |x|, |y| < |z| since |x| = |z| implies y = ±1, which is invertible, and same argument for |y| = |z|. For the polynomial algebra R = K[X], we take deg(P ) instead of the absolute value. Then if P = QR, we have deg(P ) = deg(Q) + deg(R). If P is not irreducible, there is such a factorization with 0 < deg(Q), deg(R) < deg(P ), because deg(Q) = 0 implies that Q ∈ K, and same argument for deg(R) = 0. Again, we have induction on degrees. √

Example 26 For Z, we know some small primes, such as ±2, ±3, ±5, ±7, ±11, ±13, . . . ,

and we have already shown in Section 9.1 that there are infinitely many primes here. For R = R[X], all linear polynomials aX + b, a = 0 are irreducible as their © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_25

213

214

25 Primes

degree is 1. Also, all quadratic polynomials aX 2 + b, a, b > 0 are irreducible. In fact, suppose a = 1, and then if we had a factorization X 2 + b = (uX + v)(rX + s), we may also suppose u = r = 1, and then we get X 2 +b = X 2 +(v+s)X +vs. But v + s = 0 implies that either v or s is negative while the other is positive, as vs = 0. Then vs cannot be positive. Definition 66 If R is a commutative ring, it is called an integral domain if the equation xy = 0 implies that either x or y is zero. Proposition 22 In a principal ideal integral domain R, an ideal I is maximal iff I = (p) for a prime p ∈ R. In fact, if (p) is maximal, then p = qr implies (p) ⊂ (q), so either (q) = R, hence q ∈ R∗ , or (p) = (q), hence p = qr, q = ps, i.e., p = rsp, (1 − rs)p = 0, hence 1 = rs as R is an integral domain. Therefore r is invertible and p is prime. If (p) is not maximal, there is a strictly larger intermediate ideal (q), therefore p = rq with both r and q not invertible, i.e., p is not prime.

ˇ “*

Example 57 The quotient rings Zp for primes p are fields, a fact already mentioned in Section 24.1. This applies to the musical case Z3 . Also, it is important to note that both Z12 and Z4 are not fields, because there are sequences 0, x, 2x, . . . for x = 0 that don’t cover the entire rings, e.g., 0, 2, 4, 6, 8, 10 in Z12 . The next result deals with unicity in the prime factorization. Proposition 23 If R is a principal ideal integral domain, then for a prime factorization x = p1 p2 . . . pk and any prime divisor p|x there is a prime pi such that p = epi , e ∈ R∗ . Since R/(p) is a field, the equation x = p1 p2 . . . pk implies that in R/(p), pi , one of the factors, must vanish, i.e., pi ∈ (p), therefore p = epi with e ∈ R∗ . Corollary 4 If R is a principal ideal integral domain, then two prime factorizations x = p1 p2 . . . pk = q1 q2 . . . ql must have the same number k = l of factors and there is a permutation π of [1, k] such that qj = ej pπ(j) for all j ∈ [1, k]. In particular, since we know that Z and K[X] for a field K have prime factorizations for any non-invertible element, their factorizations are unique up to permutations and multiplications of their prime factors by invertible elements. √ This unicity was used already in Chapter 12 about the irrationality of 2 when calculating the kernel of the group homomorphism Z12 → Z3 × Z4 in Section 21.1.

25 Primes

215

ˇ “* Example 58 The probably single most important musical consequence of the prime factorization theorem, Corollary 4, is that in just tuning, the exponents of primes 2, 3, 5 are unique. More precisely, Western (and more specifically just) tuning considers frequencies of shape f (o, q, t) = f0 2o 3q 5t for rational exponents o, q, t ∈ Q and a basic frequency f0 . The claim is that the map (o, q, t) → f (o, q, t) is injective. In fact, suppose f0 2o 3q 5t = f0 2u 3v 5w for o, q, t, u, v, w ∈ Q. Then we infer 1 = 2o−u 3q−v 5t−w . We have to prove 1 = 2a 3b 5c , a, b, c ∈ Q iff a = b = c = 0. But write a = aa12 , b = bb12 , c = cc12 , then we get 2a1 3b1 5c1 = 2a2 3b2 5c2 for integer exponents. Multiplying this equation with a sufficiently high positive power of the three primes, we may suppose that all exponents are positive. But then, by the uniqueness of prime factorization, the exponents of 2, 3, 5 must be equal, and therefore they were also equal before our multiplication with that high power of the three primes. This fact is crucial for the musical understanding of all tunings, just and tempered. See Chapter 28 for details.

26 Matrices

Summary. Matrices are a very classical tabular form to represent data, for example in accounting. They are built from columns that are juxtaposed and can be split horizontally into a stack of rows. The novelty in mathematics is that matrices that are built from numbers can be used to perform calculations that are of general benefit to mathematics. –Σ–

Fig. 26.1. A matrix built from columns in Greek temples. For every column j, we have its building blocks aij , referring to row i. The third dimension of depth is not dealt with in this book, but there are also three-dimensional matrices in mathematics.

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_26

217

218

26 Matrices

Matrices are the backbone of algebra. They are indispensable for most concrete calculations, but they also share the structure of a category (we shall discuss categories in Chapter 29) in a particularly transparent way. Moreover, matrices provide us with examples of important algebraic structures of modules (we shall discuss modules in Chapter 27). Finally, matrices also provide us with core examples of non-commutative rings.

26.1 Generalities on Matrices For natural numbers n ≥ 1, we denote by [1, n] the set {1, 2, . . . n} of the first n positive natural numbers. Definition 67 Suppose we are given two positive natural numbers m, n and a commutative ring R. A m × n-matrix with coefficients in R is a set map M : [1, m] × [1, n] → R. The images M (i, j) are denoted with indices, M (i, j) = Mi,j . Matrices are typically represented in matrix form with m rows and n columns: ⎛ ⎞ M11 M12 . . . M1n ⎜ ⎟ ⎜ M21 M22 . . . M2n ⎟ ⎟. M = (Mij ) = ⎜ ⎜ ⎟ ⎝ ... ⎠ Mm1 Mm2 . . . Mmn The set of m × n-matrices with coefficients in R is denoted by Mm,n (R). The matrix transposition is a bijection ∼

?t : Mm,n (R) → Mn,m (R) : M → M t with (M t )ji = Mij ; we have (M t )t = M . For an element λ ∈ R and a matrix M ∈ Mm,n (R), we have its scalar-multiplied λM that has the coefficients (λM )ij = λMij . The identity matrix En of rank n is the matrix in Mn,n (R) with (En )ij = δij , where δii = 1R for all indices i, and zero otherwise (δij is called the Kronecker delta). There are several core algebraic operations on matrices. To begin with, if M, N ∈ Mm,n (R), then we define their sum M + N by (M + N )ij = Mij + Nij . This turns Mm,n (R) into a commutative group, and we have the isomorphism ∼ of groups Mm,n (R) → Rmn , the cartesian product of mn copies of the additive group of R. The product of matrices is slightly more involved: If M ∈ Mm,n (R) and N ∈ Mn,l (R), then we define their product M N ∈ Mm,l (R) by

26.1 Generalities on Matrices

(M N )ik =



219

Mij Njk .

j

This means that the coefficient (M N )ik at row i and column k is the sum of the products Mij Njk of the ith-row coefficients of M with the corresponding kth column coefficients of N . To show the number of rows and columns of a matrix M ∈ Mm,n (R), we also write it as a symbol of a function (and you will see soon that this has good reasons): M : En → Em .

ˇ “*

Example 59 If we want to list the number of instruments of an orchestra with n instrument types (violins, violoncellos, bassoons, etc.) with respect to the m movements, we can describe this by a m × n matrix M , where Mi,j denotes the number of instruments of type j in movement i. Sorite 10 We have the following properties for all λ, μ ∈ R, and M, N : En → Em . Also, let A : En → Em , B : Em → El , C : El → Ek be three matrixes over R. (i) Scalar multiplication is homogeneous, so we have λ.(μ.M ) = (λ.μ).M , therefore we may write λ.μ.M for this expression. (ii) Scalar multiplication is distributive, i.e., we have (λ + μ).M = λ.M + μ.M and λ.(M + N ) = λ.M + λ.N. (iii) Scalar multiplication and transposition commute: (λ.M )t = λ.M t . (iv) (Associativity) (C.B).A = C.(B.A), which we therefore denote by C.B.A. (v) (Distributivity) If C  : El → Ek and B  : Em → El are two matrixes over R, then (C + C  ).B = C.B + C  .B and C.(B + B  ) = C.B + C.B  . (vi) (Homogeneity) If λ ∈ R is a scalar, then λ.(C.B) = (λ.C).B = C.(λ.B), which we therefore denote by λ.C.B. (vii) (Neutrality of identity matrixes) We have A.En = Em .A = A. (viii) (C.B)t = B t .C t . The proof of these properties is straightforward, so we omit it. √

Example 27 In particular, the triple (Mn,n (R), +, .) of square n × n matrices over R is a non-commutative ring for n > 1. The ring R is embedded in Mn,n (R) by R  Mn,n (R) : λ → λ.En , the diagonal matrices. It can be shown that these matrices are the only ones that commute with the entire ring Mn,n (R), i.e., M.C = C.M for all M ∈ Mn,n (R) iff C = λEn . The subring of these commuting matrices in a ring is called the center of the ring. In short, the center of Mn,n (R) is R.

220

26 Matrices

Here is the justification for our functional notation of matrices. Let M : ∼ En → Em be a matrix with coefficients in R. Observing that Mn,1 → Rn , it defines a map − → M : Rn → Rm : v → M v. −−→ − → − → −→ − → Clearly M = N iff M = N , En = IdRn , and λM = λM , the latter being the − → − → map λM (v) = λ(M v). Moreover, whenever we have two matrices M : En → Em , N : El → En , then −−→ − → − → MN = M ◦ N − → This means that the map M(R) → Ens : M : En → Em → M : Rn → Rm on the set M(R) of all matrices over R conserves composition of matrices and maps the identities En to identity maps of sets. This is what we call a functorial map from matrices to sets. We shall come back to this situation in Chapter 29. √

Example 28 It is important to have some examples of matrix operations that can be visualized on the plane. Let us look at the case M ∈ M2,2 (R). Such a matrix is a square 2 × 2 array   ab M= cd

− → and its associated map M : R2 → R2 maps a vector (x, y) that is interpreted as a matrix   x v= y to the vector − → M (v) =



ax + by cx + dy



− → which we rewrite as M (v) = (ax + by, cx + dy) to keep notation of vectors consistent as pairs of real numbers. Observe that property (i) in the above sorite means that the line R(x, y) in R2 that is defined by (x, y) is mapped to − → the line that is defined by (ax + by, cx + dy). And also M (0) = 0. This is why − → the maps M are called linear; more precisely, they are group homomorphisms 2 2 R → R that preserve scalar multiplication. Let us give a geometric interpretation of some typical matrices, see also Figure 26.2.

26.1 Generalities on Matrices

221

Fig. 26.2. Six frequent transformations on R2 that are induced by 2 × 2 matrices.

 M1 = M2 =

 −1 0 0 1   1 0

horizontal reflection

vertical reflection 0 −1   −1 0 M3 = = M1 M2 180o rotation 0 −1   11 M4 = horizontal shearing 01   0 −1 M5 = 90o rotation 1 0 ⎞ ⎛ M6 = ⎝

−1 √1 √ 2 2⎠ √1 √1 2 2

45o rotation

Observe that matrices M1 , M2 , M3 , M4 , M5 are also in M2,2 (Z), while M6 is strictly in M2,2 (R).

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26 Matrices

ˇ “* Example 60 Let us give some standard musical interpretations of the above transformations. 1. Horizontal reflection M1 . On the plane of onset and pitch, this transformation reverses onsets and leaves pitches fixed. It is often associated with retrograde. However, that is not exactly what retrograde does. In fact, the durations of notes play a role in retrograde. For example, if a long note follows a short note without rest, the reflection M1 on the onset parameter does not change the difference of the two onsets, and this causes the reflected long note to overlap with the short reflected successor. We need some additional transformation for a correct retrograde. We shall solve this problem using the horizontal shearing. 2. Vertical reflection M2 . The vertical reflection on the onset-pitch plane is what is known as inversion. That this inversion is a reflection at pitch zero is not relevant now—it depends only on the gauging of pitch—but we shall see the full formalism for general inversions in Section 27.1. 3. 180o rotation M3 . This rotation is the composition of inversion M2 and retrograde M1 . It is known as the retrograde inversion. We already discussed this understanding of the 180o rotation as a composed transformation in Music Example 45. 4. Horizontal shearing M4 . This transformation can be viewed in the plane of onset and pitch as an arpeggio, which in music means to play the notes of a chord one after the other from the top or the bottom. Notes that have the same onset define a chord, and their M4 -transformed notes are then played one after the other, starting on the lowest pitch. But a second interpretation is more substantial. If we consider the  plane of onset and  duration, the retrograde is defined by the matrix M7 =

−1 −1 0

1

. The onset

o is transformed into the new onset −o − d, where d is the note’s duration. This matrix is the product M7 = M1 .M4 , the “false” retrograde M1 times the shearing M4 . transformation can be viewed as the product 5. 90o rotation M5 . This   M8 .M2 , where M8 =

0 1 1 0

is the exchange of onset and pitch. The latter

was discovered by Karlheinz Stockhausen while building sound transformation devices at the lab of Herbert Eimert in Cologne. This means that M5 is generated by parameter exchange and inversion. 6. 45o rotation M6 . This transformation was used by serial composers, such as Maurizio Kagel.

26.2 Determinants We shall not discuss the full theory of determinants, but restrict our attention to determinants in the case of the above example, i.e., M ∈ M2,2 (R). Determinants

26.3 Linear Equations

223

− → are numbers that help determine when the associated map M : R2 → R2 is a bijection. 

 ab

Definition 68 If M =

cd

∈ M2,2 (R), then we define det(M ) = ad − cb.

Sorite 11 For M, N ∈ M2,2 (R), we have det(M N ) = det(M )det(N ). − → The map M : R2 → R2 is a bijection iff det(M ) ∈ R∗ . In that case, the inverse − →−1 is defined by the matrix M     d −b 1 d −b det(M ) det(M ) −1 . = M = a −c det(M ) −c a det(M ) det(M )

√

Example 29 

0 −1 1 0

−1

 =

 0 1 , −1 0



 −1  1 −1 11 = . 0 1 01

These matrices also have their determinants in Z∗ , which is essential for the existence of the inverse with integer coefficients.

26.3 Linear Equations This is just a minor addendum to demonstrate the benefit from matrix calculus for the management of classical equations. We are given a number of m equations with n unknowns xi and coefficients bij in a commutative ring R: a1 = b11 x1 + b12 x2 + . . . b1n xn a2 = b21 x1 + b22 x2 + . . . b2n xn ... am = bm1 x1 + bm2 x2 + . . . bmn xn

This configuration can be restated using matrices. We have three matrices A = (ai ) ∈ Mm,1 (R), B = (bij ) ∈ Mm,n (R), X = (xj ) ∈ Mn,1 (R) and consider the equation of matrices A = BX. In general, one may assume that m = n and B is invertible, so that the solution is X = B −1 A.

224

√

26 Matrices

Example 30 Let us look at a simple example of a linear equation 3 = 5x1 − 2x2 −2 = 3x1 + 6x2

This defines the matrix equation   3 −2

 =

5 −2



 

3 6

x1

.

.

x2

We have 

−1

5 −2 3 6

1 = 36



 6 2 −3 5

 =

2 6 36 36 −3 5 36 36

 





 =

1 1 6 18 1 5 −12 36

Therefore we get 

 x1 x2

 =

1 1 6 18 5 1 −12 36

3 . −2

 =

7 18 −19 36

 .

 .

27 Modules

Summary. Many core structures in algebra are richer than groups but poorer than rings. For example, an ideal I ⊂ R in a commutative ring is an additive subgroup, but not a ring because it has no 1 in general. However, one may multiply elements of I with any ring elements. Also, the set Mm,n (R) is an additive group, but not a ring for n = m. Its structure as a cartesian product ring Rmn is rarely considered. But again, one may multiply a matrix by a “scalar” from R. These structures remind us of vector calculus in high school. This is what we now want to investigate for the sake of music theory. The structure of this type is called a “module”, and we want to give a short and very incomplete account of the theory of modules, which plays a major role in mathematical music theory. –Σ– Definition 69 Given a commutative ring R, an R-module is a triple (R, M, ∗), where M is a commutative group (with additive operation) and ∗ : R × M → M is the scalar multiplication such that (i) (homogeneity) r ∗ (s ∗ m) = (rs) ∗ m for all r, s ∈ R and m ∈ M , (ii) (distributivity) (r + s) ∗ m = r ∗ m + s ∗ m and r ∗ (m + n) = r ∗ m + r ∗ n for all r, s ∈ R, m, n ∈ M . The elements of M are usually called vectors. For two modules (R, M, ∗M ), (R, N, ∗N ), an R-linear homomorphism f : M → N is a homomorphism of additive groups such that f (r∗M m) = r∗N f (m) for all r ∈ R, m ∈ M . The set of R-linear homomorphisms f : M → N is denoted by ModR (M, N ). Clearly, linear homomorphisms can be composed composed in much the same way as group homomorphisms, and the identity IdM is R-linear. √

Example 31 The simplest construction of R-modules is by cartesian products of R. Given a natural number n, we consider the cartesian product group © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_27

225

226

27 Modules

Rn , and the scalar multiplication is defined component-wise: If r ∈ R and (ri )i ∈ Rn , then one defines r ∗ (ri )i = (rri )i . This is the R-module structure ∼ that we defined in Chapter 26.1 for matrix sets Mm,n (R) → Rmn . For this − → structure, the maps M : Rn → Rm become R-linear homomorphisms. This − → means that the map M → M is functorial not only into Set, but even into ModR , the “category” of R-modules and R-linear homomorphisms. We shall flesh out the concept of a category in Chapter 29, but here it is simply the system of R-modules together with their R-linear maps that can be composed and have identities. If M is an R-module and S ⊂ M any subset, there is a minimal submodule N ⊂ M such that S ⊂ N , it is called the module generated by S and denoted by S. This one is defined as the intersection of all submodules of M that contain S. It is the  set of all linear combinations of elements of S, i.e., of elements of the form i ri si where ri ∈ R and si ∈ S. If S = ∅, we defineS = (0). If (Mi )i∈I is a family i Mi is the set of R-modules, their direct sum of all families (mi )i ∈ i Mi that have mi = 0 for all but a finite number of indices i. Addition is defined factor-wise, (mi )i + (ni )i = (mi + ni )i , and (rmi )i . For each index j we have an R-linear scalar multiplication by r(mi )i = : surjective homomorphism pr i i Mi  Mj : (mi )i → mj as well as an  injection inj : Mj  i Mi that sends m ∈ Mj to the family (mi )i with mj = m and mi = 0 for i = j. If Mi = M , the same module M for all indices i, we write M ⊕I for their direct sum. √

Example 32 Every commutative group G can be viewed as being a Zmodule, the scalar product z ∗ g being defined as the z-fold sum z ∗ g = g + g + g + . . . g, z times for z ≥ 0, and z ∗ g = −g − g − g − . . . g, |z| times for z < 0. Homomorphisms of commutative groups then become Z-linear module homomorphisms.

If we have a subring R ⊂ S of commutative ring S, then S as a commutative group, together with the ring multiplication by R-elements, turns S into an R-module that one denotes by R S. For example, Q R is a Q-module that is very important in music theory. Or R C is an R-module that is identified with R2 , while conjugation on C is a R-linear automorphism of R C.

ˇ “*

Example 61 A common module in music theory is REHLD , the R-module of functions f : {E, H, L, D} → R, where E means symbolic onset, H means symbolic pitch, L means symbolic loudness, and D means symbolic duration1 . This module is isomorphic to R4 , and its vectors are the note events f = (fE , fH , fL , fD ) with onset fE in units of quarter notes ♩, pitch fH in units of semitones, loudness fL in units of cents Ct, and duration fD in units of quarter notes ♩. For performance, one accordingly uses the module Rehld of 1

E stands for Einsatzzeit, H for Höhe, L for Lautstärke, D for Dauer, all German words.

27 Modules

227

functions f : {e, h, l, d} → R, where the four coordinates now represent physical onset fe (in units of seconds sec), physical pitch fh (in units of Hertz Hz, representing the logarithm of frequency), physical loudness fl (in units of decibel dB), and physical duration fd (in units of seconds). For more sophisticated investigations involving glissandi and crescendi, one works in the larger spaces REHLDGC , Rehldgc , of functions f : {E, H, L, D, G, C} → R, f : {e, h, l, d, g, c} → R whose vectors also have a glissando and crescendo component fG , fg , fC , fc , respectively. We shall come back to these spaces in Chapter 32.

ˇ “*

Example 62 In Section 21.1, we have dealt with the isomorphism ∼

t : Id × (−1) ◦ (pr3 , pr4 ) : Z12 → T3×4 . This is an isomorphism of Z-modules, but not of rings.

ˇ “*

Example 63 This example is crucial for all tuning investigations. We shall use it in Chapter 28. We saw in Musical Example 58 that Western tuning systems use the frequency formula f (o, q, t) = f0 2o 3q 5t , where o, q, t ∈ Q. We learned in that example that the map Q3 → R : (o, q, t) → f (o, q, t) is injective. Since the logarithm is also an injective function, the map Euler : Q3 → R : (o, q, t) → o log10 (2) + q log10 (3) + t log10 (5) = log10 (f (o, q, t)) − log10 (f0 ) is also injective (the basis 10 is irrelevant, it works with any basis). But this map is also Q-linear if we take the Q-module structure Q R of R. This means that we have an identification of pitch (modulo a basic pitch f0 which we ignore for the time being) with triples of rational numbers. It is crucial here to recognize that the same function with real numbers Euler : R3 → R : (o, q, t) → o log10 (2) + q log10 (3) + t log10 (5) would not be injective. In fact, we have 0 = Euler(0, 0, 0) = Euler(− log(3), log(2), 0). Therefore, the theory of tuning includes the Q-linear injection Euler : Q3 →

QR

that we call the Euler embedding in honor of the great mathematician Leonhard Euler, who was the first to define tuning systems in terms of logarithms. The image Q-module Im(Euler) =: EulerSpace is called the Euler space. We shall discuss it in Chapter 28. The set ModR (M, N ) of R-linear homomorphisms f : M → N is an Rmodule by these operations: For f, g ∈ ModR (M, N ), f + g : M → N : m → f (m) + g(m), and for r ∈ R, rf : M → N : m → r(f (m)).

228

27 Modules

Proposition 24 (Universal Property of Direct Sums) For a family (Mi )i of R-modules and a R-module N , there is an isomorphism of R-modules   ∼ Mi , N ) → ModR (Mi , N ) p : ModR ( i

 It is defined by mapping the homomorphism f : i Mi → N to the family of homomorphisms (f ◦ ini ), where ini is the injection defined above. Proposition 25 If m, n > 0 are natural numbers, then we have an isomorphism ∼ M (?) : ModR (Rn , Rm ) → Mm,n (R) of R-modules. It is defined as sending the homomorphism f to the matrix M (f ) with M (f )i,j = f (ej )i , where ej = (0, . . . 1, 0, . . .) is the vector that vanishes for k = j and has coordinate 1 at position j, and where f (ej )i is the ith coordinate of f (ej ) ∈ Rm . Proof 16 We use the universal property of direct sums for the situation Rn =  m i R. A homomorphism f is equivalent to the family (fj ) ∈ M odR (R, R ). But the factors fj are precisely the functions f (ej ), as the homomorphisms g : R → Rm are given by their values on 1R . In particular, if n = m = 1, we ∼ have the isomorphism ModR (R, R) → R that is defined by f → f (1R ).

ˇ “*

Example 64 If we work in EulerSpace, we may consider Q-linear functions f : EulerSpace → Q from the Q-module to the coefficient ring Q (interpreted as Q-module too). The general context of this situation is the space ModR (M, R) of so-called linear forms on an R-module M . A very important such form on ModQ (EulerSpace, Q) is the form Γ0 (o, q, t) = 1.o + 2.q + 4.t = (2 − 1).o + (3 − 1).q + (5 − 1).t. It is this form that essentially defines what Euler had called gradus suavitatis, his quantitative measure for the degree of consonance of an interval in EulerSpace. We see that he uses the three prime numbers 2, 3, 5 to define his linear form. Euler proposed that the function 1+Γ01(o,q,t) would measure the consonance of the interval o = o1 −o2 , q = q1 −q2 , t = t1 −t2 between pitches (o1 , q1 , t1 ), (o2 , q2 , t2 ). Figure 27.1 shows the gradus function values (the 10-fold) for frequency ratios within the just-tuned octave.

ˇ “*

Example 65 The isomorphism of modules ∼

ModZ (Z2 , Z2 ) → M2,2 (Z) is a central structure in music theory of just tuning. We shall deal with this one in Chapter 28.

27 Modules

229

10 / Γ

10

5

0

1 1

16 15

9 8

6 5

5 4

4 3

45 32

3 2

8 5

5 3

16 9

15 8

interval

Fig. 27.1. The gradus suavitatis function values (the 10-fold) for frequency ratios within the just-tuned octave.

The group of R-linear automorphisms of an R-module M is denoted by GL(M ). By proposition 25, if M = Rn , we have an isomorphism of groups ∼ GL(Rn ) → Mn,n (R)∗ , and we have the notation Mn,n (R)∗ = GLn (R). Similarly to quotient groups and rings, there are also quotient R-modules. Their definition is straightforward: Given a R-module M and a submodule N , the quotient module M/N is the quotient group, together with the scalar multiplication r ∗ (m + N ) = r ∗ m + N . This is well defined since for another representative m + n of the coset m + N , we have r ∗ (m + n + N ) = r ∗ m + r ∗ n + N = r ∗ m + N . Also, the group-theoretical kernel Ker(f ) of an R-linear homomorphism f : M → N is a submodule. Theorem 22 If f : M → L is an R-linear homomorphism, then we have the following commutative diagram, which represents f by its kernel and image: M

f

- L 6

f

proj ∼f

→

--

? ? M/L

inj

6 Im(f )

230

27 Modules

where proj is the canonical surjection, inj is the embedding of Im(f ), and f is the isomorphism of R-modules that sends the coset m + Ker(f ) to f (m).

27.1 Affine Homomorphisms Very often, linear homomorphisms are too narrow as they map zero to zero. One also wants to be able to move zero to any element of the codomain module but keep the linear properties. These generalized homomorphisms are called affine homomorphisms. Here is the precise definition. Definition 70 If M, N are two R-modules, a set function f : M → N is said to be an R-affine homomorphism if there is t ∈ N such that f = T t ◦ f0 , where f0 ∈ HomR (M, N ) and T t : N → N is the translation T t (n) = t + n by t. Proposition 26 Affine homomorphisms f = T t ◦ f0 : M → N, g = T s ◦ g0 : N → L can be composed as set functions and again yield an R-affine homomorphism g ◦ f = T s+g0 (t) ◦ (g0 ◦ f0 ). The R-affine homomorphism f = T t ◦f0 is a bijection (also called isomorphism) iff f0 is an isomorphism of R-modules, and then we have −1

f −1 = T −f0

(t) −1 f0 .

The set of R-affine homomorphisms f : M → N is denoted by AffR (M, N ), and we have ModR (M, N ) ⊂ AffR (M, N ).

ˇ “*

Example 66 In Musical Example 20, we explored the group T I of functions on Z of the form T t ◦ ±1. These are examples of affine homomorphisms of the Z-module Z. In Section 21.1.1, we also discussed the group T Z∗12 , which is a group of affine automorphisms of the Z12 -module Z12 , and an example of a group of affine automorphisms of the R-module R defined by the invertible factors r ∈ R∗ for the affine homomorphisms T t ◦ r. We often identify the matrix group Mm,n (R) with the corresponding group of R-linear homomorphisms M → N and then write T Mm,n (R) for the set of − → affine homomorphisms f : Rn → Rm defined by f = T t ◦ M , M ∈ Mm,n (R), and we also write f = T t ◦ M or even f = T t M if no confusion is likely. √

Exercise 29 Prove that for an R-affine homomorphism f = T t ◦ f0 , its translation element t and its R-linear homomorphisms f0 are uniquely determined by f .

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Example 67 In music theory, most morphisms are not linear, but affine. For example, the pitch function P itch(o, q, t) = log(f0 ) + o log(2) + q log(3) +

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t log(5) is affine, with transposition log(f0 ) and linear part o log(2) + q log(3) + t log(5). Also, Euler’s gradus suavitatis function is Γ (o, q, t) = 1 + Γ0 (o, q, t), with transposition part 1.

ˇ “* Example 68 Let us discuss a famous compositional strategy by Pierre Boulez, which he applied in his composition Structures pour deux pianos. Refer to [84, Chapter 25] for a thorough discussion. Boulez composes this work in the serial style—that is, he not only considers dodecaphonic series for pitch but also for duration, attack, and loudness. He starts with one such series for each parameter: SP itch , SDuration , SAttack , SLoudness . In his procedure, Boulez uses different transformations of the pitch series. For example, the retrograde series of transpositions of SP itch = (SP itch,0 , SP itch,1 , . . . SP itch,11 ) within the pitch class module Z12 . But he also wants to apply these transformations to the other parameter series. This causes a serious conceptual problem of serialism since, different from pitch classes, there are no naturally selected classes of duration, attack, or loudness, let alone natural transformations on these spaces. Boulez applies a very mathematical trick to solve the problem of carrying over pitch-class transformations to the other parameters. Denote by ei = (0, 0, . . . 1, 0 . . . 0), i = 1, 2, . . . 11 the vector in Z11 that has zeros except for the ith coordinate, which is 1. To begin with, the pitch series SP itch is represented by a Z-affine homomorphism S P itch : Z11 → Z12 which sends the vector ei to SP itch,i for i = 1, . . . 11 and 0 to SP itch,0 . This can be done by the homomorphism S P itch = T SP itch,0 Q, where Q(ei ) = SP itch,i − SP itch,0 is the linear part. If we apply a permutation p of the pitch classes, we get a new series p ◦ S P itch : Z11 → Z12 . But this series can also be obtained by first permuting the vectors ei , i = 1, 2, . . . 11 and 0 and the applying the original ∼ series S P itch , i.e., by an Z-affine isomorphism P : Z11 → Z11 defined by the permutation of ei , i = 1, 2, . . . 11 and 0, such that p ◦ S P itch = S P itch ◦ P. Boulez now uses this formula to transform his other series SDuration , SAttack , SLoudness , which he rewrites as affine homomorphisms S Duration , S Attack , S Loudness by prepending P , i.e., producing S Duration ◦ P, S Attack ◦ P, S Loudness ◦ P . Using this smart method, he avoids opening any discussion about musically reasonable symmetries on these other parameter spaces. Of course, Boulez was not using these mathematical methods explicitly, but his procedure was exactly the one that the theory of modules and affine homomorphisms describes. The idea to rephrase a series in Z12 as an affine homomorphism Z11 → Z12 is a very powerful new method in mathematical music theory, called “addressed objects.” It will be discussed in Chapter 29.

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27.2 Free Modules and Vector Spaces An R-module M is said to be free if it is isomorphic to a direct sum R⊕I . The index set I is essentially uniquely determined, which means that for a finite set ∼ ∼ I → n, Rn → Rm iff n = m; the cardinality of I is unique. This number is called the dimension of the module, denoted by dim(M ). In general, modules are not free, but for fields R = K of scalars, this is always true. A K-module for a field K is called a vector space over the scalars K. We want to describe sub-vector spaces of K-vector spaces, in particular kernels of linear homomorphisms. Definition 71 A sequence of elements  (xi )i = (x1 , x2 , . . . xk ) of a K-vector space M is linearly independent if i ri xi = 0 implies ri = 0 for all i = 1, 2, . . . k. If (xi )i is not linearly independent, it is called linearly dependent. In other words, (xi )i is linearly independent iff the K-linear map Rn → M : (ri )i → i ri xi is injective, i.e., its kernel is trivial. A sequence (xi )i of vectors generates M iff the homomorphism Rn → M :  (ri )i → i ri xi is surjective. A sequence (xi )i of linearly independent vectors that generate Mis called a basis of M . In other words, the homomorphism Rn → M : (ri )i → i ri xi is bijective. It follows that if (xi )i is linearly independent, then xi = 0 and xi = xj for i = j. Also then (xπ(i) )i is linearly independent for any permutation π ∈ Sn . √

Example 33 In the free K-vector space K n , the sequence (ei )i=1,...n of vectors ei = (0, . . . 1, 0, . . . 0) with zeros except a 1K at position i is a basis, the so-called canonical basis.  In K n , the sequence (fi )i=1,...n with fi = (1, 1, . . . 1, 0, . . . 0) = j=1,...i ej is a basis. √ Example 69 In Q R, the vectors √ 1, 2 are linearly independent. This means that there is no equation r + s 2 = √ 0 with rational coefficients r, s. In fact, such an equation would imply that 2 is rational, which is wrong. Therefore, the 12-tempered tritone interval is irrational, and a fortiori the 12-tempered semitone is so, too. We know this, but it is remarkable to learn that this fact is a statement about linear independence in Q R. In Q R, the vectors log(2), log(3), log(5) are also linearly independent. To prove this statement, recall that we used the unique representation of natural numbers as products of prime numbers. We have shown in Musical Example 63 that o. log(2) + q. log(3) + t log(5) = 0 iff o = q = t = 0 for rational coefficients o, q, t. This mathematical independence is in perfect correspondence with the musical understanding of the three intervals octave, fifth, and third, related to 2/1, 3/2, 5/4, respectively, in just tuning. It is remarkable that what music

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theorists had understood musically was confirmed many centuries later on the level of mathematics. Proposition 27 If a K-vector space M is finitely generated, i.e., there is a finite sequence (xi )i of generators, then M has a finite basis that is a subsequent of (xi )i . indeProof 17 The proof goes by induction. Suppose that (xi )i are linearly  pendent, then we are done. Otherwise, there is an equation 0 = i ri xi with one coefficient rj = 0. We may supposethis is the first one (after a permutation of indices). Then we have x1 = −1 i>1 ri xi , and the shorter subsequence r1 (xi )1 0 a natural number, and this collection is a set! The morphisms En → Em are the matrices in Mm,n (R). Composition of morphisms is the product of matrices, and the identity matrices are also the identical morphisms. This category is particularly nice since it has no special objects; the objects are just the identity morphisms. Category theory can in fact be defined without distinguishing objects from morphisms. This is the modern point of view: Everything is a morphism. And matrices are a beautiful and simple example of such a completely “morphic” category. ModR is the category whose objects are the R-modules for a commutative ring R, while the morphisms are the R-linear homomorphisms. AffR is the category whose objects are the R-modules for a commutative ring R, while the morphisms are the R-affine homomorphisms. ModR is a subcategory of AffR , ModR ⊂ AffR ; the objects are the same, but ModR has only the r-linear homomorphism as morphism sets, a subsystem of all R-affine homomorphisms. If C is a category, its opposite category C opp is defined as follows: We set opp Ob(C ) = Ob(C) and C opp (X, Y ) = C(Y, X). The composition is the same as for C, but in reversed order, i.e., for f ∈ C opp (X, Y ), g ∈ C opp (Y, Z) we set f ◦C opp g = g ◦C f .

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Example 71 One of the most important categories in mathematical music theory is the category LocR of local compositions in R-modules. The objects are pairs (K, M ) of a subset K = ∅ of an R-module M . The typical examples are sets K of notes in a score, where the score is thought to be an R-module, such as REHLD (see Musical Example 61). If (K, M ), (L, N ) are two local compositions, a morphism f : (K, M ) → (L, N ) is a set map f : K → L such that there is an affine homomorphism F : M → N with F |K = f . We have seen many such morphisms, e.g., symmetries, such as inversions or retrogrades that

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map a local composition to its symmetric image. Composition of morphisms is the set-theoretic composition, and the identity is the set-theoretical identity. The theory of local compositions has many important results, in particular the list of all isomorphism classes of local compositions (K, Z12 ) in LocZ , see [75, Appendix L.1]. These are in fact the chord classes that we have already discussed earlier. But there is also a complete classification of musical motives in 12-periodic onset time and pitch, i.e., (K, Z212 ) for card(K) = 1, 2, 3, 4, see [75, Appendix M.1-M.4], and three element motives in Z5 ⊕ Z12 , see [75, Appendix M.5]. It is interesting that the number of the latter classes is 45, while the number of classes of three element motives in Z212 is only 26, even though it is larger. This is due to the fact that the number of prime numbers is larger in Z5 ⊕ Z12 as compared to Z212 . This is one of the reasons why improvising in 5/8 meters is much more difficult then improvising in 3/4, 4/4, 6/8 meters. When category theory was invented in 1945 by Samuel Eilenberg and Sounders Mac Lane, their focus was not on particular categories, but on “morphisms” between different categories. They are defined as follows. Definition 73 Let C, D be two categories. A (covariant) functor F : C → D is a function that associates with each object X ∈ Ob(C) an object F (X) ∈ Ob(D), and for each morphism f : X → Y a morphism F (f ) : F (X) → F (Y ) such that F (IdX ) = IdF (X) and F (g ◦ f ) = F (g) ◦ F (f ), whenever g ◦ f exists. A functor F : C → Dopp is called a contravariant functor. √

Example 36 For example, the forgetful functor Grp → Set is the functor that associates with each group G the underlying set G and with every group homomorphism f : G → H the underlying set map. A similar functor can be defined for ModR instead of Grp. If C = M(R), D = ModR , the functor which associates with each matrix object En the R-module Rn and with each matrix M : En → Em the module − → homomorphism M was defined in Section 26.1.

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Example 72 There is a useful functor R? : LocR → ModR

defined as follows. If (K, M ) is a local composition, take the module RK := {x − x0 |x ∈ K} generated by the differences x − x0 with respect to a chosen element x0 ∈ K. It is obvious that this module does not depend on the chosen x0 . Moreover, if we have a morphism f : (K, M ) → (L, N ) with an affine homomorphism F : M → N such that f = F |K, then take the R-linear homomorphism F0 |RK : RK → RL with the linear part F0 of F as image of f . Is this reasonable? Take any difference x − x0 in RK. Then F0 (x − x0 ) = F0 (x) − F0 (x0 ) = F (x) − F (x0 ) = f (x) − f (x0 ), so the linear homomorphism F0 |RK is only a function of f and maps RK into RL. This functor associates

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with every local composition a module over the composition’s ring R. This helps us verify the necessary conditions for local compositions to be isomorphic, since if they are so, then their modules must be isomorphic, too. For example, the triads {0, 3, 6}, {0, 4, 7} ⊂ Z12 cannot be isomorphic since Z{0, 3, 6} = 3Z12 , while Z{0, 4, 7} = Z12 . Definition 74 If F, G : C → D are two functors, a natural transformation n : F → G is a collection of morphisms n(X) : F (X) → G(X) for all objects X in C such that the following diagram commutes for all morphisms f : X → Y in C. n(X) F (X) −−−−→ G(X) ⏐ ⏐ ⏐G(f ) ⏐ F (f )  n(Y )

F (Y ) −−−−→ G(Y ) Natural transformations n : F → G, m : G → H can be composed to a natural transformation m ◦ n that maps X to m(X) ◦ n(X). This composition is associative, and there is the identical natural transformation IdF : F → F for every functor F : C → D. This means that we have the category of natural transformations Nat(C, D) for every couple of categories C, D. For every category C, we have the category C @ (@ for “address,” see below) of contravariant functors F : C → Setopp as objects and natural transformations as morphisms. For every object X of C, we define a contravariant functor @X : C → Setopp by @X(Y ) = C(Y, X). It maps a morphism f : Y → Z to a set map @X(f ) : C(Z, X) → C(Y, X) : g → g ◦ f . If h : X → Y is a morphism in C, the natural transformation @h : @X → @Y maps C(Z, X) to C(Z, Y ) via k → h ◦ k. This is a functor Y : C → C @ . It is the Yoneda functor that was defined by Japanese computer scientist Nobuo Yoneda in 1956. Its relevance is the following lemma: Theorem 23 (Yoneda’s Lemma) If X, Y are two objects in a category C, ∼ then the Yoneda functor defines a bijection C(X, Y ) → C @ (@X, @Y ), and in ∼ ∼ particular, X → Y iff @X → @Y . The proof of this lemma is quite technical, but not difficult, so we omit it. The significance of the lemma is that objects X in general categories can be represented by their contraviariant functors @X, and this means that an abstract object X can be completely described by the functor @X, i.e., by the system of all morphisms f : Z → X. This is what we call the Yoneda philosophy: Objects can be understood by the system of all “perspectives” f : Z → X from any “address” Z.

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29.1 The Yoneda Philosophy The Yoneda lemma has caused a revolution in mathematics, since the replacement of usually abstract objects X in categories by their functor @X changed the way we can look at mathematical structures. Instead of X one now could look at a system of Z-addressed “perspectives” f : Z → X, and X would be completely understood from such a system. To understand an object from its different perspectives is quite common in the visual arts. When you look at a sculpture, you may observe the front and also walk around it, changing your perspective. This method was also relevant to the history of mountain climbing. The famous Swiss mountain Matterhorn was thought to be too difficult to be climbed. This was due to the usual perspective of the mountain when you look from the nearby village of Zermatt, see Figure 29.2, right part. The mountain looks very steep. But one day, Edward Whymper, an English mountaineer, changed the perspective and observed the Matterhorn from the Theodul glacier (left part of Figure 29.2). From this new perspective, the slope of the north-east fin appeared much less dangerous than from the traditional perspective. This motivated Whimper to climb the mountain on July 14, 1865, an adventure that was successful but caused the death of four of his partners during the descent.

Fig. 29.2. The Matterhorn from the usual perspective (right) and from the Theodul glacier (left).

Another application of the Yoneda philosophy is recognized while performing and interpreting a work of art, a poem, a musical composition. Each such rendition can be understood as an interpretation of the work, i.e., as a perspective on that object. The artist produces a performance that expresses how the interpreter views the work. According to the work’s complexity, no single perspective will ever reveal the complete message of the work. Art lovers generally agree that to understand a work, one must take the sum of all interpretive performances, including the bad ones, which can be helpful too. In fact, you then know how not to interpret the work, which is an awkward point of view. For example, Glenn Gould’s paralyzing interpretation of Beethoven’s Appassionata Sonata op.57 is, when compared to Vladimir Horowitz’s perfor-

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mance, a deep lesson about the gestural dimension in Beethoven’s approach to music, see Figure 29.3.

Fig. 29.3. Glenn Gould (left) interprets Beethoven without addressing his gestural creativity, as opposed to Vladimir Horowitz (right), who fully realized this dimension.

In more mathematical terms, if we view a composition in a score as a local composition (K, M ) in an adequate module M of musical parameters, the Yoneda Lemma tells us that (K, M ) is completely understood when considering all morphisms (L, N ) → (K, M ). In particular, we have to consider all small parts (L, M ) of (K, M ), such as motives, chords, rhythms, etc. This is exactly what we all do when trying to understand a composition: We look at the system of parts and glue them together into a global picture of the work under consideration.

Part VII

Continuity and Calculus

30 Continuity

Summary. Despite the rich algebraic formalism of monoids, groups, rings, and modules, we lack a type of analysis that does not compare objects by their transformational relations, such as symmetries or module homomorphisms, but by their similarity—referring to the paradigm of deformation. This type of relationship is what topology, the mathematics of continuity, is about. –Σ– Modern mathematics of the 20th century was above all characterized by advances in topology. Even category theory was invented to deal with functors on categories of topological nature. The great mathematician Yuri Manin predicted that future mathematics will have its foundations in topological, elastic, and deformable objects, not by abstract entities such as sets and categories. In music theory, the topological paradigm becomes increasingly important, in particular because gestures are a topological concept and because in performance, continuity of movements is crucial and cannot be boiled down to abstract algebraic structures. This chapter and the two following on differentiability and gestures are somewhat more advanced in their mathematical style, but we believe it is important to have a rough idea about these new mathematical tools. Definition 75 A topology T on a set X, in short: a topological space X, is a collection T ⊂ 2X of subsets of X, called open sets of the topology, such that (i) ∅, X ∈ T ,

(ii) for any family (Oi )i of open sets, the union i O is in T ,  (iii) for any finite, non-empty family (Oi )i of open sets, their intersection i Oi is in T . For a topology T on X, a subset C ⊂ X is closed iff X − C is open. One may also state the topological axioms using closed sets. This means that one requires that (i) ∅, X are closed, © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_30

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Fig. 30.1. The musical glissando on a piano and a violin are different. On a violin, the pitch changes smoothly, while on the piano we have a discrete set of pitches, similar to walking down steps as opposed to skiing down a mountain.

 (ii) for any non-empty family (Oi )i of closed sets, the intersection i O is closed,

(iii) for any finite family (Oi )i of closed sets, their union i Oi is closed. If x ∈ X is an element of a topological space X, an open neighborhood of x is an open set that contains x. √

Example 37 The most important example is X = R, where the open sets O ⊂ R are the subsets such that for every x ∈ O, there is an open interval ]a, b[:= {x|a < x < b} with x ∈]a, b[⊂ O. In particular, open intervals are open sets, but closed intervals [a, b] = {x|x ∈ R AND a ≤ x ≤ b} are not. See Figure 30.2, left part, for an open interval. If X, Y are topological spaces, the cartesian product topology is the topology on X × Y where the open sets are those O ⊂ X × Y such that for every x ∈ O, there is a pair of open sets U ⊂ X, V ⊂ Y such that x ∈ U × V ⊂ O. A similar definition holds for the cartesian product topology of a finite family (Xi )i of topological spaces Xi . The usual topology on Rn is the cartesian product topology induced from the above topology of R. The open sets are those subsets O ⊂ Rn that contain for every point x = (xi ) ∈ O a cartesian product ]a1 , b1 [× . . .]an , bn [ of intervals with xi ∈]ai , bi [ for all i = 1, . . . n. See Figure 30.2, right part, for an open set in R2 . If X is a topological space, a subset Y ⊂ X is turned into a topological space by assigning the open sets U ⊂ Y to those sets of shape U = Y ∩ O for

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Fig. 30.2. Open sets in R and R2 . In R2 , each element x of the open set U is in a rectangular open neighborhood that is contained in U.

an open set O ⊂ X of the comprising space. This topology is called the relative topology. Definition 76 If S is a topology on X and T is a topology on Y , then a set function f : X → Y is said to be continuous if the inverse image f −1 (O) of every open set in Y is open in X. This means that for every element x ∈ X and every open neighborhood V of f (x), there is an open neighborhood W of x such that f (W ) ⊂ V . Clearly, the composition g ◦ f of two continuous maps f : X → Y, g : Y → Z is continuous, and the identity IdX on a topological space is continuous. This means that we have a category Top of topological spaces with morphism sets Top(X, Y ) = {f |f : X → Y continuous}. Isomorphisms of topological spaces are called homeomorphisms. Example 38 For the cartesian product topology i=1,...n Xi , the projec tion prj : i Xi → Xj is continuous for every index j. In fact, for a neighborhood V of prj (x), the cartesian product W = X1 ×. . . Xj−1 ×V ×Xj+1 ×. . . Xn projects into V . Many important functions that we introduced earlier are continuous. For example, polynomial functions R → R : x → P (x) for P ∈ R[X], or affine functions Rn → Rm , or conjugation for the cartesian product topology on C in its identification with R2 . All arithmetic operations on R, C, such as addition or multiplication, and the exponential function x → ax , a > 0 on R, and the logarithm x → loga (x), a > 0 on R+ are continuous. Also the classic trigonometric functions x → cos(x), sin(x) on R are continuous. A traditional restatement of the continuity of a function f : R → R reads as follows: For every x ∈ R, and every ε > 0, there is δ > 0 such that whenever |y − x| < δ, then |f (y) − f (x)| < ε. This follows immediately from the definition of open sets in R as unions of interval neighborhoods of their points. √

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Example 73 In musical performance, continuous functions are mandatory for good rendition. If we want to produce a crescendo between onset time t0

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and onset time t1 from a p to a ff , we have to increase loudness as a continuous function of time, not as a discontinuous step function, see Figure 30.3. Often, the function is defined by a polynomial, which is a frequently used continuous function, especially in performance software, such as RUBATO ’s Performance rubette [75, Section 41.1]. Other performance functions are also continuous, such as tempo changes (see Chapter 32), glissandi, or intonations, i.e., local changes of tuning, especially for strings or singers.

Fig. 30.3. Stepwise and continuous increase of loudness.

Definition 77 A topological space X is compact iff any covering family (Xi )i ,

i.e., i Xi = X, has a finite covering subfamily k=1,...m Xik = X. For example, a cartesian product of closed intervals [ai , bi ], i = 1, . . . n in Rn is compact.

30.1 Generators for Topologies Similar to monoids, groups, ideals, and modules, there are for continuity some standard techniques to “generate” a topology from some “generating” open sets. If we are given a set S ⊂ 2X , we may consider the intersection S of all topologies T on X that contain S as a subset. There is always at least one such a topology, namely, the discrete topology 2X . This intersection is called the topology generated by S. A subbase S of a topology T on X is an set S ⊂ 2X such that S = T , i.e., a generating set. This topology consists of all unions of finite intersections of members of S. A base for a topology is a generator such that every open set of the topology is a union of members of that generator, i.e., the finite intersections are not necessary for a base. For example, the open intervals in R are a base for the usual topology on R. A frequent construction of a base is given when a space has a metrical structure, i.e., a distance function between pairs of elements of that space. More precisely, we have the concept of Euclidean distance on Rn .

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Definition 78 The Euclidean distance function is the map Rn × Rn → R  2 defined by d(x, y) = i (xi − yi ) . We have these characteristic properties for a distance function: (i) d(x, y) = d(y, x) ≥ 0, (ii) d(x, y) = 0 iff x = y, and (iii) (the triangle inequality) d(x, y) + d(y, z) ≥ d(x, z). This function defines a topological base on Rn by open balls around points x ∈ Rn defined by Be (x) := {y|y ∈ Rn AND d(x, y) < e} for any x ∈ Rn and radius e > 0. It can be shown that the topology on Rn defined by open balls coincides with the product topology defined above.

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Example 74 Distances in Rn play a role in the theory of motivic similarity. In this context, we consider a musical motive of k notes or k-motive to be a sequence m = (m1 , m2 , . . . mk ) of elements (the notes) mi ∈ REHLD in a symbolic space REHLD of onset, pitch, duration, and loudness with real values. We suppose that the Euclidean distance is selected on REHLD . The set M otk of all k-motives is then associated with the space (REHLD )k . On this space, we define a distance between motives m, n by d(m, n) := maxi (d(mi , ni )). It is easy to verify the axioms of a distance (Definition 78) for M otk . With this distance among motives, one may consider motivic similarity either directly via a measurement of distances or by investigation of balls Be (m) of k-motives around a given k-motive m. Melodic similarity compares motives by their topological position, not by symmetries that could transform motives into each other. This splits musical paradigmatic analysis into two quite different subfields: the “para” part—topological similarity, “side by side,” versus the “deigmatic” part—“pointing to”: transformational relations. See [75, Chapter 22] for more details about the now active research field of motivic topologies. The topological aspect was unfortunately ignored by music psychologist Christian von Ehrenfels when he characterized “gestalt” as something that has supersummativity, the whole is more than the sum of its parts, and transposability, you can transpose a gestalt without changing it. He explained his concept with the example of a melody. But he should have added that a gestalt is invariant under similarity, you may deform it slightly without changing it. For function sets Top(X, Y ) one considers the following set of subsets of functions. One selects a compact set C ⊂ X and an open set O ⊂ Y . The subset of the compact-open topology on Top(X, Y ) consists of all sets (C, O) = {f |f ∈ Top(X, Y ) AND f (C) ⊂ O}. The compact-open topology has the universal property that for any three topological spaces X, Y, Z, where Y is locally compact Hausdorff, the composition map Top(X, Y ) × Top(Y, Z) → Top(X, Z)

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is continuous for the compact-open topologies on these function spaces and the cartesian product topology on the left. A space is Hausdorff iff any two different points have disjoint open neighborhoods. It is locally compact if every neighborhood of any point contains a smaller compact neighborhood of the point. For example, Rn is Hausdorff and locally compact. Compact-open topologies are crucial for mathematical gesture theory that will be discussed in Chapter 33.

30.2 Euler’s Substitution Theory As explained in Chapter 28, Leonhard Euler defined a consonance degree with his gradus suavitatis function. He needed a substitution in the human brain, which replaces an arbitrary music interval by a neighboring just interval, because his function only works for just intervals. Restating Euler’s theory, this boils down to starting at any pitch p ∈ Q R and then finding the next possible pitch in the just-tuning grid of EulerSpace defined by integer coordinates, i.e., points stemming from Z3 under the Euler injection. That Euler’s idea does not work follows from this topological consideration: In a topological space X, for any subset S ⊂ X, we may consider the  intersection S := S⊂C & C⊂X closed C of all closed subsets C ⊂ X that contain S. This is called the closure of S. It is the smallest closed subset of X containing S. A subset S ⊂ X is said to be dense iff S = X. For example, Q = R, the rationals are dense in R. This is easy to verify. In the other case, we would have a non-empty open set R − Q. But every such set contains an open non-empty interval, and any such open interval contains rational numbers, so this leads to a contradiction. The problem with Euler’s proposal is that the grid of just-tuning pitches Euler(Z3 ) is dense in Q R for the standard topology on the reals. See [75, Appendix A.2.3] for proof. This means that for any pitch x ∈ R, and for any open neighborhood ball Be (x), there are just-tuned pitches in Be (x). But there are also infinitely many such pitches since there are infinitely many mutually disjoint small balls Be/2(n+2) (x − e/2n ) in Be (x), and each such small ball contains just-tuned pitches. Therefore the selection of the nearest candidate is impossible, and Euler’s substitution cannot work for topological reasons.

31 Differentiability

Summary. Differentiability is stronger than continuity in that for a differentiable curve, we need to have a slope line at every point, i.e., the curve must not have corners. This chapter deals with this concept and its application to music. –Σ– For differentiability, we need to describe functions f : U → Rm , where U ⊂ Rn is an open set. Definition 79 Let x ∈ R and f : U − {x} → Rn be a function defined in a neighborhood U of x, except for x. We say that f converges to a vector z ∈ R iff for every ε > 0, there is a δ > 0 such that for |y − x| < δ, |f (y) − z| < . We then write “f (y) → z for y → x”.

Fig. 31.1. The slope line of the derivative 2x of function x2 at x = 1.

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Definition 80 Let f : U → Rm be a function. Then f is said to be differentiable at a point x ∈ U if there is a linear homomorphism D ∈ ModR (Rn , Rm ) (x)−D(y−x)| → 0 as y → x. The function such that the function Δ(y) = |f (y)−f |x−y| D is called the derivative of f at x and is denoted by Df (x). If D exists, it is uniquely determined. In the special case of n = m = 1, the derivative is represented by a number Df (x) ∈ R, the single coefficient in a 1 × 1-matrix that is df . The function identified with Df (x), and one often denotes this number by dx is said to be differentiable if it is differentiable in every point of its domain U . See Figure 31.1 for the derivative slope of f (x) = x2 at x = 1. If a function f is differentiable, then it is also continuous. But the contrary is false, as is seen for the function f : R → R : x → |x|, which is continuous but not differentiable in x = 0, see Figure 31.2.

Fig. 31.2. The absolute value function |x| is not differentiable for argument x = 0.

 Example 39 If f = P : R → R is a polynomial P (X) = i ai X i , then it  is differentiable and Df (x) = i>0 iai xi−1 . The R-affine functions T t ◦ M are differentiable, and we have Df (x) = M . The function sin(x) has D sin(x) = cos(x), the function cos(x) has D cos(x) = − sin(x). √

Proposition 28 If f, g : U → Rm are both differentiable at x, then we have D(f + g)(x) = Df (x) + Dg(x). Proposition 29 (Chain Rule) If f : U → Rm , and g : V → Rl with Im(f ) ⊂ V are both differentiable, then we have D(g ◦ f )(x) = Dg(f (x)) ◦ Df (x).

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ˇ “* Example 75 In music theory of performance, basic concepts are based on differentiation. A classic example is given by the concept of tempo. What is tempo? It is usually measured in M.M. ♩/min., i.e., Maelzel Metronome quarter notes per minute. So it relates physical time (minutes) and symbolic score time (♩). The situation here is that in any performance, there is a transformation from symbolic time to physical time. Figure 31.3 shows this functionality.

Fig. 31.3. Tempo relates to the function that sends symbolic score time to physical time. It is the inverse differential quotient of this function.

Tempo relates to the slope of this time function ℘E : E → e. It is defined by1 Tempo = T (E) =

1 1 . = D℘E de/dE

A similar formalism holds when we want to define intonation precisely. We have a function ℘H : H → h from symbolic pitch (in semitone units) to physical pitch (in Cents). The intonation function is defined similar to tempo by Intonation = S(H) =

1 1 . = D℘H dh/dH

This function (S stands for German “Stimmung,” intonation) is important because intonation can vary locally in quite dramatic ways—for example, for string or voice performance—much as tempo can vary locally, and it is called agogics or microtiming. See Chapter 32 for more aspects of these functions.

1

The symbol de/dE is a traditional writing of D℘E , the derivative then being called differential quotient.

32 Performance

Summary. Performance is understood to be the transformation of a symbolic musical object of notes—as represented in a score of Western tradition—to a physical object composed of sound events. Going beyond the common description of performance, we shall present a mathematical theory of this type of transformation. –Σ– This description is a huge abstraction since performance includes the performing artist, gestures, and real instruments, all being realized in a concert hall in front of an audience. But it is the approach that is traditionally taken when addressing “expressive performance research.” We want to address this abstraction here too, but we are aware that it is not the whole story. In the next chapter we shall also discuss the gestural aspect of performance, adding another aspect to a most complex total image. However, the present abstraction to notes and sound events can be helpful as an introduction when trying to piece together the whole picture. It will turn out that gestural performance can be built upon this note-oriented abstraction, in the sense of getting a feel for what is being played as sound events. This being said, let us provide a short summary of how expressive performance research is structured. To begin with, it deals with the structure theory. The action of performing notes by sound events must be clearly defined and ideally described in precise mathematical terms. This latter precision is not surprising since notes and sound events have been identified as being points in adequate parameter spaces, such as REHLD for notes and Rehld for sound events, see Musical Example 61. Once one has gained control over the structure of performance, the second part of the theory is the question of why a specific performance is chosen, in the sense that some message has to be expressed. Expressive performance is a semiotic action, a rhetorical endeavor of transmitting messages (of gestural, symbolic, or emotional nature) via instrumental interfaces to a more or less interested audience.

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We shall not deal here with the semiotic aspect of performance, but simply describe the most elementary ingredients of structure theory. A more complete description can be found in [75, Parts VIII-XII].

32.1 Mathematical and Musical Precision Expressive performance research was first received with much skepticism because it was believed, above all by traditional performance experts in musicologist circles, that mathematical and musical precision were incompatible. Or that some performance (machine-made or human) could be mathematically precise but musically invalid, and that differences in the quantitative performances could be irrelevant to the precision of the musical message. This turned out to be a misunderstanding due to a confusion between structural determinants and the semiotic contents. To put it simply: The relationship between form and content is a complex one. But it is not true that structural precision does not matter. It just doesn’t guarantee the transfer of a deeper content. Already Theodor Wiesengrund Adorno in his correspondence with Walter Benjamin [2] had recalled1 : “Walter Benjamin hat das Vermögen der Phantasie die Gabe, im unendlich Kleinen zu interpolieren definiert. Das bedeutet blitzhaft die wahre Interpretation. (...) Im dicht gewobenen Zusammenhang des Notentextes sind die minimalen Hohlräume zu entdecken, in denen sinnverleihende Interpretation ihre Zuflucht findet. (...) Das Medium künstlerischer Fantasie ist nicht ein Weniger an Genauigkeit, sondern das noch Genauere.” This wonderfully arcane text can be clarified if we recall that there is an exquisite science of infinite precision, namely differential and integral calculus. We take this reference as a starting point for a mathematically rigorous theory of musical performance.

32.2 Musical Notation for Performance Musical scores are not yet music: They indicate to performers the points to reach. Let us briefly discuss some performance-related components of Western scores. A classic Western score mainly contains information about pitch, onset, duration, loudness, and timbre. The names of the notes indicate their pitch class, i.e., c d e f g a b in AngloSaxon notation, c d e f g a h in German notation, and Do Re Mi Fa Sol La Si 1

Walter Benjamin has defined the power of fantasy as the talent to interpolate in the infinitely small. This immediately means true interpretation/performance. (...) In the densely woven context of the score’s text minimal cavities are to be discovered, wherein meaningful interpretation has found its refuge. (...) The medium of artistic fantasy is not a diminution, but an augmentation of precision.

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in Italian2 and French notation (except Do → Ut and Re → Ré). Alterations (sharp , moving a semitone up, and flat , moving a semitone down) complete this information. The position of the notes in the musical stave determines their precise octave. The diapason gives the correspondence between the names of notes and their frequencies. The current reference is a at 440 Hz. Information about diapason is not given in the score; it is an external parameter established before musical performance. It deals with the tuning systems and is implicit in the indicated set of instruments. In a score, (symbolic) time is expressed by onset and duration. Onset of a note (or a rest) is also expressed by the duration of previous and following notes (and rests). Relative values of durations are indicated as sub-multiples (whole note, half note, quarter notes, and so on). These values are converted into precise time indications in seconds via a Maelzel metronome. In modern scores (from Beethoven to present day), the tempo at which the metronome is set is indicated at the beginning of the score. Often a metronome is also used during the preparatory study of a score. Mechanical structures are just a starting point for performance. They cannot enhance musical expressivity. Expressivity comes from variation of mechanical structures. Also, it is not true that tempo must be the constant: There may be score indications such as accelerando or rallentando. A much more ambiguous indication is loudness. Verbal expressions such as f , mf , and p, give indications about the relative loudness of a note or an entire phrase or section of a score. While performing, they must be converted into decibel (dB) levels. Variations in loudness levels are realized by performers (or the conductor of an orchestra), depending on the style of the composition and the instruments involved. The name of the instrument (violin, flute, piano, etc.) defines what should be the timbre (or sound color) of music. Timbre is determined by the physical characteristics of the instruments: their vibrating body (chord, tubes, and membranes), the resonance system, and the way they are played. For example, the timbre of a violin can be completely changed in a pizzicato. Timbre is also strongly influenced by technique and the personal touch of the performer. This crucial topic deals with musical acoustics, which is an entire branch of mechanics in physics. Other verbal indications, such as con fuoco, con dolcezza, con espressione, don’t have any obvious mathematical or mechanical correspondence. They must be translated by the performer into gestural variations that imply sounding variations, and gestures (not only musical but any bodily gestures) can be mathematically described, as we will see in Chapter 33. 2

Italian names are derived by Guido d’Arezzo from the first syllable of each verse of St. Johannes hymnus Ut queant laxis, in Latin.

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32.3 Structure Theory of Performance The mathematical approach to performance starts with a local composition (K, REHLD ), the performance kernel that is mapped by a function ℘ : K → L to a local composition (L, Rehld ). We don’t suppose that the map here is induced by an affine homomorphism, but that there is a neighborhood of K, called the frame of the performance, F =]aE , bE [×]aH , bH [×]aL , bL [×]aD , bD [ such that ℘ is defined on F and that it is a differentiable map with invertible derivative D℘(X) in each point X ∈ F . This map ℘ is called the performance transformation. Recall that the derivative is an R-linear isomorphism, being represented by an invertible 4 × 4-matrix J℘(X), the Jacobian matrix. We now want to describe the performance transformation ℘ in terms of its Jacobian. We follow the idea of describing physical onset times using tempo. Recall from Musical Example 75 that tempo T (E) at onset E was the inverse of the derivative D℘E = de/dE. To know the physical onset e = ℘E (E) at symbolic (score) time E, one has to know the physical initial time e0 = ℘E (E0 ) of a starting symbolic time E0 , and then—this is the experience of every musician— the tempo curve T (E) tells us when in physical time we arrive at symbolic time E. The mathematical part thereof is a bit more involved, because it needs integration of functions. We don’t explain integration here, but the formula is this (see [75, Section 33.1.1] for details): 

E

℘(E) = ℘(E0 ) + E0

1 , T

the integral over the inverse tempo function T . This one-dimensional case is used to set up a formalism to solve the general case. The inverse derivative −1 D℘−1 . E that defines tempo can be generalized to the inverse Jacobian J℘(X) But tempo is a number and a matrix is not a number. To solve this problem, the mathematical analysis shows that we may apply the matrix J℘(X)−1 to the vector Δ = (1, 1, 1, 1) ∈ Rehld . This defines a vector

‫( ℘צ‬X) = J℘(X)−1 .Δ in REHLD . Similar to the tempo, this is a vector for every argument of the symbolic space. Such a structure is called a vector field, and in our performance theory, we call it the performance field ‫ ℘צ‬of ℘. Figure 32.1 shows a performance field in the two dimensions E, H of onset and pitch. It is defined from the one-dimensional tempo and intonation fields T, S. For onset and duration, one gets a tempo-articulation field in the plane of E and D, see Figure 32.2. This is a consequence of the fact that with tempo, duration as the difference of onset and offset of a note can also be defined. But in general, with staccato and legato, duration has a more complex shape. Similar to the integration of T1 to get the performance onsets, one can integrate the field ‫ ℘צ‬and get the performed point x = ℘(X). This is quite difficult to explain in detail, but there is an intuitive approach that everybody

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Fig. 32.1. The performance field for onset and pitch that is defined by the tempo and intonation fields.

Fig. 32.2. The performance field for onset and duration that is defined by the tempo field with duration being derived as a difference of offset and onset of a note.

can understand. The integration in the general case involves looking at integral curves of such a performance field. Imagine the field as being the velocity field of a water current on the surface of a river. If you are in a boat, your trajectory will be a curve that is determined by the velocity field of the river. This is what they call an integral curve. One may calculate the integral curves of performance fields that go through a selected note X, and the curve will hit a point X0 within the frame F where one knows the image ℘(X0 ). Together with this initial value, the curve will then allow us to calculate the image ℘(X). This has been implemented in the software RUBATO , allowing us to calculate performance of musical scores, once the performance fields were given.

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32.4 Expressive Performance The musically delicate part of this theory is the question of how to define performance fields. In the one-dimensional case, you may ask yourself about the operators that define the tempo or intonation field. There is an extensive theory that investigates these performance operators, see [75, Part IX] or [83]. We also investigated performances by pianists and were able to prove that certain standard performance operators (derived from harmonic, melodic, and rhythmic analysis) can be correlated to agogic, i.e., the measured tempo curves generated by these pianists, see [75, Part XI], [17], and [18].

33 Gestures

Summary. Gestures are complex in their common understanding. The concept of a gesture has never been thoroughly defined to this date, although gestures are very important in humancomputer interface design, human expressivity in and Fig. 33.1. Michelangelo’s Divine Gesture from God to beyond common language, Adam. and above all in the arts. Painting, dance, music, theater, and film would not be understood without gestural concepts and processes. We therefore will give a short introduction to the first mathematical theory of gestures. –Σ–

33.1 Western Notation and Gestures The first Christian liturgical music was the Gregorian chant. It was developed early in the Middle Ages and influenced by the Greek modes. It is a form of initially unaccompanied monophonic chant. The Western musical notational system is derived from Gregorian chant, whose origins, as we will see, stem from gestures. In 4th- and 5th-century Europe, there was no musical notation for melodies. Cantors in monasteries learned them by heart. Later, some signs were added to words written in manuscripts, called neumes (from the Greek word for sign), see Figure 33.2. Initially, these neumes were positioned above syllables of the text, without any reference to a precise pitch. Successively, neumes © Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_33

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Fig. 33.2. From neumes to notes.

have been introduced in a four-line musical stave (the Italian tetragramma), the ancestor of our modern five-line stave (pentagramma). The use of a stave

Fig. 33.3. The Gregorian musical figures.

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established the passage from non-diastematic to diastematic notation, where the precise pitch is shown by the vertical position on the musical stave. We have these transitions: 1. 2. 3. 4.

from from from from

acute accent to virga grave accent to punctum (with the stem) circumflex accent to clivis anti-circumflex accent to pes or podatus

Some of the Gregorian musical figures are the following (see Figure 33.3): • • • • •

Punctum quadratum (simplest symbol) Punctum inclinatum (lozenge, used in a group of descending notes) Virga (like punctum quadratum, but with a stem) Scandicus, three ascending notes Quilisma (the modern mordente)

All of these musical figures have the same duration, and they are usually transcribed in modern notation using eighth notes. By combining these figures, it is possible to obtain more complex neumes. For example, by combining two ascending or descending notes, we respectively obtain the pes and the clivis, or the torculus/porrectus by using three notes. What is the origin of neumes? The neumes reproduced the ascending-descending movements used by the choral conductor to represent variations of the shape of the melody. These hand movements that indicate the shape of the melody constitute the cheironomic movements, from the Greek χε´ιρ, cheir, hand, and ν o´μoς, nomos, law. Due to its origin, we can state that the Western musical notation is derived from gestural indications. There is a transition from a continuous shape, characteristic of gestures, as we shall see in the following sections, to a discrete set, characteristic of the notes in the symbolic score. We argue that the idea of freezing gestures into Fig. 33.4. The earliest music notation simple signs is exactly what a composer does when he or in China was creshe is writing a new score. Mathematically, that corre- ated for an instrusponds to a procedure of discretization. ment called chin.

33.2 Chinese Gestural Music Notation Chinese language is based on gestures and visual symbols. For example, the left side of Figure 33.5 shows the development of the Chinese word for “mountain.” The contemporary Chinese word is on the lower right side of the mountain image. Other Chinese words with the same genealogy are shown to the right part of Figure 33.5. Western music notation represents the sound (result), but

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Fig. 33.5. Left: Chinese word development for “mountain,” right: some Chinese words with the same genealogy.

old Chinese music notations represents gestural instructions (production) for the player. The earliest music notation in China was created for an instrument called chin (古琴). Old Chinese texts are organized in columns, from the right to the left. Figure 33.6 shows Chinese drum notation. In addition, because Chinese instruments were created for solo performance, there is no expectation or requirement for precise beats. The analogy might be this: In tai chi, individuals move in response to their own inner mood; however, in Western dance, the individuals respond to the rhythmic pattern from the music outside of their own bodies. This makes the gestural message so difficult to grasp from modern Western notation.

Fig. 33.6. An ancient Chinese drum notation.

33.3 Some Remarks on Gestural Performance Gestures are strictly related to music. Using a metaphor proposed by one of the authors (Guerino Mazzola), musical notes are the discrete points on the ground touched by a dancer while he or she is continuously moving. These continuous movements are gestures. So, to have a complete understanding of

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a musical performance, we need to extend the previous analysis (mapping of symbolic parameters of a score to physical quantities) to gestures. Why are gestures important in music? Every musician is able to read a score; however, not every musician is able to play all instruments. A great part of musicians’ training deals with learning specific gestures to play a selected instrument. In general, performance of a score combines a collection of little gestures (articulations, scales, staccato or legato, etc.) that can be glued together. A gesture of gestures is called hypergesture, and it will be described in Section 33.6. There are also gestures not directly tied to sound production but that help orchestral musicians select the best specific gestures to bring forth a general musical idea. This is the role of an orchestral conductor. We call symbolic gestures the movements suggested by a score that ideally allow a perfect realization of the content of score. We call physical gestures the real movements by performers. To understand the difference, suppose one has a piano score containing two consecutive quarter notes with the same pitch, for example middle C. Ideally, these notes must be played one after the other without any interruption. Practically, the pianist has to raise a finger. Ideally, there should be a zero-time interval between them (with infinite speed, see Figure 33.7). In practice it is only possible to have finite-speed movements. If the onset-position graph of the ideal movement implies a straight line for infinite speed, the real movement can be represented as a smooth curve.

Fig. 33.7. The beginning of Beethoven’s “Hammerklavier” Sonata is an example of a gesture that is impossible to play. For the left hand, for example, there is no pause between the first b and the successive chord. This would require an infinite speed.

However, infinite speed of the first case corresponds to a finite speed in imaginary time. In fact, there are two ontologically different kinds of time: real for physical gestures and imaginary for symbolic gestures. Distinction between imaginary and real time reminds us of the dichotomy of potentiality and actuality in Aristotelian philosophy. For more details, see the definition of complex numbers in Chapter 14 and the description of complex time and Descarte’s dualism in Section 33.7.

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The formalism developed for this simple case can be extended and applied to any musical performance. It is clear that there is an infinite variety of physical gestures associated with the same symbolic gesture. For the same symbolic gesture, in fact, there can be some physical realizations that are more difficult than others, some physical gestures that correspond more or less to the musical idea, and so on. Musical composition requires the inverse process, from physical reality to symbolic reality. The composer can start from real physical movement— such as piano improvisation—or “virtual” physical gestures imagined to be the final result when the music is performed. Starting from physical gestures, the composer has to find the optimal symbolic approximation of his or her ideas. It means he or she will answer the question, what are the symbolic indications whose physical realization is closer to my idea?

ˇ “*

Exercise 12 Play a song/phrase on a piano keyboard three times, choosing a different gesture each time. Then try to write a score indicating each gesture. A detailed analysis of gestures is beyond the scope of this book. However, because of their dramatic importance in music and future development of music, we will give mathematical definitions and some bibliographic references. The gestures are curves in space and time. To formally connect the symbolic to the physical, we use methods involving surfaces (world-sheets) (worldsheets). These surfaces were inspired by the string theory in theoretical physics, as described in [68]. A more complete discussion of the implication of gestures in music is given in [85]. An example of transformation from symbolic to physical reality is portamento for string instruments. A glissando, as we know, is a continuous transition from a starting note to an ending note. A portamento also occurs between two different notes, but there are some differences in the execution. Ideally, if the transition from one note to another is instantaneous, as a step-function, a portamento implies a more smooth transition. This smoothness is realized with a quick pass through more pitches than simply the starting and ending ones, similar to the lower and upper level of a step. However, it does not involve all intermediate pitches, as happens in a glissando. Portamenti are often used by voice and string instruments. The duration and pitches involved depend on the style of the chosen music.

ˇ “* Exercise 13 Compare a baroque song and an aria of a classic opera, and try to find differences in portamento. ˇ “*

Exercise 14 Sing two different notes with precise intonation, but with no pitch transition between them. Then try to perform a portamento. Measure time duration of the first and the second case, and try to find the intermediate notes that you are singing. Finally, make a graph of your first and your second attempt, and compare the two images.

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33.4 Philosophy of Gestures Jean-Claude Schmitt, in La raison des gestes dans l’Occident médiéval [101], has given the most complete and important contribution to a history of the concept, philosophy, and social and religious roles of gestures during the early centuries of our modern Western culture. It starts with a summary of the antique Greek and Roman tradition—Plato, Aristotle, Cicero, and Quintilian and then draws a trajectory through the Middle Ages, from the early writings of Martianus Capella (between 410 and 470) to in the sophisticated and detailed writings of Paris-based Christian theologist Hugues de Saint-Victor (1096-1141), and terminating with a detailed discussion of the transition of the Medieval Christian culture during the 12th and 13th centuries to an “intellectual Renaissance” in which new technologies, the new medical paradigm of surgery, the first universities, and the rediscovery of ancient traditions generate new perspectives on the phenomenon of gestures. Hugues de Saint-Victor gave the classical non-mathematical definition of a gesture [101]: “Gestus est motus et figuratio membrorum corporis, ad omnem agendi et habendi modum.” (“Gesture is the movement and figuration of the body’s limbs with an aim, but also according to the measure and modality proper to the achievement of all action and attitude.”) Gestural expressivity is reflected in the etymology of emotion: e-movere, to move inside out. Gestures are movements. To our knowledge, a philosophy of gestures was not developed until the 20th century, except for Tommaso Campanella’s insight in the 17th century, stressing the pointer gesture as the most certain of all certitudes. The more recent attempts at a philosophy of gestures are mainly of European—and even more precisely, of French—origin. Michel Guérin in his Philosophie des gestes [48] characterizes gestures as non-semiotic phenomena, specifying four elementary gesture types, but without giving a definition of the very concept. These types are: faire (make), don(ner) (donate, gift), écrire (write), and danser (dance). Guérin’s approach is comparable in style to Vilem Flusser’s essay Versuch einer Phänomenologie der Gesten [37]; however, Flusser views gestures as signs. His definition is essentially negative: freedom, that part that cannot be satisfied in the causal determination of the human body’s intentional movements and its associated tools. He searches desperately for the meaning of that movement. In fact, to him, what is meaningless cannot be understood. This is the semiotic trap that Gilles Châtelet, Charles Alunni and their French associates avoided. We discuss these philosophers below. As observed above, the main contributions to a contemporary philosophy of gestures have been created by French scholars. Their works also differ from the Anglo-Saxon linguistic philosophy of gestures developed by Adam Kendon and David McNeil [65] and [87], who focus on gestures that are co-present in linguistic utterances. From this perspective, their concept of a gesture is strictly semiotic: Gestures are special signs that support the building of linguistic syn-

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tagms and contents. And they are always related to the body’s actions; no abstract concept of a gesture, such as a gesture in a musical melody or a thought gesture, is addressed. The French tradition of gesture philosophy is characterized by the thesis that gestures constitute a proper ontology that is independent of semiotic systems. It typically precedes them, or is pre-semiotic. As in many other cases of French philosophy of the 20th century, Paul Valéry is the figurehead of gesture theory. His famous inscription on the front of Palais Chaillot in Paris says:

Fig. 33.8. Paul Valéry is the figurehead of French gesture theory (1871-1945).

Dans ces murs voués aux merveilles J’accueille et garde les ouvrages De la main prodigieuse de l’artiste Égale et rivale de sa pensée L’une n’est rien sans l’autre (In these walls devoted to the marvels I receive and keep the works Of the artist’s prodigious hand Equal and rival of his thought One is nothing without the other.)

Again it is the thinking hand that Valéry invokes, a hand that in its gestural originality is a full-fledged partner of human thoughts. It is not surprising that Valéry wrote an essay on the philosophy of dance [111] in which he concludes not with a scholarly description of dance, but a suggestion to start dancing our thoughts instead of thinking about dance. Ahead of his time, French mathematician and philosopher Jean Cavaillés in 1938 stated in [26] a core property of gestures that bypasses any semiotic basis: “Comprendre, c’est attraper le geste et pouvoir continuer.” (“Understanding is catching the gesture and being able to continue.”) Cavaillés’ dancing thought (also shaped in Boulez’s reflection on gesture in music [24]) was stated with respect to mathematical theories, and as such it was one of the very first principles of gestural embodiment in mathematics, an idea now quite fashionable through the work of George Lakoff and Rafael Núñez [58] but also anticipated in Châtelet’s observation [27] that the Fregean (Gottlob Frege (1848-1925) was German logician) concept of a function f in mathematics is a dramatic (and questionable) abstracFig. 33.9. tion that replaces the moving gesture from argument x to Mathematician its functional value f(x) by a kind of disembodied “teleand philosopher portation,” where the evidence of the functional relation is Jean Cavaillés wrapped and hidden, if not destroyed. (1903-1944). Gestures—except when “tamed” by social codes—are not signs in a semiotic environment. They are not a realization of Ferdinand

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de Saussure’s classical signification process. Châtelet [27] is very clear in this point: “Le concept de geste nous semble crucial pour approcher le mouvement d’abstraction amplifiante des mathématiques. (...) Un diagramme peut immobiliser un geste, le mettre au repos, bien avant qu’il ne se blottisse dans un signe, et c’est pourquoi les géomètres ou les cosmologistes contemporains aiment les diagrammes et leurs pouvoirs d’évocation préemptoire.” (“The concept of a gesture seems crucial to approach the amplifying movement of abstraction in mathematics. (...) A diagram can immobilize a gesture, put it to rest long before it is hidden in a sign, this is why geometers and contemporary cosmologists love diagrams and their power of preemptive evocation.”) A gesture can be immoblilized by a diagram (which in this French theory is a kind of disembodied gesture) before it becomes a sign. And Alunni [6] confirms this creative pre-semiotic role of gestures: “Ce n’est pas la régle qui gouverne l’action diagrammatique, mais l’action qui fait émerger la régle.” (“It is not the rule that governs diagrammatic action, but it is action that causes the rule to emerge.”) Summarizing, we learn that gestures are generally understood as pertaining to a proper ontology that is not subordinate to semiotic lines of thought. In particular, the dominant French diagrammatic philosophy exhibits a sharp di33.10. chotomy between “wild” and “tamed” gestures, the former Fig. Philosopher being independent or antecedant of semiotic realms, while the latter are serving semiotic purposes as special types of Charles Alunni signs. Conceptual creativity is exibited in the layer of wild (1951- ). gestures. The communicative characteristic of (wild) gestures stresses their “howness” as opposed to their substantial “whatness.” Gestures are understood in their behavior, not in their absolute being (such as Immanuel Kant’s Ding an sich).

33.5 Mathematical Theory of Gestures in Music We now transform into mathematical terms Saint-Victor’s definition of a gesture. → − Definition 81 Given a topological space X, the spatial digraph X is the digraph, whose vertex set is X, while its arrows are the continuous functions f : I → X defined on the closed unit interval I = [0, 1] ⊂ R, which we call curves in X. We define h(f ) = f (1), t(f ) = f (0), so we may represent a curve f as an arrow f : t(f ) → h(f ) or t(f ) - h(f ). If m : X → Y is a continuous map, the associated digraph morphism → − → − → − m : X → Y maps vertices to vertices via m and transform the curve f : I → X to the curve m ◦ f .

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Definition 82 Given a directed graph Γ and a topological space X, a gesture → − is a morphism g : Γ → X of graphs. Γ is called the gesture’s skeleton, while X is called its body.

Fig. 33.11. A gesture with body in a topological space X.

→ − Intuitively, a gesture g : Γ → X is a system of curves in X that are connected according to the arrows of the skeleton, see Figure 33.11.

ˇ “*

Example 76 The simple finger gesture is the most important example of an elementary gesture of a pianist’s hand, see Figure 33.12. It is used in fundamental research about the transformation of symbolic gestures into physical gestures, see Section 33.7. → − Definition 83 If g : Γ → X , d : Δ → h : g → d is a pair h = (t, m) where t : Γ m : X → Y is a continuous map, such that Γ ⏐ ⏐ t Δ

− → Y are two gestures, a morphism → Δ is a digraph morphism, and the following diagram commutes. → − g −−−−→ X ⏐ ⏐→ m − → − d −−−−→ Y

Clearly, morphisms can be composed in the expected way, and this composition → − is associative. Every gesture g : Γ → X has the identity Idg = (IdΓ , IdX ) as morphism. This setup defines the category Gest of gestures. → − Example 40 For every arrow a in the skeleton Γ of a gesture g : Γ → X , we may restrict the gesture to that arrow. This defines a morphism of gestures as shown in the following diagram. It is the restriction of g to the arrow a, and we shall use it in Section 33.6.

√

33.6 Hypergestures

283

Fig. 33.12. An elementary fingertip gesture.

→ − X ⏐ ⏐−−→ IdX → − g Δ −−−−→ X g◦a∗

[1] −−−−→ ⏐ ⏐ a∗

33.6 Hypergestures → − The set Γ @X of gestures g : Γ → X can be given the structure of a topological space as follows. We first deal with the case of Γ = [1], the chain digraph with − ∼ → two points and one connecting arrow. Clearly, in this case [1]@X → X . The → − set X of arrows in X is the set Top(I, X), and in this case, we have the compact-open topology that turns it into a topological space. The general case Γ @X is obtained by looking at all of its arrows a and looking at the morphism a∗ : [1] → Γ sending the arrow of [1] to a. Each such a∗ defines a map pa : Γ @X → [1]@X by restriction of a gesture g to g ◦ a∗. There is a unique topology on Γ @X that turns all these restriction maps pa into continuous maps. This is the topology we impose on Γ @X. This construction has the important consequence that we may now consider the gesture space Δ@Γ @X = Δ@(Γ @X). Gestures in this space are gestures of gestures, which we call hypergestures. This construction may be iterated to yield higher hypergestures Γ1 @Γ2 @ . . . Γn @X.

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Theorem 24 (Escher Theorem) If Γ1 , Γ2 , . . . Γn is a sequence of digraphs, if X is a topological space, and if π ∈ Sn is a permutation, then there is a canonical homeomorphism ∼

Γ1 @Γ2 @ . . . Γn @X → Γπ(1) @Γπ(2) @ . . . Γπ(n) @X.

Fig. 33.13. First species counterpoint can be viewed either as a hypergesture from cantus firmus to discantus (left) or as a hypergesture from the first to the subsequent intervals.

ˇ “* Example 77 The first species counterpoint can be understood as being either a hypergesture from cantus firmus to discantus (left in Figure 33.13) or as a hypergesture from the first to the subsequent intervals (right in Figure 33.13). The first interpretation is common in counterpoint courses, but it is historically wrong [100]. The second is adequate in the sense that the tension of the concept “punctus contra punctum” (“point against point”) cannot be vertical because the intervals are all consonances and is instead horizontal, expressing the tense step from interval to interval. But the Escher Theorem puts these two hypergestures into a one-to-one correspondence. ˇ “*

Example 78 In improvisation, it often is important for a creative step to change one’s perspective of what is being heard. Here the Escher Theorem can

33.7 Hypergestures in Complex Time

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help change, for example, a chord sequence in a given rhythm into a rhythm of chord changes, i.e., to change one’s perception of the generative musical shape.

33.7 Hypergestures in Complex Time In this concluding section we sketch some of the most recent developments in gesture theory of music. It deals with the transition from the symbolic data on a Western score to the gestural realization by a pianist’s hands. To begin with, one should be aware that the score cannot be played as-is in many cases. The left side of Figure 33.14 depicts some chords of the initial fanfare of Beethoven’s “Hammerklavier” Sonata, op. 106. The sequence of the same notes without any intermediate rest is physically impossible, because the fingers must go down until the offset of a note and then move up and come down for the next note in zero time, which would need infinite velocity.

Fig. 33.14. Playing these notes (left) is physically impossible. The symbolic information (rectangular curve of “imaginary” finger movements, left) must be deformed into a smooth physical curve (right).

The pianist has to deform this set of symbols to produce a physical rendition. This is mathematically achieved using a hypergesture from the finger gesture in symbolic space to the finger gesture in physical space. This hypergesture is similar to what in modern physics of strings is called a world-sheet. It is a surface that represents a whole gesture of gestures, starting from the symbolic

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gesture that is physically impossible (see its non-differentiable horizontal and vertical step-lines) and ending at the physical gesture. This approach uses two different realities: the symbolic and the physical. But how can we view them as being parts of a comprising larger ontology? Our solution is to work in a space-time where the three spatial coordinates (in the vector space R3 ) are added to a complex time space, i.e., the space C of complex numbers. We have described this approach in Musical Example 33.

Fig. 33.15. Two world-sheets with a complex potential are shown, the left upper one for the transition from symbols to physics, the right one for the inverse process from physical performance to symbolic data as represented in a score. The bottom drawing summarizes the double process from symbols to physics and back.

Our solution uses energy potential transitioning smoothly over time coordinates. We start with imaginary times on the symbolic gesture and move smoothly to real time for the physical gesture. See [68] for details. The shape of the world-sheet of such a transition depends on the energy potential that controls the efforts made by the pianist to achieve this performance. The top left part of Figure 33.15 shows such a world-sheet of a complex potential. But we should be aware that the composer may also move in the opposite direction, starting with a physical action of the hand(s) and then transforming it into symbolic data, a process that is also achieved when recording a pianist’s performance via MIDI technology. The top right part of Figure 33.15 shows this inverse world-sheet. And the bottom drawing summarizes the double process from symbols to physics and back.

Part VIII

Solutions, References, Index

34 Solutions of Exercises

Summary. This chapter contains the solutions of the mathematical and musical exercises. Each solution number corresponds to the exercise number. –Σ–

34.1 Solutions of Mathematical Exercises √

Solution 1 The logical function A IMPLIES B is false iff A is true but B is false. The function (NOT A) OR B is false iff both NOT A and B are false, i.e., iff A is true and B is false, and this is the same as the first function. A NOT A B (NOT A) OR B A B A IMPLIES B F F

T

F

T

F

T

FT TF

T

F

T

T

T

F

T

F

F

F

TT

T

T

F

T

T

√

Solution 2 Using the involution property of addition, the equation x+y = z has the solution x = z + y since x = x + y + y = z + y. Therefore, we have x = {s, w} + {r, s, w} = {r}. √

Solution 3 The elements of 2a are 0 = ∅, 1 = a. We have x y x+y

x y x.y

00 01

0 1

00 0 01 0

10

1

10 0

11

0

11 1

© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_34

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√

Solution 4 We have (x, y, z) = ((x, y), z) and (u, v, w) = ((u, v), w), therefore (x, y, z) = (u, v, w) iff (x, y) = (u, v) and z = w, iff x = u and y = v and z = w.

√

Solution 5 We have (x, y) ∈ 22

a∪b

. Therefore a × b ∈ 22

2a∪b

.

√

Solution 6 Looking at the values f × g(x, y) = (f (x), g(y)), if every pair (u, v) ∈ b × d is hit, then every u ∈ b and every v ∈ d is hit, and vice versa. So epi for the cartesian product is equivalent to epi for each factor function. A similar argument proves the mono statement, and since iso is epi AND mono, the third statement follows. √

Solution 7 The object Set(a, b) consist of triples (f, a, b), where f ⊂ a × b is a graph. This means (f, a, b) ∈ 2a×b × {a, b}2 . Therefore Set(a, b) ⊂ 2a×b × a×b 2 {a, b}2 or Set(a, b) ∈ 22 ×{a,b} , which is a set. √

Solution 8 The fact that q and a × b are equipollent follows from these facts: From the universal property and the hypothesis about q, we imply that ∼ Set(c, q) → Set(c, a × b) for all c. This is true in particular for the singleton set c = 1 = {0}. But for any set x, Set(1, x) is in bijection with the set of elements of x. This defines a bijection of q and a × b. More precisely, let us also show that f is a bijection. Exchanging the roles of q and a×b, we have a unique g : a×b → q such that fa ◦g = pra , fb ◦g = prb . The composition f ◦ g : a × b → a × b must be the identity since it is the unique function guaranteed by the universal property of a × b. Therefore f ◦ g = Ida×b . Exchanging the roles of q and a × b, we also get g ◦ f = Idq . This means that these two sets are in bijection with each other. Solution 9 If x∈a fx is well-ordered, take of any index x ∈ a the subset [fx ] ⊂ x∈a fx of elements that have a fixed coordinate for all indices y = x and any value at index x. This set is in bijection with fx , and the induced ordering on [fx ] coincides with the ordering on fx . But as the induced ordering on [fx ] is a well-ordering, the same holds for