Chesnut Musical Ideas and Measure

The Shape of Meaning in Music Mathematics & Musical Rhetoric

John Chesnut

Copyright © 2012-2014 John Chesnut This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.

The cover art was generated with computer assistance.

This paper is updated frequently. Before you refer to this paper, make sure that you have the latest version.

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Dedicated to my wife JERRI

Without whose unfailing support This work would not have been possible

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Contents Foreword

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Acknowledgements

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About the Author

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Preface: The music says, “Listen! I have something important to say.”

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Chapters 1 – What is a Distinctive Musical Idea? Taking the Measure of the Cantabile

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2 – Texture and Rhetoric in Debussy’s Syrinx: A Computer-Assisted Analysis

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Appendices A – Five Groups of Five Simulated Arches

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B – Analysis of the Cantus Firmi by Jeppesen

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C – Stochastic Models for Limited Growth and Perpetual Cycles

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Suggested Reading

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Notes

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Foreword This monograph is an extract from a work in progress, The Shape of Meaning in Music: Mathematics & Musical Rhetoric, in which musical rhetoric – the interweaving of meaningful contrast with skewed symmetry, by which a composer makes a convincing case for complex musical ideas – is understood in terms of mathematical shapes and patterns. Encapsulating a lifetime of reading and thinking, this study is as much concerned with interpretation as with theory. As currently planned, this study will examine selected examples of Western art music ranging from the Gregorian chant, Rorate Caeli, to Debussy’s composition for solo flute, Syrinx. Although music is, in certain respects, a sui generis art form – the sound of a perfect fifth or major seventh, for example, is not just like anything else – an appreciation of music requires abstract intellectual competencies that are common to all human endeavors. It follows that change in the way that melodies were put together during the selected time frame did not take place in a cultural vacuum, but were paralleled by changes in intellectual perspectives. Critical analysis of this music and its forms requires some reference to sociological developments in Western society, having to do with such things as the Reformation and Counter-Reformation, the chaos of the Thirty Years War, the rationalization of social structure, the decline of ancient traditions, social disruption, the rise of the self-created individual, and the beginnings of the ongoing struggles to create new social institutions. This study is written in an accessible style, requiring only that its readers be familiar with music theory and be comfortable with mathematical formulas, graphs, and probabilistic reasoning. An acquaintance with computer-programming algorithms would be helpful. This study will make extensive use of computer-assisted analysis. The mathematics is concerned with how to think about melodic contours, various aspects of musical texture, and how to think about musical time. This study is more concerned than most with the question of when events occur in the flow of a musical design. A number of indicators will be used to characterize different aspects of melodic movement, temporal relationships, and rhythmic strength. Among other things, applications of the following equation of motion will be explored: x = atC (1 – t) – btD (1 – t) + et + f.

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This formula encapsulates the deterministic form of a Limited Growth Model, describing a type of rhetorical symmetry that reconciles opposite tendencies through time. This symmetry applies to melodies (and other time series of musical interest) possessing a simple, closed form, in which the deviation from the long-term linear trend has at most one distinct high point, or one distinct low point, or both. The model proposes that, for this specific class of musical shapes (not additive or cyclical forms, nor layered self-contrapuntal designs), the broad outline of the form can be understood as the product of a single wave-like gesture rather than a succession of independent incidents – that the tendencies of such a musical process to rise and fall, expand and contract, are present throughout, and that the relative strength of these tendencies determines the outline of the shape that emerges over time. In other words, to an approximation, the unifying invariance of the form is to be found in a set of constant dynamic properties, which unfold at different rates. The equation (for time t scaled to range from zero to one) is a generic, parameterized schema for the shapes to which it applies. The equation is a polymorphous abstraction from theories of melodic shape by Kurth, Meyer, Narmour, Gjerdingen, Adams, and Huron. There is also a stochastic version of the Limited Growth Model, in which the rising, falling, expanding, and contracting tendencies of a melody are simulated by randomizing procedures. In an age of increasing specialization, it seems necessary to say that the wide-ranging nature of this study does have precedents. Without being directly comparable to any of its forebears, this study is heir to the intellectual example of musical scholars such as Leonard Meyer, Leo Treitler, Nicholas Cook, Alan Lomax, and Godfried Toussaint. Although each of these authors has emphasized some subjects more than others, none fits easily into a narrow specialty. They have selected from a menu that ranges from historical musicology to cultural history, music theory, ethnomusicology, and mathematics. It is not a coincidence that the last two scholars named on my list are mathematical ethnomusicologists. My formative exposure to music has been culturally expansive, ranging from Greek folk music that I heard neighbors singing in San Antonio when I was a little boy, to mariachi and Bluegrass, and onward to Stravinsky, the Louisville Orchestra Commissioning Project, the European classics, atonality, and transcendent jazz. I was educated in traditional historical musicology and music theory, with a background in mathematics. Mathematics is appealing, because it is a cultural unifier, which provides a layer of thought that transcends cultural relativism. Mathematics is, in a sense, the ultimate universal language – not because everyone speaks it, but because the language of mathematics was developed over millennia in radically different cultures, both ancient and modern.

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The subject matter of this study, therefore, is a natural fit for me, which I find personally absorbing. My primary motivation is simply to satisfy my own curiosity, to make sense of my own life experience. To follow the trail marked by one’s own curiosity is much like following the development of a musical composition – it is an introspective Odyssey, the adventure of wending one’s way home. I hope that, as I recount my adventures, I will encounter like-minded wayfarers, who will enjoy joining into the conversation. For specialists in music theory, this study serves as an introduction to a Neo-Kurthian wave interpretation of Meyer’s theory of expectations, re-imagined here as a theory of exploratory behavior.

The Preface, the first chapter, and what is planned to be the last chapter of the

prospective book are included here. Abstracts of other potential chapters are found in the notes, some of which are quite extensive. Examples of computer-generated cantus firmi will be found in Appendix A. Appendix B contains an Exploratory Factor Analysis, which compares Jeppesen’s treatment of the modes in his cantus firmi. To give the reader some indication of where this study is heading, I have also included Appendix C, which introduces a stochastic variant of the Limited Growth Model and a stochastic interpretation of Simple Harmonic Motion. The appendices are written more tersely than the main body of the text, assuming more technical knowledge on the part of the reader. The most important ideas discussed in the first chapter are the following: The study begins with an investigation of melodic motion, arguing that the idea of the cantabile vocal line is central to our traditional understanding of what is normative and what is distinctive. A distinctive melody has to take risks, so we want to get a clearer idea of what is a risk. First, we look at what David Huron learned about the shape of folk melodies. Then we take a look at six cantus firmi by Fux, melodies that form the basis of his counterpoint method – melodies that are presumably subject to similar mathematical laws as the middle-ground lines that appear in Schenkerian reductive analysis of large forms. The cantus firmi can be understood as not merely lessons, but genuine haiku-like compositions that express definite musical ideas.

Fux’s idea of how to

elaborate a melody is different from Jeppesen’s, and even more different from, for example, a doxology. These differences can reasonably be interpreted as meaningful. Although Gregorian chant appears to have served as Fux’s model, Fux’s cantus firmi have a forceful rhetoric, which – as pointed out by Richard Crocker – is not typical of Gregorian chant. Fux’s cantus firmi are almost perfect examples of pure form; but they express definite ideas about order and disorder, contrasting an Age of Reason desire for complete closure in Fux (not consistently found in Jeppesen) versus an open-ended Medieval contemplative’s yearning for things unseen in a

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doxology. Fux’s cantus firmi are compared with computer simulations, to discover how Fux’s specific preferences resemble what is possible and probable for a wide range of generally similar melodies. We will also discuss Schenker’s idea of the so-called fundamental line, or Urlinie. This concept, which appears to be foretold in Fux’s cantus firmi, may result from a confluence of preexistent cultural ideals with the statistical tendency toward smoothing that results when a random walk is subjected to the geometric constraints of goal-directed motion.

The location of the

climactic high point of an arch-shaped melody, such as a cantus firmus, has structural implications that suggest a meaningful aesthetic choice. The later the high point, the fewer the possibilities for melodic development that remain for the final approach to the cadence, the more the closing motion will resemble an Urlinie, the stronger the perceptual closure of the ending, the more purposeful the closing motion will seem, and the greater will be the implied claim of certainty. The second (that is, last) chapter contains an extended analysis of texture and rhetoric in Debussy’s composition for solo flute, Syrinx, with commentary on the relationship of the closing passage to the recently re-discovered text that inspired the music. This is a computer-assisted analysis, making use of nine time-series indicators (mostly new): Duration-Weighted Average Pitch, Pitch Range, Interval Volatility (the average absolute size of the intervals), Extreme VHF (departures from the central value of an indicator concerned with the complexity of melodic contour), Duration-Rising (the amount of time given to rising versus falling intervals), Rate of Attack, Metric Hierarchy (a duration-weighted average of the levels of the notes in the hierarchy of the meter, scaled by the highest hierarchic level achieved), Pitch-Class Concentration (a measure of how the pitch-classes are distributed around the Circle of Fifths), and Pitch-Class Dominance (a special-purpose, probabilistic indicator created to capture Debussy’s idiosyncratic treatment of pitch relationships in Syrinx).

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he specific subjects just listed are examples that represent more general concepts. Since the first chapter lays the foundations for the book as a whole, the topics raised there are chosen

to illustrate steps in the development of a larger theoretical project. Topics are discussed in the order of their logical dependency. Each topic builds on the one before it, in need-to-know order, following a chain of reasoning. It may help the reader to know something about the larger context into which this first chapter belongs and toward which this chapter points. The book as a whole is concerned with the what, how, and why of what could be called, in the broadest possible sense of the word, development in musical form – but what would be

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more precisely described as dynamic change in the course of musical ideas and spatial relationships. To address why questions, I believe we have to think about cultural meaning. To address what and how questions, I believe the generality of mathematical reasoning is useful, because it helps us to identify common features between musical compositions that are ostensibly unique. The development of musical ideas, as it is most familiarly known to us, is often described in terms of motives and sequences – that is to say, categorical entities. These entities, however, are limited in scope, vary from one composition to another, and are not always clearly identifiable in the free-form spinning-out of a melodic line. The general principles of development, if they exist, are left somewhat unclear. Based on my experiments with computer-generated melodies, I think that special techniques may be needed to understand the development of musical ideas considered as categorical entities; but that part of my research is still in progress, and I will not say anything more about it at this time. Heinrich Schenker addressed this issue by disregarding rhythm, and reducing the significance of motives and sequences. He interpreted these features as ornamental superstructures imposed on an underlying architectonic plan, the Urlinie mentioned above, which, in combination with its standard bass harmonization, is called the Ursatz. One of the implications of the present study, in my view, is that the conventional understanding of the Urlinie is somewhat off the mark. I have become skeptical about the concept of the Urlinie as a categorically delimited entity, comparable to an element of spoken language. I would suggest that the Urlinie may be better understood as the idealization of an evanescent, statistically emergent, aggregate spatial property, occurring under certain conditions in a melody’s narrative arc. I think that to understand the Urlinie we have to unravel both a mathematical problem and a problem in the history of cultural aesthetics. This will lead us to look at musical structure and its meaning from a new perspective. Ernst Kurth, in my view, had a more generalized understanding than Schenker of complex, dynamic flow. Unfortunately, he lacked the tools to explain his ideas precisely. For that I would suggest that we need to use mathematics in the service of cognitive theory. Initially, I will devote a lot of attention to a special case, the melodic arch. Although melodies are not universally arch-shaped, arches are rather common. The study of other shapes – such as varieties of S-curves – will be considered later. To motivate our interest in melodic arches, I will point out their similarity to the narrative arc that is often found in literature. The course of narrative tension moving in time from exertion to exhaustion, when envisioned as an

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arch, is the spatial analogue of a Classical Greek rhetorical device used in legal and political arguments (a type of chiasmus), which, according to Apostolos Doxiadis (2012), is the basic form of a syllogism.

As closed, symmetrical forms, the syllogism, chiasmus, and arch evoke a

subjective sense of certainty. The rhetorical purpose of the forms is to encourage the auditor by indirect means to accept the truthfulness of what is said. I will begin with simple ways to characterize melodic motion. The basic objective of the first chapter, then, is to examine certain mathematical indicators that can be used to compare and differentiate melodies; and to discuss the significance of the results obtained from applying those indicators to specific musical examples. From the standpoint of the general theory of complex, dynamic processes, Fux’s cantus firmi are of interest because they are ideal types, in which important issues appear in a concise form. They are a microcosm of the universe of arch-shaped, complex dynamic processes of every imaginable kind, not limited to melodies. Since the cantabile melody is only a special case of the general problem of complex, dynamic processes in music, we also want to hint at the larger context of arch-shapes in musical tension curves in general, such as arise in Lerdahl’s study of tonal pitch space, and can be derived from the Cooper-Meyer theory of rhythmic structure, not to mention what may ultimately be discovered through the Riemann/George/Steblin/Brower sharp-flat principle, Neo-Riemannian theory, Tymoczko’s geometry of pitch relationships or Toussaint’s geometry of rhythm. We do not want to delude ourselves into thinking that what we learn about the cantus firmi automatically establishes universal laws of motion, however. It is important, therefore, to look at the cantus firmi in a larger context, to gain some impression of what they are and what they are not. First, we want to see that the cantus firmi are not alone, because they have certain things in common with the larger world of folk melodies. On the other hand, there are important differences between Fux’s cantus firmi and certain melodies that bear a general resemblance to them, such as Jeppesen’s cantus firmi and doxologies. A lot of space is given in this paper to Heinrich Schenker because his theory of the Urlinie appears to be directly inspired by Fux’s cantus firmi. Schenker is an influential figure who must be reckoned with; so, it is necessary to explain why it will be advantageous to go in a somewhat different direction. Schenker is such an influential thinker that he and his followers have practically taken ownership of the word structure, by which they mean a harmonically organized hierarchy of tones; but, this is not the only sense in which the word is used by music theorists. Cooper and Meyer (1960) spoke of a hierarchical rhythmic structure. George (1970) referred to relationships between keys based on the sharp-flat principle as structure. Narmour (1990) talks about structure

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in terms of melodic implications and their realizations. Zbikowski (2002) uses the term to refer to cognitive patterning in general. Cope (2009) uses the term to refer to musical organization that can be discovered by computer analysis of music. The word structure is easily over-used, but hard to avoid when one wants to refer to a specific musical pattern that contains some degree of internal organization and one does not want to use the word form, which implies a culturally established schema, such as the standard forms of the Classical style – theme and variation form, binary form, rondo form, sonata form, and certain mixed types. There is some danger, however, that the word structure may imply a systematic, rationalized, sanctioned, iconic practice, when one actually wants to refer to something more ad hoc, and one prefers not to use the word gesture, which has deliberate extra-musical associations. To minimize confusion, I will use other terms when they are clearly more appropriate: pattern, organization, relationship, shape, geometry, limit, constraint, and frame, or framework.

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t is difficult, if not impossible, for one person to know everything there is to know about music. Specialized study is necessary to achieve deep understanding of any aspect of music, and it is

inevitable that musical studies will become more and more specialized with time. My own work depends on the accumulated knowledge of dedicated scholars, and every citation in this paper is a note of gratitude. Even so, the process of discovering new ideas does not always follow a straight path that is confined within a pre-determined specialty. Some facts assert themselves; but other discoveries depend on what questions are asked, questions that may be posed from outside a given specialty. Some relationships are uncovered by examination of newly discovered facts. Other relationships are revealed by competition between ideas that cross accustomed intellectual boundaries. We listen to music with our hearts and minds, as well as our ears. Furthermore, we hear music in the context of our own life experience, which takes place within a larger cultural context. Listening, in the full sense of the word, therefore, is a many-layered psychological process. For this reason, I would suggest that it is difficult to have a complete thought about music unless one is willing to cross over the boundaries that divide one habit of discourse from another – from the descriptive to the interpretative, or from the mathematical to the aesthetic, for example. I would say more than that, for I believe that the identity of a thing is most vividly defined by comparing it with its opposite and that we obtain a more complete picture of an object when we view it from alternative vantage points.

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One might take as ones organizing theme a philosophical or political ideology, an aesthetic or psychological theory, a principle of musical form or structure, the life and works of a composer or an historical period or style – all of these being legitimate topics of musicological study. Unfortunately, in the last three or four decades, the lines between these topics have become more sharply divided. The scholarly discipline of musicology has become fragmented, due to the splitting off of music theory from historical musicology and the advent of the socially conscious New Musicology. The basic fault-lines of Anglo-American musicology are described by Joseph Kerman (1980, 1985), Nicholas Cook (1998), and Eugene Narmour (2011). Every reader and every writer comes to terms with this fragmentation in his or her own way. On the whole, I think that the issues raised by music are broader and more far-reaching than a fragmented discipline can easily discuss. I do not imagine that I will be able to reconcile deep-seated ideological differences of opinion about music, but I do believe that diversity is the life-blood of creative thinking. There is a creative tension between specialization and generalization, either side of which can be productive. I prefer to err on the side of inclusiveness, because this is where most of my ideas come from. Furthermore, I think that this policy helps me to keep my work in perspective. It reminds me that there is no Holy Grail of musical commentary. The main emphasis in this paper is on analysis, which I think of as a pragmatic commonality of the just-mentioned organizing themes. By analysis, I mean analysis in general – reasoned thinking about music, addressing questions about first principles – questions concerned about what we are trying to accomplish when we discuss a piece of music. I do not limit the term analysis to refer to a specific, established system of musical interpretation, such as the fullydeveloped late-Schenkerian method of Der Freie Satz. The fundamental question the analyst must ask is what analysis is for. Everything else follows from that. What I mean by the term analysis is not unlike what Joseph Kerman (who objected to the Schenker-dominated analysis of his time) preferred to call criticism, with the provision that I have a special interest in a subject not known to Kerman, the mathematics that describes spatial relationships and dynamic processes in music. Despite my interest in mathematics, I am not a strict positivist. I believe in using the methodology that is most appropriate to the subject matter at hand. The approach to analysis followed here will be analogous to the diagnostic procedure of a CT-Scan, which builds up a three-dimensional picture of an object, such as the human body, by assembling two-dimensional pictures that are effectively slices through the object. The procedure, called tomography, creates images by sectioning. A more accurate analogy – because it is as appropriate to the humanities as to the sciences – would be that of an archaeologist’s exploratory

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trenches, which crisscross a many-layered historical site to obtain a broad overview of the architectural structures and cultural artifacts that lie beneath the surface. In principle, the approach I take here is not unlike the Cantometrics project undertaken by the comparative musicologist, Alan Lomax (1978), discussed by Cook (1987, pp. 199-202), although my subject matter differs. Lomax’s group compared several thousand songs, selected to be representative of all the world’s cultures. Correlations were found between measurable aspects of melody and social structure. The fact that the Cantometrics project required the co-ordinated efforts of a large group of specialists, however, tells us that my own project will be highly selective by comparison. The analysis of musical examples in this paper (which, you will remember, are chosen to illustrate an underlying theory of complex, dynamic flow) follows a distinctive issue-oriented methodology – or tomography, if you like – in which the goal is to identify the salient issues raised by a musical composition when it is examined from the perspective of a significant binary opposition of the sort that might be raised by any one of the foregoing themes, such as: ideology, aesthetics, psychology, formalism or history.

(I will avoid biography, unless it provides a

compelling insight into the music itself. Even if I thought that I could do justice to the complexity of individual personalities in this space, it is not always clear what the private lives of composers tell us about the public meaning of their music.) The analysis evolves in a step by step manner, ultimately leading to results that were completely unanticipated at the beginning of the research. I hope that showing readers traces of the path that I have followed in getting to the point where I now stand will help them to understand how I came to my current conclusions. This paper can be read as a conversation, centered on a small number of especially pertinent musical examples. The discussion considers a succession of contrasting points of view, but a persistent theme – the cultural significance of complex, dynamic flow, particularly as that occurs in arch shapes and the related S-curves – acts as a connecting thread throughout the paper. No harm will be done if readers select topics from this paper according to their special interests; but readers will profit most from the paper if they follow the connecting thread all the way through.

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Acknowledgements I am indebted to many others for help and inspiration, none of whom can be faulted for any flaws that remain in this paper. Leonard Meyer, William Klenz, Iain Hamilton, Robert Cogan, Ernst Oster, and Robert Marshall – listed in approximate chronological order – made me aware of fundamental issues in the sphere of music. David Welland was my principal guide through the subtleties and rigor of college mathematics. George Welsh’s insights on the psychology of the creative process were as enlightening to me as they were informative. This paper would not have been possible without the many interlibrary loans obtained for me by Jimmy Smith at the local public library. I am grateful to those who have given me useful advice, helpful information, encouraging words, or words of wisdom concerning this manuscript (in alphabetical order): Thomas Fiore, Kyle Gann, Robert Gjerdingen, Robert Marshall, Brian McLaren, Eugene Narmour, Andrea Porth O’Connor, Lyle Sanford, Susan Shields, Nan Shostak, David Smith, and Dmitri Tymoczko. I have not yet acted upon all of the words of wisdom, but I am still at work, and I have not forgotten. Robert Gjerdingen and Eugene Narmour deserve special thanks for their many good-humored and insightful comments concerning an early version of this manuscript.. My undying thanks go to my wife, Jerri, not only for her expert editorial assistance, but also for her long-standing encouragement.

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About the Author My earliest musical education took place in Louisville, Kentucky, the home of the Louisville Orchestra Commissioning Project, which performed regular concerts of new works during the 1950s. A festival of contemporary chamber music, which I attended avidly, took place every Spring in the beautiful setting of Gardencourt, then the home of the University of Louisville School of Music. My instrument was the oboe. I became a National Merit Scholar. Having placed out of all the undergraduate music theory courses at the University of Chicago, my first formal instruction in music theory was a graduate seminar in analysis with Leonard Meyer in my senior year. After obtaining a degree in mathematics at Chicago, I studied Schenkerian analysis with Ernst Oster and contemporary theory with Robert Cogan at the New England Conservatory. Back at Chicago, I wrote a dissertation on Mozart’s teaching of composition with Robert Marshall.

Marshall suggested that I look into Ernst Kurth’s germinal ideas about melodic

structure, a suggestion not immediately acted upon that in retrospect has proven to be remarkably prescient. In a side-trip to Duke University, I studied counterpoint with William Klenz, and composition with Iain Hamilton. Klenz taught the Palestrina style from Jeppesen, but his general recommendations concerning a well-balanced melodic line were not unlike Schenker’s. Hamilton emphasized the importance not only of being open to new experiences but also of being true to ones instincts. Salvatore Maddi, at the University of Chicago, introduced me to the University of North Carolina psychologist, George Welsh, whose experimental research on preferences for visual art stimulated my thinking about the cultural meaning of abstract patterns in general. It was clear in those days that the computer was going to change the way people think about things, and I wanted to be part of that.

Professionally, I now work in information

technology as a systems-integration specialist, among other things. I am accustomed to thinking of information systems as structures that can be turned inside out to match other systems that are conceived by their originators as being organized quite differently. Understanding is achieved by looking for mappings that reveal what is invariant in seemingly different interpretations of structure. The systems-integration point of view is pervasive throughout this book. Although I am not a professional mathematician, my thinking was profoundly affected by the study of mathematics. Despite the association of mathematics with the sciences, as Paul Lockhart points out, mathematics is better understood as one of the arts. Mathematics is an art of abstract patterns, a potentially complex art, but one in which simplicity is regarded as the crowning virtue.

I hope I can convey to an audience of non-mathematicians the beauty of

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mathematical reasoning as a method of understanding relationships of all kinds, even – in surprising ways – the richness and majesty of music.

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Preface: The music says, “Listen! I have something important to say.” Music focuses the mind. In ordinary speech, we do not sustain controlled pitches with measured durations, as we do in singing. Singing requires greater mental discipline than ordinary speech. Either controlled pitch (not necessarily fixed pitch), or measured duration (not necessarily metric rhythm), when attached to a message, comprises a symbol that the message is of great importance. When we choose to sing a text or tap on a drum while we recite a text, it is because we believe that we have something worthwhile to say; and we want people to pay close attention. The symbols of controlled pitch and measured duration convey the importance of our message even in the absence of words.

Instrumental music lifts the symbolism of important

communication away from concrete meaning into the realm of perceptual, cognitive, and affective abstraction. Music is stylized. Music, like any other art form, is more selective than everyday life experience. We have more control over the content of art than we do over ordinary experience. Selectivity makes the difference between an ordinary life experience and a message.

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experience of art differs from the experience of life because art delivers us from distractions. By delivering us from distractions, art reveals to us a particular view of what is central to our lives. Life is complex, even in mundane matters. Household appliances work well for a time; then, they wear out, break down, and stop functioning.

In our careers and personal lives,

fulfillment is mixed with loss. Active struggle alternates with quiet contemplation. Even at rest, we are never inert. We are not like rocks that have rolled down a hill and lie motionless at the bottom. We consume food and air just to stay alive. When we stand upright, our muscles make fine adjustments to keep us from falling down. When we close our eyes, we do not see a flat gray screen at the back of our field of vision. We see an uneven, fluctuating, granular display. It is difficult to sit completely motionless for any length of time. If we attempt to quiet our minds, we are interrupted by stray thoughts. All is pulsation, energy, and vibrancy; light flickers in the shadows. When we look around us, in the city or in nature, we see much the same thing. We contend with the merging, swirling, and interweaving of automobiles on the street. We observe the leaves in the grass and on the trees moving quietly in the breeze, each contributing a different movement to the whole, some fluttering quickly on short stems, others swaying masterfully on sturdy branches.

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If we think about the events of a life in terms of their structure, movement, tension, and pattern, life, in fact, is not very different from music, except that music only exists in the realm of sound. In music, brightness, darkness, organization, complexity, movement, quietness, pulsation, energy, and vibrancy are exclusively qualities of aural perception, cognition, and affect. It is true that there is nothing in life that corresponds exactly to any particular aural quality, such as the sound of a perfect fifth.

Conversely, there is nothing in music that

corresponds exactly to the appearance of physical phenomena, such as fire or storms at sea. To imagine that there exist any specific associations between aural perceptions and physical phenomena requires, as Coleridge said, a “willing suspension of disbelief for the moment, which constitutes poetic faith.” This is all true, but we do find similarities between music and life outside music in the realm of psychological dynamic properties, tensions, and structure that are common to all affect and cognition, musical and non-musical alike. This is the best explanation for the fact, which has been verified abundantly by experimental psychologists, that – despite individual differences of opinion – people in our culture tend to agree in the aggregate about the feelings that are expressed by music. See Juslin and Sloboda (2011). Briefly stated, that is what I believe to be the case. A complete argument, considering the pros and cons of every variant interpretation, would be exponentially longer. Readers who would like to explore these issues in more detail will enjoy Meyer (1956), Newcomb (1984), Kivy (1989, 1990), Davies (1994), Hatten (1994, 2004), Robinson (1997), Cook (1998), Scruton (1999, 2009) , Lidov (2005), and Treitler (2011). This short list of references will lead the reader into a rich and complex literature, full of subtle distinctions and invigorating differences of opinion. Music, in my view, has meaning to the extent that it resembles life in general; and we recognize and value the resemblances. Music is informative to the extent that it can be understood as a stylized representation of life in general, which pares away irrelevancies. For this reason, I believe myself to be justified in holding the opinion that the contemplation of music is an essential component of the examined life. We understand music by examining its place in the context of our life experience. Music, however, cannot arouse affect or suggest meaning until we first have some sense of what it is. Identity is the basis of meaning. Therefore, the study of musical meaning begins with the analysis of musical structure. To think about musical structure in terms of its meaning is to approach music from a philosophical point of view. The philosophical point of view raises questions about definition and purpose. For example, what does it mean in analytic terms to say that a piece of music is

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expressive? What does it mean to say that a musical idea is distinctive? What does it mean to say that a piece of music has a balance of unity and variety? What does it mean to say that a piece of music is complete, with a beginning, middle, and end?

What is the purpose of coherence,

development, or narrative form? A serious attempt to answer questions such as these requires close analysis. To examine these questions closely requires, in my opinion, a mixture of conventional and unconventional methods. The unconventional methods include some mathematical techniques that first came into existence in the middle of the twentieth century, techniques borrowed from the disciplines of information theory and fractal geometry. The questions that will be asked here are rather different from the questions that are asked by a physicist. Physics looks for universal laws, such as the law of gravity, and invariant physical constants, such as the speed of light. It is important to our understanding of music, in fact, for us to be aware of general laws of perception, cognition, and affect. For example, it is important to be aware of cognitive biases, such as the primacy and recency effects and the distinction bias. Nevertheless, it is very difficult to generalize about music. Musical styles differ greatly from one another, not only from one culture or historical period to another, but even within the lifetime of a single composer. Furthermore, individual compositions are required by our culture to be original and unique. In the study of music, the exceptions are as important as the rules; and variation is as important as central tendencies. What might be considered “noise” in the data stream by a physicist might be important information for our studies. As a music theorist, I am working on the opposite end of the questions asked by the physicist, since my work begins with the analysis of musical artifacts that are the final product of the composer’s creativity. We are not alone in this endeavor, however, because there are other disciplines where researchers are accustomed to working with data that is hard to explain, either because of random variation, or because the observed data is the result of complex interacting processes. Like the biologist, we can identify and categorize the variations we find in our data. It happens that an equation used by biologists to describe the growth of the population of microbes in a medium can be adapted to describe the particular, idealized musical processes found in Fux’s cantus firmi. Like the financial analyst, we can develop batteries of indicators to evaluate our data from multiple perspectives.

Like the psychologist, we can look for underlying dimensions of

variability. Also, we can build theoretical models. To build a model is to create a theoretical representation of a tangible phenomenon. The intention is not to explain every aspect of the

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phenomenon in detail, but only to capture its broad outlines. We look for a compromise between theoretical simplicity and explanatory power. We prefer for the terms of the model to have some concrete meaning. A model is considered useful if it clarifies our thinking and if it leads to fruitful questions for further research.

In the final analysis, the main objective of this study is to

accomplish what we hope for a model to accomplish.

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Chapter 1 What is a Distinctive Musical Idea? Taking the Measure of the Cantabile The vocal style as a norm – the Fichtean curve – the prevalence of small, falling intervals and arches in folk song – Fux’s cantus firmi, and Schenker’s concept of melodic fluency – the prevalence of small, falling intervals in the cantus firmi – an indicator for the cumulative duration-rising with respect to time – rising intervals more common in major cantus firmi than minor – the cantus firmi as arches – meaningful discrepancies between modality and tonality – an indicator for melodic smoothness and sinuosity – closing motions in Fux’s cantus firmi smoother than opening motions, suggesting the Urlinie – the emergence of smoothness from the statistics of a random walk and the geometric constraints of goal-directed motion – the Urlinie – structural hierarchy – Kurth vs. Schenker. Leo Tolstoy wrote in the opening of Anna Karenina, “All happy families are alike; every unhappy family is unhappy in its own way.” Uniqueness – or so Tolstoy would have us believe – is inseparable from suffering. The philosophy of life embodied in Tolstoy’s novel, which I will take as a benchmark for comparing value systems in general, is that happiness is found, not in the glitter of fashionable and sophisticated society, but in the simple, unreflective lives of the peasantry. Tolstoy’s aphorism states a paradox that is central to life and art, placing into direct opposition the virtues of simplicity and complexity. The conflict between the two virtues is fundamental and cannot be resolved. It is true that, whether we are writing a melody or making a life, there are more ways to go wrong than there are to go right. This would be an argument in favor of simplicity – sticking to the tried and true. One might question, however, the extent to which we really want to live simple, unreflective lives. Most of us want more freedom than was experienced by a nineteenth-century Russian peasant. The character in Anna Karenina who best reflects Tolstoy’s philosophy is Constantine Dmitrich Levin. Ironically, Levin’s life is a complicated spiritual quest. Levin struggles through a period of dissolute behavior before he discovers his personal understanding of Tolstoy’s ideal of simplicity. It would have been artistically impossible for everyone in the novel, even Levin himself, to have always been happy in accordance with Tolstoy’s terms. A novel exclusively concerned with unalloyed happiness would have been meaningless – not simply because there

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would have been no conflict, and therefore no plot – but also because meaning is expressed most vividly through comparison and contrast. To clearly convey the idea of happiness, one most also convey the idea of the opposite of happiness. In general, vivid music is not that different in principle from vivid prose. To make a clear and convincing statement – whether musical or literary – is not possible without engaging in contraries. Pursuing this suggestion, even though the parallels are not exact, let us consider two examples of simple melodies. The hymn, “Amazing grace,” and the bugle call, “Taps,” both evoke a mythic, primordial sense of unity, strength, and peace. These melodies are rather similar to each other. They are not elaborate. Both use a limited number of notes in the major mode. The notes of “Taps” are entirely confined to a single triad. “Amazing grace” is pentatonic, but its principal tones outline the tonic triad. Both are limited to a range of one octave. They employ no chromatic alteration or dissonant leaps. They have similar wavelike contours, making broad arches climaxing on the dominant scale degree, primarily differing in how they approach the lower dominant. These melodies could be considered as representing a single archetype. They are a happy melodic family, so to speak. To say that a melody is distinctive, however, is to say that it is unconventional – the more distinctive, the more unconventional. Unconventionality is risky, to a greater or lesser degree. The norms of melody writing – adhered to more closely by some styles of music than others – are primarily intended to make melodies easy to singi. A melody cannot violate these norms without taking risks. A melody that is risky to the singer raises both psychological and physical tensions. Tension is expressive.

Risky melodies are edgy or poignant.

So, expressiveness and

distinctiveness are closely allied. Both the expressiveness and the distinctiveness of a melody depend on the approach it takes to risk-taking. If a melody is sufficiently difficult to construe, it will appear to be in some sense mysterious. The other-worldly quality of Gregorian chant, for example, is probably due in large part to its free rhythm, which does not fall into regular meters. The diminished seventh chord and the whole tone scale are mysterious, because – dividing the octave into equal parts – they are tonally ambiguous. There is a legitimate place for musical expression to be mysterious. A life without a sense of mystery is greatly impoverished. Without the whole-tone scale, for example, we would not have Debussy’s Syrinx for solo flute; and we would be much deprived. The range of possible musical expression defined by the norms of melody writing is very broad, and every point on the spectrum has value.

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There are at least three ways to think about the distinctiveness of a melody. If a melody is too irregular, it will be perceived as shapeless, lacking any identifiable feature that makes the melody different from other melodies. To begin at this point, however, would be to seize the most problematic end of the question. There are techniques for evaluating the statistical randomness of a series of events, but statistical randomness is not always the same thing as perceived shapelessness. A series of notes generated from a list of random numbers will now and then contain sequences that remind us of common scales and harmonies. Furthermore, to show that a musical idea is random, we would have to show that it cannot be explained by any conceivable notion of orderliness. That is, we would be attempting to prove a negative; and we would have no way of knowing when our task is finally accomplished. It is possible to think about the distinctiveness of a melody in terms of the distinctiveness of its parts, identifying especially prominent features belonging to the melody or striking contrasts found within the melody. To do this, however, we first need to develop standards of reference. The fastest way into the problem is to begin by identifying what makes a melody conventional. Historically, the normative melody is vocal in style, whether it is actually sung or it is performed on an instrument. The notion of a cantabile line has a role in the history of Western art music that is somewhat analogous to the representation of the human figure in visual art. The history of the cantabile is longer than the history of the major achievements of the major-minor tonal system, since it begins earlier and – with a few exceptions, such as works of Richard Strauss – lasts longer. The cantabile can not only be found in plainsong. Schoenberg’s atonal melodies are strongly influenced by traditional notions of the cantabile. The notion of the cantabile is not, to be sure, a single concept. The long exploratory line of the Palestrina style is, for example, quite different from the more or less regular cyclical waves of a dance, strophic hymn, or folk song. The former corresponds to what Ernst Kurth called Fortspinnung and the latter to what he called Gruppierung (see Rothfarb [1989], p. 33; the term Fortspinnung originated with Wilhelm Fischer in 1915; see, for example, Swain [2003], p. 184). The two categories are not mutually exclusive, because great musical compositions can be appreciated at many levels and from a multiplicity of perspectives. Bach, for example, wrote both dances and long developmental lines. In his long lines, Bach employed flexibly varying motives, which are, of course, small, regular groupings of notes. I will introduce the cyclical type of melody by drawing from David Huron’s (1996) study of folk song and the long line by drawing from Heinrich Schenker’s study of Fux’s cantus firmi.

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Our experience in singing melodies gives us a deep-seated sense of what is musically easy and what is musically difficult. As we have seen in “Taps” and “Amazing Grace,” all easy melodies have certain things in common, which tend to make them generic. Easy melodies have a narrow range; and they move diatonically in predominantly small intervals, with consonant leaps and regular rhythms.ii On the whole, low notes are easier to sing than high notes, assuming that one has a flexible technique with equal resonance in all registers. Although there are specific exceptions, such as dissonant intervals and the falling sixth, it follows that it is, generally speaking, easier to sing a falling interval than a rising interval. All of these factors have their influence on melodic shape. It has been a long time since music was strictly limited by the constraints of vocal production, of course.

In the violin family, developed in the sixteenth century, cross-string

technique allows the performer to play large leaps easily. The invention of the fortepiano in the eighteenth century extended the capability for making both subtle and dramatic contrasts of loud and soft volume at the keyboard. This invention did not lead to a sudden and cataclysmic change in the kind of music that was written, however. Although the piano is a percussion instrument, the fortepiano was considered an improvement over the harpsichord in part because its sustained tone allowed the performer to emulate the style of vocal music. Improvements in the mechanism of the brass instruments in the nineteenth century made it possible for them to play melodies in the vocal style using all of the tones of the chromatic scale in every register. With modern computer software, such as Finale or Sibelius, it is now possible to play music that no human performer – vocal or instrumental – would be able to play. There is very little practical limitation on the styles of music we can make today. It is understandable that composers do not want to be limited by the standard of vocal music, but the vocal style still sets an important benchmark by which we evaluate what is normative and what is distinctive. Since Western music historically originated as vocal music, the constraints of vocal production have had an influence not only on how vocal melodies are shaped but also on how instrumental melodies are shaped. We empathize with the singers that we hear, and we even transfer our experience in singing to our understanding of instrumental music. The vocal style does not set a limit on what composers can do, but it helps to define a dimension of variability by which we understand the character of melodies. Later, we will find it instructive to compare very specialized collections of vocal melodies – cantus firmi by Fux and Jeppesen – with similar melodies generated by computer that are created under varying constraints. Under certain conditions, we shall see that the time series

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geometry of a melody tends to induce a certain smoothing effect as a melody approaches a predetermined goal and the possibilities for expansion are reduced.iii In other words, a melody (or a segment of a melody that has a closed form) can be modeled as a circumscribed random walk with diminishing variability (a concept that will be discussed later). In order to carry out this analysis it will be necessary for us to compare melodies as having relative degrees of smoothness or sinuosity, a concept of melodic motion that is neither categorical nor hierarchical. The results of these studies will have implications for our understanding of Schenker. To the extent that the smoothing effect bears some resemblance to Schenker’s concept of the Urlinie, it implies a somewhat nuanced view of Schenker, because the smoothing effect is not strictly deterministic. The smoothing effect suggests that, under the right circumstances, there is a logical basis for the Urlinie; but it leaves room for interpretation by the composer, who has a choice whether to enhance the smoothing effect (as in Fux), or back away from it (as often happens in Jeppesen). If the style of the composer is such that the choice to enhance or back away is made consistently, it suggests that the composer is acting out of fundamental aesthetic values. The decision to enhance the effect suggests a somewhat rationalistic preference to emphasize the difference between order and disorder and to assert the dominance of the former over the latter. The decision to soften the effect suggests a less structured perspective in which order and disorder are more equally and freely intermingled. From this point of view, the Schenkerian interpretation of melodic structure arguably represents a rationalistic aesthetic, which may or may not be fully appropriate for every musical example. It happens that Jeppesen made a deliberate effort to immerse himself in the style of sixteenth-century counterpoint to a greater degree than Fux, and it is arguable that Fux’s cantus firmi are more obvious candidates for Schenkerian analysis than Jeppesen’s (not to say that it is impossible to interpret Jeppesen in the Schenkerian manner, but only that Jeppesen’s closing motions are more variable in smoothness than Fux’s). It is well known that Schenker had a strong aversion to the sixteenth-century style, however. We should not expect that Schenker’s system of interpretation would fit a loosely structured style quite as well as it fits a more highly structured style. Before we go any further, if we are to avoid any misunderstandings about methodology in the following discussion, we need to distinguish between fact, theory, and interpretation. A fact can be proved to be either true or false. A theory, in the narrow sense of the word, is a rational explanation of a body of facts, which – in Karl Popper’s view, at least – cannot be proved true, but can be proved false. Theory in this narrow sense should be distinguished from speculative theory, which is primarily concerned with logical analysis of systems of thought.

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Popper’s understanding of the scientific method has been called into question by Thomas Kuhn, who distinguished between normal science and scientific revolutions that result from paradigm shifts. In actual practice, it is possible for rival theories to account for the same facts to a similar degree of approximation. The evidence for or against a theory may depend on questions that are difficult to resolve, concerning methodology or the existence of confounding factors. It follows that the choice between theories may not be as clear-cut as stated by Popper. The choice between theories may depend on subjective evaluations of their relative simplicity, extensibility, and explanatory power. An interpretation is a rational explanation of a body of facts that depends on individual judgment and is, therefore, inherently subject to differences of opinion. An interpretation cannot be proved either true or false, but arguments can be presented to justify the reasonableness or unreasonableness of an interpretation. As the terms are defined here, much of the analytical tradition that musicians call theory would be better understood as interpretation, because skill, judgment, and musical sensitivity are required to create an interesting and insightful analysis of a composition. More objective, scientific knowledge is important if one is trying to resolve issues concerned with the nature-nurture debate or the underlying psychological motivations of listeners; but interpretation is essential for critical analysis that is directed toward questions of meaning and aesthetic judgment. I hope that it will be clear from the context in my own writing when I am speaking of fact, theory, or interpretation.

Folksong I will use Charles Adams’s (1976) classification of melodic contours, discussed by Cook (1987, pp. 196-198). See Fig. 1.00. Adams typology is based on the relative pitch height and location in time of four notes in a melody: the first, last, highest, and lowest. There are only two or three notes in some of Adams’s classifications, because of duplications. In theory, I am equally concerned with all of the types; but, in practice, I will mainly be interested in the second line of the chart, which depicts falling, level, and rising arches, and the last line, which depicts falling, level, and rising Open-Low-High-Close forms. The latest version of my dynamic model, the Limited Growth Model (still under investigation), attempts to describe both arches and OpenLow-High-Close forms, together with their inversions, and also transitional forms between the canonical types, for suitably normalized cases. In the melodies of many cultures, it has been found that falling intervals are more common than rising intervals.iv With the exception of a small sample of north Indian classical

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music, this tendency has been found in Albanian, Bulgarian, Iberian, Irish, Macedonian, Norwegian, and African-American folk songs, as well as in Australian aboriginal music, Chinese folk songs, traditional Korean music, Ojibway, Pondo, Venda, and Zulu songs. Since there is an overall balance between the total rising and falling motion (what goes up must come down), rising intervals tend to be large, and falling intervals tend to be small.

Fig. 1.00. Adams’s classification of melodic contours.

Huron found from his study of European folksong – drawn from over six thousand melodies in the Essen Folksong Collection – that the arch-shaped phrase (called convex by Huron) is the most common type, amounting to almost forty percent of the whole. v Arch shapes are not a majority of the melodic shapes, but they are a plurality. Arch shapes are four times as common as their symmetrical opposites, the troughs (called concave by Huron). Ascending and descending phrases are the next most common types, accounting for nearly fifty percent of all phrases. While ascending phrases tend to be followed by descending phrases, the opposite is not true. The preference for arch shapes can also be found in whole melodies. If one represents each phrase by the average pitch-height of all the notes within the phrase, over forty percent of the melodies were found by Huron to have an overall arch-shaped contour. Visual inspection of the graphs provided by Huron suggests that phrases tend to ascend comparatively quickly at the beginning and to descend more slowly toward the end. (This does not necessarily imply that the highest note occurs in the first half.) From the point of view of vocal production, this suggests that phrases are typically constructed to allow the voice to spend more time relaxing than tensing. We seem to tolerate a quick increase in vocal tension, if we do

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not have to sustain it for very long. Perhaps, from a temporal standpoint – though not necessarily from the standpoint of the total expenditure of vocal energy – the affect of a typical phrase is predominately one of relaxation.

Arches The motivation underlying arches would appear to be more complex. One must take care in discussing melodic arches to avoid overemphasizing the highest note, or climax. First, this reduces a potentially complex melody to the attributes of a single note, which is a disproportionate oversimplification. Second, to focus on the climax is to suggest that the motivation underlying the arch shape is primarily theatrical, which may or may not be the case. The climax must be understood in context. The climax is only the most salient feature of a larger pattern.

Fig. 1.01. Gardner’s depiction of the “Fichtean curve” (from ingridsnotes.wordpress.com).

It happens that the arch-shaped profiles of Fux’s six cantus firmi (Ex. 1.1) bear a general resemblance to the course of a novel’s emotional development (Fig. 1.01) as described by Gardner (1984, pp. 187-8). Gardner calls this the “Fichtean curve,” presumably referring to the Fichtean dialectic between Self and Not Self.

We should not expect a direct, one-for-one causal

relationship to exist between Fux’s melodies and Gardner’s curve, because Gardner’s curve is only an illustration of a general concept. What bears investigation, however, is why there should be any resemblance at all. The base-line (a) of Gardner’s chart, corresponding to the tonic scale degree of an archshaped melody, is described by Gardner as the “normal” course of action, the course that the central character would take if he cared only for safety or stability and, so, did not assert his independent will. The course of action that the character actually does take is represented by a rising and falling line (b), which is not smooth but moves through a series of increasingly intense

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climaxes as the character struggles against odds and braves conflict. Finally, in the denouement, the curve of the action descends back to normal (c), either because the will of the character has been overwhelmed or because he has won, and his situation is once more stabilizing. We expect musical phrases to end with cadences, which establish a greater or lesser degree of closure. An arch (or its inverse, a trough) is an indirect approach to the cadence. The broad course of such an arch, and each of the individual short-term fluctuations that occur in the course of the arch, can be compared to what Meyer (1956, 1973) called a “deviation from expectation,” where the concept of “expectation” is analogous to what Gardner called the “normal” course of action.

Meyer thought of music as an essentially autonomous art-form,

however. He argued that musical expectations and deviations ought to be understood in purely musical terms, as music referring to itself, not referring to external meanings, as would be the case in Gardner’s example of a novel, where the central character is in conflict with external forces, a conflict involving moral choice, and therefore profound meaning. The meaning of music, admittedly, is abstract. Even representations by music of concrete events, such as fire or storms at sea, are not photo-realistic. However, the structural parallelism between the narrative arc of a novel, as described by Gardner, and the arch of a melody, as seen most graphically in Fux’s cantus firmi, is highly suggestive. The underlying forces governing the emotional development of a novel and the rise and fall of a melody must be very similar, if not in specific detail, at least in their general dynamic relationships. In other words, even if the tensions between the characters in a novel and the tensions between tones in a melody have a different concrete basis (people being physically different from musical tones) the tensions appear to work themselves out in similar patterns. For this reason, I question that music is an autonomous artform at its deepest levels. Correlation, of course, does not prove causation. I would not contend, as a general rule, that music imitates literature, or vice versa. On the other hand, I would suggest that the narrative arc often found in literature and the arch shape often found in melody may have a common cause. This is not the place for a detailed comparison between Fichte, Gardner, and Meyer. I will only say this: to find a connection between literary form and musical form it is useful to turn our attention away from the agency of independent actors, such as the central character and his or her antagonist in a novel. Perhaps, there are special cases where contrasting themes or motives in a melody might be comparable to independent actors, but not all melodies are constructed like that. Most instrumental music, for example, cannot reasonably be interpreted as telling a story about concrete events in the physical world and is, therefore, not best understood as program

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music. The point is not that a melody represents a tangible story about events happening in the real world, necessarily – although that might be the case when music is explicitly associated with a text or program. The point is that both a melody and a story appear to be expressions of similar psychological processes. I question Meyer’s choice of the word deviation, because I think the negative connotation of the term does not fully convey the underlying motivation for an arch form. We learn from population biology and technology forecasting that the summation of two terms, at least one of which is non-linear, is needed to explain an arch as a closed form. One of the terms is expansive, the other contractive. Although the word deviation can be used to describe the expansive factor, I prefer the word exploration.

The contractive factor can be called consolidation.

We can

understand a melodic arch as resulting from the combination of simultaneous tendencies to expand and contract, which I call exploratory behavior.vi Whether we are talking about population biology, technological developments, the course of a melodic arch, or the plot of a novel, the arch shape emerges out of important, meaningful struggles, including life and death struggles, over finite resources, strengths, or capacities.

Somehow, despite radical differences in concrete

circumstances, all of these struggles follow analogous paths.

Fux Johann Joseph Fux (b. 1660-d. Vienna ,1741), Kapellmeister to three Austrian emperors, was a prolific composer, best known to us today as the author of a treatise on counterpoint, Gradus ad Parnassum.vii Published in 1725, the Gradus – a successor to earlier works by theorists such as Zarlino, Artusi, Diruta, Bismantova, Berardi, Banchieri, Zacconi, Bononcini, Lippius, and Mattheson (see Bent [2002], esp. pp. 561-570) – was written in Latin as a dialogue between a student, Josephus, and his master, Aloysius, who was intended by Fux to represent the voice of Palestrina. Each era has adapted Fux’s treatise to its own purposes. The Gradus was used by Franz Joseph Haydn to teach himself counterpoint. W.A. Mozart’s father, Leopold, owned an annotated copy of the work. Mozart himself used the Gradus in teaching, as did Beethoven, who studied Fux’s method under Haydn, Johann Schenk, and Johann Georg Albrechtsberger. Albrechtsberger adapted the exercises to major and minor tonalities.

Albrechtsberger’s

interpretation of the teaching was influential throughout the nineteenth century, until Brahms returned to Fux’s work in its original form. The Gradus was used by Richard Strauss and Paul Hindemith. Heinrich Schenker’s (1910) teaching of counterpoint explored the artistic issues raised by Fux. Knud Jeppesen (1892-1974), best known for his close study of the Palestrina style,

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recreated Fux’s method in a form that is more faithful to Palestrina. Jeppesen was a composer, who dedicated his counterpoint text to his teacher, Carl Nielsen. All of Fux’s teaching is concerned with adding counterpoint to a given cantus firmus. There are five species of counterpoint, in which different rhythmic values in the counterpoint are set against the cantus firmus, which remains unchanged: (1) one note in the counterpoint against one note in the cantus firmus; (2) two against one; (3) four against one; (4) syncopation and suspensions; (5) mixed note values. Fux gave us six cantus firmi, one in each of the church modes (Ex. 1.1).

Ex. 1.1. The six cantus firmi by Fux.

The cantus firmus is unusual in the training of a composer because it imposes a compact, well-rounded form, which, for all its brevity, is complete in itself. Other exercises worked by student composers are more loosely structured: improvising accompaniments to a numerically coded, figured bass, for example; and the writing of canons, where one voice follows another in strict imitation, the form being determined by what makes feasible counterpoint as one measure succeeds another. In only the last few years, a remarkable exercise of the more freely structured type has been discovered: the partimento, where the student composer improvises imitative counterpoint, and even fugue, over schemas set forth in a bass line (see Christensen, Gjerdingen, Sanguinetti, and Lutz [2010]). Partimento, developed mainly in Naples, was taught to highly influential composers of eighteenth- and early nineteenth-century Italy, such as Paisiello,

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Cimarosa, and Zingarelli. The discovery of partimento reveals that the level of improvisatory and contrapuntal skill developed by Italian composers of the period was much higher than we might have imagined. The discovery suggests a fundamental reappraisal of our assumptions about the craft of musical composition and the mental habits of composers. For our purposes, the structure of a well-made cantus firmus is of interest, not because species counterpoint was uniquely important in the education of composers, not because the structure of Fux’s cantus firmi would have had any direct influence on the structure of free-style compositions in the repertory, and not merely because Schenker thought the subject was important. The cantus firmi are of interest to us because they provide us with a laboratory where we can study the principles of melodic construction. The cantus firmi are not merely lessons. Although the cantus firmi are as formally constrained as haiku, they are genuine compositions, which express definite musical thoughts. Though small in number, this collection of six cantus firmi occupies a pivotal role in our historical understanding of the concept of a well-shaped melodic line, being written in the old church modes, but having been adopted by some of the most important of the later tonal composers for the study of counterpoint. The melodies are simple enough that they allow us to focus our attention on well-defined issues. At the same time, they are complex enough to raise challenging questions about the dynamic processes by which a melody is formed. That is why the cantus firmi have been successful models for teaching, and that is why they are useful for our purposes. Before we study truly complex melodies, we need to develop a clear understanding of the analytical techniques that we will employ. For that we need prototypical examples. In choosing the cantus firmi for my examples, I am joining a long tradition.

Ex. 1.2. Cantus firmus in the Dorian mode by Jeppesen.

Nevertheless, Fux’s cantus firmi are a substantially unique genre.

They are not a

representative statistical sample from any larger body of works. There are even subtle stylistic differences between cantus firmi by Fux and those by other theorists. In one of Jeppesen’s cantus firmi in the Dorian mode, for example (Ex. 1.2), the upper-neighbor B-flat decorates a tone – the fifth scale degree – that is approached by a large leap. The effect is much more expressive than that of the comparable upper-neighbor in Fux’s Aeolian cantus firmus, where the decorated note is approached by the leap of a third. Jeppesen’s closing motion contains a descending fourth, which

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makes it more irregular than a comparable passage by Fux. None of Fux’s closing motions contain leaps greater than a third. Fux and Jeppesen tended to follow different developmental narratives. If we follow Fux’s example, we will be led to a different concept of the ideal of unity, balance, and flow than we learn by following Jeppesen’s example. The location of the climax in Jeppesen’s cantus firmi is highly variable, covering the complete range of possibilities; but almost half (9 out of 19) of the high points occur in the first quarter of the melody. In the cases where Jeppesen’s cantus firmi reach their main climax early (most commonly in the Dorian mode), the cadences are likely to be expanded into secondary elaborations.viii Not infrequently, therefore, Jeppesen’s cantus firmi appear to be somewhat looser in construction than those of Fux. A stylistic review of Jeppesen’s cantus firmi, using Exploratory Factor Analysis, will be found in Appendix B.

Ex. 1.3. Doxology from the Introit of the Mass in the first mode.

Jeppesen was well aware of the possibilities for variety in the construction of melodic lines. He compared the well-balanced melodic line of Palestrina against the early-climaxing line of a Gregorian melody and the late-climaxing line of Bach (p. 85). He described the form of modal music as freer than that of tonal music in several respects, such as the possibilities for harmonization and the distribution of cadences. The doxologies from the Introit of the Mass that Jeppesen cited as examples of Gregorian chant (see the Liber Usualis, Benedictines of Solesmes [1963], pp. 14-16) are distinguished by a tendency to push elaboration outward from the middle of a formal unit to the beginning and end. For instance, compare the Jeppesen cantus firmus just cited with an example of a doxology (Ex. 1.3) in the first mode (which corresponds to the Dorian mode). Jeppesen’s cantus firmus resembles Fux in style, of course, much more than it resembles a doxology. What the example by Jeppesen shares with the doxology is a relatively discontinuous flow through the denouement. The differences are subtle; but, compared to the absolute closure

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found in Fux, the cantus firmus by Jeppesen and the doxology both show a certain tolerance for ambiguity. Perfect closure does not appear to be an objective of either idiom. A doxology is divided into three parallel passages. The middle of each passage is taken up by the recitation tone. Elaboration is confined to the opening that leads up to the recitation tone and the closing that leads back down from the recitation tone. It happens that the close of the third passage, which is the close of the entire doxology, is the most elaborate of all. Notice that the concluding tonic in the example is approached by leap – something that never happens in the cantus firmi – so it is not tied closely to the structure of the melody as a whole. The main body of the melody and the tonic are as distinct as heaven and earth, the one floating lightly over the other. We should be cautious in concluding that any particular ideal of the well-made melody is a universal law. It follows that we should not expect a direct transfer of knowledge gained from Fux’s cantus firmi to the rest of the musical repertory. True, the cantus firmi can be thought of as prototypes for the middle-ground lines described by Schenker’s theory of structure; and, in this sense, they raise fascinating questions. Nevertheless, the main purpose of studying Fux is to develop problem-solving techniques that are flexible enough that we can adapt them to new circumstances as we find them.

Schenker The name of Heinrich Schenker (1868-1935) is synonymous with the concept of musical tonality as an intricate, hierarchically nested, honeycomb structure. Even theorists who do not agree with Schenker, I think it is fair to say, will agree that he raised some of the most important and thought-provoking issues that were investigated in the twentieth-century theory of tonality. Schenker discussed at length in Kontrapunkt I the question of how to write a good cantus firmus.

ix

Although the discussion appears in an early work by Schenker, some of Schenker’s most

important ideas can already be found there in germinal form. Here, we see Schenker struggling with the difficulty of describing products of the creative imagination in terms of specific rules. He expressed his ideas in two different forms: narrowly specific prohibitions and general, holistic guidelines. Both types of statement are crafted to instruct while leaving much to the imagination of the composer. In this section of his book, Schenker can be thought of as being more concerned with rhetoric than syntax. In other words, with few exceptions, he is concerned with the effective presentation of musical ideas (even if stated in terms of strict but otherwise open-ended prohibitions), rather than specific, prescriptive rules.

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Schenker’s prohibitions actually have characteristics of both syntax and rhetoric. They resemble syntax in their specificity, but they resemble rhetoric by being open-ended.

The

difference between syntax and rhetoric, for present purposes, is the difference between rules and guidelines. Syntax is specific; rhetoric is general. Syntax is concerned with correctness; rhetoric is concerned with effects. Syntax imposes constraints; rhetoric enables.x Neither one is creative in itself, but they both serve the purpose of paring back irrelevancies and giving focus to the creative impulse. The Schenker of Kontrapunk I did not hesitate to express strong views. He criticized the church modes unrelentingly and at great length. Although Schenker was a galvanizing, epochmaking thinker, those readers who take a broad view of music history will not find Schenker’s arguments against the church modes convincing. It is clear that where there is a discrepancy between modality and tonality, Schenker would make an unequivocal value judgment in favor of tonality. If we are to take modality seriously, we cannot avoid questioning Schenker’s claims. Within the realm of what Schenker found acceptable, however, he was surprisingly indulgent, in my view, when discussing his first love, counterpoint. At the most general level, Schenker wrote (pp 17-18): “We must aim for a complete equilibrium of the tones in relation to each other, in contrast to the predominance of individual, independent fragments characterized by rhythmic variety and a harmonic common denominator.” He explained that “everything must be avoided in the cantus firmus that would give it an individual character – that is, turn it into a kind of real melody in the sense of free composition.” According to Schenker, the objective in writing a cantus firmus is to create an overall shape without the internal melodic groupings such as one encounters in free composition (pp. 94-95). Schenker’s list of prohibitions was very long, but we should not think that his recommendations were primarily negative. Given that one accepts the general aims of a cantus firmus, Schenker was not prohibiting anything that a composer would strongly insist on doing. The importance of Schenker’s prohibitions, I would argue, is that they are non-directive in the sense that they establish limits without giving specific instructions.

Schenker was thereby

showing respect for the possibilities of the creative imagination. The combined effect of the prohibitions allows for a great variety of fully formed cantus firmi. Schenker actually did make positive recommendations, in addition to those already mentioned; but first we should review the prohibitions. I have added a few to those listed by Snarrenberg:

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

All rhythmic variety must be avoided within the cantus firmus (p. 18).



A cantus firmus should not be more than fifteen or sixteen notes long (p. 20).



The cantus firmus must move only within the space of a tenth (p. 41).



No tone shall be presented twice in succession in the cantus firmus (p. 42).



The apex of the melody should not be repeated (p. 100).



Avoid successions of tones that would be conspicuous as an arpeggiation or figuration of a single chord (p. 19).



Avoid configurations that “circumscribe an individual tone by means of its neighbor note” (p. 19).



Chromatic motion is prohibited in the cantus firmus (p 46), but modulation is permitted (p. 101).



The cantus firmus must avoid all dissonant leaps, either directly or as the sum of a succession of tones (pp. 52-53, 68, 71, 73).



The cantus firmus must avoid several leaps in different directions, because they are too expressive (pp. 92-93).



The cantus firmus must avoid several leaps in the same direction, because they would tend to exceed the permissible range (p. 93).



Give preference to the ascending octave over the descending (p. 79).



The fourth should not be used too often in succession (p. 81).



The closing successions 2-7-1 and 7-2-1 are considered poor (pp. 102-104). Schenker’s rules are consistent with Fux’s practice, on the whole, although there are

exceptions. Fux outlined two complete triads in both the Lydian and Mixolydian melodies. Fux employed neighboring tones in four of the six cantus firmi. Neighboring tones appear as cadence formulas in the Phrygian and Lydian modes. We will return to this subject when we study tones of the scale that have a privileged role in the tonal hierarchy. Schenker’s positive recommendations include these simple rules: the cantus firmus should begin on the tonic (p. 33) and end on the tonic (p. 102); and the second is the only horizontal dissonance that melody can use (p. 83). There are two more positive recommendations, which are stated in general, holistic terms, not as precise rules. These are important ideas in Schenker’s system of thought. In the Dorian cantus firmus – which Schenker considered to be particularly beautiful – Schenker spoke of the unfolding of the D minor triad as the melody moves first from D to F, and

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then to A (p. 54). This unfolding takes place in the opening motion of the cantus firmus. Although Schenker did not say so, the closing motion, moving from A back to D, is also an unfolding of the tonic triad. “Such unavoidable aggregates,” Schenker wrote, “cannot and should not be subject to any restriction.” Schenker only prohibited smaller and more limited aggregates that tend to form sharply defined subunits within an otherwise homogeneous context. I will discuss the projection of privileged tones – such as the tones of the tonic triad – later, when I turn to a discussion of the tonal hierarchy. Another important positive recommendation is found toward the end of Schenker’s discussion of the well-made cantus firmus (pp. 94-100). Schenker advocated what he called melodic fluency, which entails a kind of wave motion. xi Schenker’s concept of melodic fluency led to one of his most significant ideas, the latent line, and ultimately led by stages, according to Pastille (1990), to Schenker’s concept of the fundamental line (Urlinie) as we know it.xii We will return to these concepts in a moment. After giving recommendations on how to avoid the dangers produced by larger intervals, Schenker wrote (p. 94):

Such procedures yield a kind of wave-like melodic line which as a whole represents an animated entity, and which, with its ascending and descending curves, appears balanced in all its individual component parts. This kind of line manifests what is called melodic fluency, and one may confidently state that the second (cf. §21) – as the smallest interval and agent of rescue in cases of emergency – is the primary ingredient of melodic fluency. Melodic fluency, then, is a kind of compensating aesthetic justice vis-àvis the overall shape, within which each individual tone is a constituent part of the whole as well as an end in itself.

Schenker then went on to say (p. 95) that we desire to see the requirement of melodic fluency in free composition – fulfilled, for example, in a melodic line of a composition for piano or violin. “Such a line may be based on the postulates of polyphony and thus may tend to express, through itself, several latent voices in a unified fashion.” Schenker gave an extended example of such a latent line (p. 96) in the prelude of J.S. Bach’s English Suite in D Minor (Ex. 1.4). The latent line is essentially what we would now call a middle-ground reduction of the melody. Notice

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that Schenker normalized the rhythms; so, tones do not occur at precisely the same time in the latent line as they appear in the original melody.

Ex. 1.4. Schenker’s example of a latent line, p. 96.

Schenkerian analysis is concerned with more than the mere mechanics of finding implied linear connections because the connections could be random. Since every tone of the chromatic scale lies within a whole step of some tone of a perfect triad, it is a foregone conclusion that one will find smooth voice-leading connections between the implied harmonies of a melody, as long as one is willing to disregard register transfers. The most interesting questions of Schenkerian analysis have to do with the interpretation of the patterns formed by the latent lines, such as the long line that Schenker called a Zug.

Ex. 1.5. Reductive analysis of the Dorian cantus firmus in the Schenkerian manner.

Example 1.5 shows a reductive analysis of the Dorian cantus firmus, illustrating a few basic concepts of the Schenkerian method, which Ernst Oster emphasized in his teaching. The notation is flexible, since it displays both hierarchic (vertical) relationships and melodic (horizontal) relationships. Notes deemed to be merely embellishing are shown without stems. Slurs indicate the principal melodic motions.

Notice how the opening motion of a fifth is

articulated into two thirds, and the embellishing tones are connected to the main lines by subsidiary motions of a third.

I interpret the second occurrences of the notes D and F as

subsidiary, and I interpret the first occurrence of the note G as a passing tone (indicated by the symbol PT). This interpretation emphasizes the linear ascent that underlies the opening motion. Special attention is given to the closing motion by assigning its notes white note heads, because of the special importance that Schenker gave to this schema, the Urlinie, in his mature writings. If

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the tones of the Urlinie were more spread out, we might want to label them with their scale degree numbers, to call even more attention to them. The closing motion in the Dorian cantus firmus corresponds to what we now know as the fundamental line, or Urlinie, of a free composition. The Urlinie is (1) a latent line which (2) unfolds the tonic triad in (3) descending (4) stepwise motion (5) ending on the tonic scale degree. Three of the five criteria just listed are included in the positive recommendations that Schenker gave for the construction of a well-made cantus firmus. The third and fifth criteria – that the Urlinie descends to the tonic scale degree – are implicit in the understanding of the Urlinie as a closing motion. As the reader will already know, there are three forms of the Urlinie, the most common form descending from the third scale degree, the next most common form descending from the fifth scale degree, and an uncommon form descending from the octave above the tonic. The Urlinie is harmonized by the progression I-V-I, known as the Bassbrechung.

The

combination of the Urlinie and Bassbrechung is called the Ursatz. Since these are specialized technical terms, invented by Schenker, we usually employ the original German rather than English translations. It is reasonable to surmise that Schenker drew his inspiration for the Urlinie from the closing motions of these cantus firmi. (We should not conclude from this that composers who studied Fux were influenced by the cantus firmi to write the Urlinie into the deep structure of their own works in the free style. That is a completely different matter.) Given the reductive principles of later Schenkerian analysis, the main obstacle to reconciling Fux’s cantus firmi to the archetypical forms of the Urlinie is that two of the melodies unfold the subdominant triad rather than the tonic triad. The Phrygian melody is one of these, and the Mixolydian is the other. This is an inheritance from the practice in Gregorian chant, where C took the place of B as the dominant in the Phrygian and Hypomixolydian modes, and A took the place of G in the Hypophrygian mode (see Jeppesen, pp. 59-82). This change took place about the year 1000 A.D. The change from B to C was presumably made to avoid having such an important tone as the dominant scale degree be in a tritone relationship with F, and the change from G to A was a parallelism. The concept of the Urlinie raises a deeper issue, however. The very name, Urlinie, translated by Oster as fundamental line, conveys the special importance that has been attached to this structure. Since the arch, however, is the most prevalent shape for phrases and melodies in folk song (representing the plurality of shapes, though not the majority), there is a sense in which the arch could also be considered a fundamental shape. The arch is at least a special case that is

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worthy of our attention. It may not be possible to resolve decisively the question as to which shape is more fundamental, the Urlinie or the arch. From the rather broad perspective that I take on questions of melodic shape, the Urlinie seems to be best understood as a special case. The fact that the issue can be raised at all, however, suggests that we need to ask further questions about the classification of melodic shapes and the processes that underlie them. A thorough investigation of melodic types is a subject for another day. xiii

Rising and falling motion Consistent with what Vos, Troost (1989), and Huron (1996) found in folk songs, descending intervals are the majority (60 percent) of the intervals in the cantus firmi by Fux; and the descending intervals tend to be smaller than the ascending intervals. Since the cantus firmi all begin and end on the tonic, the sum of all the intervals in each melody (counting each rising interval as a positive number of scale steps, and each falling interval as negative) is always zero. It necessarily follows that the ascending intervals are larger on average (almost half again larger) than the descending intervals. The cantus firmi are not all alike, however; and the differences between them are musically important. To sing a rising interval requires an increase in vocal tension, and the opposite is true for a falling interval. The distribution of rising and falling intervals in a melody has an effect on the accumulated vocal tension required to sing the melody, which contributes to our perception of the feeling expressed by the melody. Not only do people prefer to sing melodies that have a preponderance of falling tension, but listeners may, empathetically, feel that such melodies are more relaxing to hear. The ratio of rising to falling intervals varies in Fux’s cantus firmi. The Dorian cantus firmus has, relatively speaking, more falling intervals than any of the others. The Lydian cantus firmus stands out in the opposite direction, because it has more rising intervals than falling intervals and is, therefore, contrary to the norm. melodies begin and end on the tonic.

This raises an interesting paradox.

Both

In that sense, therefore, they are ultimately flat.

Nevertheless, if we think about the direction of the melodic intervals alone, disregarding for the time being the sizes of the intervals, the Dorian melody has, in that special sense, an overall tendency to fall. The Lydian melody, by the same standard, has a tendency to rise. It is interesting to see how the rising and falling intervals are distributed throughout the cantus firmi in time. We can do this by constructing a mathematical indicator.

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Before we proceed any further, I should say a few words about the role of mathematics in this study. Although the influence of mathematics on music theory is growing, I will assume that the majority of my readers know more about music theory than they do about mathematics. Mathematics raises foundational questions. The purpose of mathematics in this study is to represent musical patterns in a way that clarifies questions concerning musical structure and the relationship between musical structure and musical meaning. These questions are ultimately philosophical, in the sense that they are questions about what it is reasonable for us to believe about large issues. The questions may be empirical as well; but they are not necessarily so, because the questions concern qualitative cultural ideals as much as they concern quantifiable observations about compositional practice. Mathematical indicators can show us places where a composer expresses distinctive preferences. We cannot read a composer’s mind, but where we find indications of preferences there is some evidence for at least the appearance of purposefulness. The appearance of purposefulness suggests the possibility of meaning. Mathematics, unfortunately, cannot be read at the same pace as ordinary English prose. The reader who wishes to get a general overview of this text might want to skim through the mathematics. Those who wish to understand how the formulas actually work are advised to temporarily make a dramatic reduction in their reading speed. To understand mathematics, it is necessary to think about the meaning of every word individually and how the meaning of every word relates to that of every other word. To avoid slowing the pace of the reading too much, explanations of the mathematics that require either detailed descriptions of calculations or a relatively high level of theoretical abstraction will be placed in notes. The first indicator we will discuss is concerned with the relative amount of time devoted to rising and falling intervals. The duration of an interval is measured as the elapsed time from the start of the note that begins the interval to the start of the note that completes the interval. Points are scored for rising and falling intervals. For each rising interval we add the duration of the interval to the point score, and for each falling interval we subtract the duration of the interval from the point score. We keep a running subtotal of the scores, adding to the cumulative score of rising and falling intervals for every note that occurs from the beginning of the melody to the end. Call this running sub-total the duration-rising indicator, for short.

It is a time-weighted

cumulative total of the rising intervals minus the falling intervals. To make one melody comparable to any other melody, we will normalize the indicator. We will divide the running subtotal of the duration scores by the total elapsed time for all of the intervals in the melody; so, the final score will range between the extremes -1.00 and 1.00. The

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final score will be -1.00 when all of the intervals are descending, and it will be 1.00 when all of the intervals are ascending. Furthermore, we will calculate the elapsed time at any given moment as a decimal fraction ranging from zero to one, starting at the beginning, and then dividing the time at which each interval is completed by the total elapsed time. The normalized graph shows the relative elapsed time on the X-axis, and it shows the relative cumulative time devoted to rising minus falling intervals on the Y-axis. Since the scores range between -1.00 and 1.00, the possible range in the scores is two points. It follows that the final duration-rising score is not the same as the percentage of time that is allotted to rising intervals. To get the percentage, add one to the final duration-rising score and divide that by two. For example, if the final duration-rising score is -0.20, add one to that to get 0.80, and divide the sum by two to get 0.40. That means that forty percent of the time was devoted to rising intervals.

Fig. 1.1. Duration-Rising in selected cantus firmi by Fux.

If more time is devoted to falling intervals, as in the Dorian cantus firmus, the indicator will have an overall downward trend, beginning at zero and ending at a negative number, spending little time in the positive range. A melody like the Lydian cantus firmus, which spends more time with rising intervals, will have an overall upward trend, ending with a positive number. This is shown in Fig. 1.1. The chart also shows the indicator for the Mixolydian melody, which is close to the average. It can be seen from visual inspection of the chart that the opening motions of the

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melodies tend to have a mixture of rising and falling intervals, with the greatest concentrations of falling intervals in the closing motions.

Fig. 1.2. Duration-Rising final scores for all of the cantus firmi by Fux.

Fig. 1.2 shows the final duration-rising scores of Fux’s melodies in order from lowest to highest. This order does not correspond to the order of the modes (not in the order: Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Ionian). What stands out is that all of the modes with minor tonics (the Dorian, Phrygian, and Aeolian) have lower final duration-rising scores than all of the modes with major tonics (the Mixolydian, Ionian, and Lydian). This means that the minor melodies have a stronger tendency to fall, and the major melodies have a stronger tendency to rise. The minor mode seems to have inspired Fux to write melodies with drooping contours suggesting a dispirited lethargy, or melancholy – raising the question of whether or not any other composers would have acted similarly. It is worth emphasizing that the cantus firmi are almost perfect examples of pure form. Nevertheless, the propensity toward small falling intervals is a characteristic of melodic structure which is potentially meaningful in itself, because it shows a preference for relaxed affect, and which, at least in Fux, is dependent on the choice of mode. The choice of mode, of course, is also meaningful, since the major mode is traditionally identified as evoking a brighter mood than the minor mode.

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Examples of rising and falling melodic lines can be found in both the major and minor modes in Fux’s six cantus firmi.

The direction of the closing motions is unambiguously

downward. The direction of the opening motions is more ambiguous; but all of the opening motions, no matter how elaborate, are ultimately ascending melodic lines. xiv Major and minor triads, of course, have the same total intervallic content. Each contains exactly one perfect fifth, one major third, and one minor third, the only concrete difference between them being the order in which the thirds are stacked above the root. Nevertheless, for reasons that are not entirely clear – probably having something to do with the importance of the major triad in the first overtones of the harmonic series xv – tonal frameworks based on major triads seem more stable than those based on minor triads. Perhaps, major frameworks are more easily processed by the mind than minor frameworks, because they are more consistent with the harmonic series. In any case, because of the greater stability of a major tonal framework, melodic movement in a major mode can be interpreted as circulating with greater assurance around and between the pitches of the framework, whereas melodic movement in a minor mode has to find its way through more uncertain surroundings.

Where upward movement in the major seems

uninhibited and aspiring, upward movement in the minor conveys a sense of struggle. Where downward movement in the major seems assertive or masterful, downward movement in the minor seems more passive or yielding. Since rising intervals – being more demanding to sing – are more adventurous than falling intervals, which are more quiescent, Fux’s preference for a greater proportion of rising intervals in the major modes suggests that he was more comfortable with the major than the minor. We cannot know Fux’s innermost thoughts, but it would seem that the greater tonal stability and resulting freedom of movement found in the major modes encouraged Fux to take more risks in the major. Similarly, the relative instability and uncertainty of movement in the minor modes may have encouraged Fux to be more cautious in the minor. These combinations of major and minor, rising and falling, are not the only possible combinations, however. Mode and direction of movement are logically independent. Melodies with more rising intervals are possible in the minor, just as melodies with more falling intervals are possible in the major. To be more adventurous in the relatively unstable minor modes might suggest an atmosphere of emotionality, however, that did not appeal to Fux. To be more inhibited in the more stable major modes might have suggested to him an excess of caution. Fux chose to follow specific constraints, which give his melodies an atmosphere of confident rationality that contributes to the definition of his musical style.

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Opening and closing motions Fux’s cantus firmi are roughly arch-shaped in the sense that they all rise from the tonic to a high note near the mid-point of the melody and, then, fall back to the tonic. However, the opening motion from the initial tonic to the high note is typically quite different in shape from the closing motion from the high note back to the tonic.

Fig. 1.3. Comparison of the opening and closing motions in Fux’s cantus firmi.

Fig. 1.3 shows all six melodies centered on their high notes. The vertical axis of the chart shows the number of scale degrees that each note of a cantus firmus lies below its own high note. The approach to the high note in the first half of each melody is circuitous and highly variable, employing a variety of melodic intervals, reversing direction over a wide range of scale steps, creating a kaleidoscopic contrast of tone color. The specific content of intervals and scale steps that gives each melody its individuality is concentrated primarily in these opening motions. The return to the tonic in the second half of each melody is smoother and more predictable, emphasizing structural coherence over content. In the closing motions, the more salient fact is always the connectedness of the tones rather than their individual identities. The arch shape, as realized by Fux, is arguably meaningful, because of specific qualities of the opening and closing motions. The opening motion, being more turbulent, arouses relative uncertainty. The closing motion, being smoother, reduces that uncertainty. The closing motion, in other words, is not only a literal conclusion, it is also conclusive in the figurative sense. The closing motion is not only what leads the melody to its final tonic. The closing motion, by its

25

configuration, is also a resolution of cognitive dissonance.

Where the opening motion is

expansive, exploratory, differentiating, and information seeking, the closing motion is contracting, reductive, integrating, and “knowledge making,” in the sense that it converts raw information into the perception of coherent understanding. The whole of the melodic arch suggests the form of a question followed by an answer. In a sense, the process of melodic unfolding found in the melodies is an allegory of problem solving, a narrative of comprehension, finding simplicity where there is complexity. The distinction in quality of movement between the opening and closing motions is greater than it strictly speaking needs to be, simply to cover the distance from tonic to dominant and back. For example, if we add up the absolute values of the intervals – disregarding the directions of the intervals – we find that the opening motion of Fux’s Dorian cantus firmus takes ten scale steps divided over six intervals to span a distance, from the first note to the peak, of only four scale steps. This is more convoluted than is absolutely required to complete the opening. On the other hand, the closing motion of the Dorian cantus firmus is as smooth as it could possibly be; but that is not the only way to cover the distance from dominant to tonic.

Ex. 1.6. Alternative Dorian cantus firmus.

To demonstrate, we can write a variation of the Dorian melody in which the opening motion is smoother than the original and the closing motion is more sinuous (Ex. 1.6). The two halves of the new melody are similar in complexity to each other. xvi The contrast between the two halves is more crisp and authoritative in the original by Fux. The alternative Dorian cantus firmus has a very high proportion of rising intervals: six out of ten. Sixty percent rising intervals is just outside the range of what is found in Fux, although it is comparable to the Lydian melody, which has fifty-five percent rising intervals. This fact illustrates the constraints that are imposed on a composer of cantus firmi. One is not allowed to do just anything, if one wishes to stay within the style. The simplest way to stay within the style would be to sing the melody backwards. That changes rising intervals to falling intervals, and vice versa. If we leave the melody as is, the aspiring quality of the many rising intervals might be considered psychologically more consistent with one of the modes with a major tonic. On the other hand, if we keep the melody in the Dorian mode, the psychological complexity of rising

26

intervals in the minor could also be valued for creating a certain atmosphere of mysterious, wistful poignancy. Fux made a definite choice to emphasize ascending leaps where the overall trend of the Dorian melody was upward and to avoid ascending intervals of any size where the trend of the melody was downward. To make such a clear choice gives an impression of purposefulness, which suggests that Fux was expressing a distinct idea. The well-defined question and answer form of Fux’s cantus firmi suggests an idea of rational clarity.xvii We normally understand tone-painting and program music to be concerned with the representation of concrete things or events. The cantus firmi do not represent anything concrete. We value them as allegories of problem-solving, however, because they satisfy a cultural ideal that we have inherited from the Age of Reason (whether or not this ideal is unique to the Age of Reason as such). The Age of Reason has taught us to place an exalted value on solved problems. If the concept of representation can be extended to include the representation of an abstraction, the cantus firmi are representational in that sense. In any case, they can be interpreted as meaningful, for the following reasons. We understand the quest of the melodic unfolding in Fux first of all as being concerned with ascertaining specifically musical facts having to do with the relationship of selected tones to their tonic, but the process of exploration and resolution found in the cantus firmi can be more broadly understood as a general behavior of the inquiring mind that is not limited to music. It is a short step, therefore, to understand the asymmetrical arch shape found in Fux as an allegory of the life of the mind in general. This makes the cantus firmi cultural artifacts as much as they are exercises in strict counterpoint. In my view, they are products of rhetoric and semantics, not products of a universal grammar. Fux’s cantus firmi would appear to be commentaries on a view of the world that places a high value on both confronting life difficulties – whatever those difficulties may be – and assimilating oneself to – or at least acknowledging ones understanding of, if not actual acceptance of – the observed state of affairs. Confrontation and assimilation are both necessary to this world view. One is not meaningful without the other, either rhetorically, or in life experience. This is an edited, idealistic vision of reality, however. There is room for alternative interpretations of life experience. In a different value system, perhaps one given to more or less existential certainty – perhaps, one attuned to listeners with a different sense of adventure – the form of the melodies might be different, either more or less highly structured.

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We are so immersed in Enlightenment thought that it may be difficult to recognize Fux’s cantus firmi for what they are, unless we stand outside of history and observe the Enlightenment in a much broader context.

On the perception of certainty According to Robert Burton (2008), our subjective sense of whether or not we know something to be true is a mental sensation, rather than evidence of fact, which he calls the “feeling of knowing.” This sensation stems from the primitive limbic system of the brain, which is responsible for emotion and long-term memory, among other functions. The “feeling of knowing” is a deep-seated psychological phenomenon, which applies to all areas of human thought. I would contend that this fact supports my claim that music is concerned with the symbols of possibility and purpose. Music is not only capable of arousing the “feeling of knowing,” but, in doing so, raises impressions of possibility and purpose in the world at large. Barbara Herrnstein-Smith was influenced by Leonard Meyer’s theory of music and made frequent comparisons between the structure of poetry and the structure of music. In her study of how poems end (1968, pp. 151-158), Herrnstein-Smith discussed the relationship between the sense of closure, finality, and stability and what she called the “sense of truth.” Certain special devices – such as alliteration and antithesis, syntactical forms (such as “not only … but also”), occasional repetition (as in irregular internal rhyme or alliteration, but not systematic repetition, such as rhyme, meter, or couplet series, which imply continuation), and allusions to death tend to have the force of closure. When they occur at the end of a poem, they have a quality that is experienced as imparting a sense of validity, or apparently self-evident truth, that goes beyond what is positively verifiable. Utterances may have the ring of truth because they have an apparent tone of authority, suggested by impersonality, brevity, or lack of qualification. These devices are used not only by poets but also by political and commercial propagandists. The point, according to Herrnstein-Smith, is not that a poem allows us the momentary entertainment of illusions, but that it allows us to know what we know, including our illusions and desires, by giving us the language in which to acknowledge it. I would suggest that the relatively smooth closing motions in Fux’s cantus firmi are examples where perceptual closure gives rise to a sense of authoritativeness, or certainty.

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Furthermore, the symmetry of an arch-shaped melody can be interpreted as the spatial analogue of a specific rhetorical device from ancient Greece, also intended to create a perception of certainty. According to Apostolos Doxiadis (2012), the methods of mathematical proof and the methods of rhetorical proof, which overlap substantially, both developed in Late Archaic and Classical Greece from narrative and poetic storytelling. The depiction of cause and effect in narrative and rhetorical proof corresponds to the demonstration of logical necessity in mathematical proof. In a classical-age trial (5th-4th centuries BC), forensic rhetoric followed a four-part template consisting of an introduction, narration, proof, and epilogue. The proof proper borrowed the forms of poetry for their beauty but kept some of them for their persuasive effectiveness. The logical structures later theorized by Aristotle as syllogisms are versions, in their basic form, of two poetic techniques: chiasmus and ring-composition.

Chiasmus is a

symmetric structure of phrases of the form A-B-B*-A*, in which the number of elements can be extended indefinitely. The elements A, B, and so forth represent words, phrases, or concepts. An asterisk stands for a repetition of an element or its antithesis, in whole or in part. In ringcomposition, the central pair is replaced by a unique central element, as in A-B-A*. Citing Rodolphe Gasché and Paul Friedrich, Doxiadis argues that structural symmetry relates to higher cognitive processes as a form of thought that allows oppositions to be bound into unity and creates an illusion of a synchronic, monocular vision of an absolute aesthetic truth, usually having radical closure. (If we do not take the parallelism too literally, the arch-form of a cantus firmus can be thought of as a spatial analogy of the ring-composition, A-B- … -N- … -B*-A*, where the opening motion of the arch corresponds to A-B- …, the climax corresponds to N, and the closing motion, understood as a freely varied, antithetical unwinding of the opening motion, corresponds to … B*-A*.) Reason has had changing roles in the history of ideas. The sociologist, Max Weber (1911), told the history of Western music as a process of rationalization regarding the tuning of musical scales, the development of rule-based harmony and counterpoint, the creation of a written notation allowing co-ordination between complex combinations of performers, technical improvements in the musical instruments, and the organization of the symphony orchestra, implicitly comparing the rationalization of Western music to the general trend toward rationalization found in other spheres of Western thought and social organization.

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This issue came to a head at the beginning of the Age of Reason in the seventeenth century, when questions about certainty appear to have been “in the air.” According to Albert Hirschman (1977/1997, esp. pp. 42-56, 132-135), political philosophers had lost confidence in the capacity of religion and reason to restrain the passions that lead to despotism, and – following the precedent of Machiavelli – argued (mistakenly, as it turned out) that the rational pursuit of material self-interest would lead to more predictable statecraft. Parenthetically, passion and reason had previously been thought to be antithetical; but, as the concept of interest was further developed in the eighteenth century, it was thought that interest would “partake of the better nature of each, as the passion of self-love upgraded and contained by reason, and as reason given direction and force by that passion” (Hirschman, p. 43). As the Enlightenment led into the modern bourgeois industrial state, however, the failure of reason to accomplish the goal of giving direction to passion was criticized by the Romantics from many points of view (see Michael Löwy and Robert Sayre, 2002).

Insofar as the end of the

Enlightenment can be given a specific date, Blanning (2010, pp. 9 ff.) marks it with the publication of Jean-Jacques Rousseau’s novel, La Nouvelle Héloïse in 1761, followed by his confessional autobiography in 1782. Blanning argues (p. 15) that the essence of Romanticism was summed up by Hegel as “absolute inwardness.” Inwardness is given a systemic explanation by Mitchell, in his study of Alexis de Toqueville (2013), as a consequence of democracy.

A

conservative, de Toqueville was concerned that democracy breaks the traditional chain of connection between citizens that is found under aristocracy and leads each man back toward himself alone, confined to the solitude of his own heart. Modernism, then, understood as what followed the gradual loss in the legitimacy of hereditary aristocracy, is the age of the self-created individual.

Durkheim amplified this concept in his classic study of suicide in 1897, which

examined the social disorder (called anomie) that accompanies social change.

Seemingly

antithetical, Enlightenment and Romanticism are only different paths toward the individualism of post-aristocratic modes of social organization and expression. Returning to the early seventeenth century, the Scientific Revolution made increasing use of mathematics, as, for example, in Galileo’s work on falling bodies and Kepler’s work on planetary motion. The mathematicians Pierre de Fermat (ca. 1607-1665) and Blaise Pascal (16231662) developed the theory of probability, which is the science of certainty and uncertainty. René Descartes (1596-1650) changed the philosophical question of what is true to the question of what is certain.

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The question of how one identifies certain knowledge was radically posed by Descartes in several works, most clearly in the Meditations on First Philosophy, published in 1641. The Meditations is the subject of a monograph by Janet Broughton (2002). In his “lunacy,” “dream,” and “deceiving God” arguments and the argument that we are so created by “fate or chance” that we are constitutionally deceived, Descartes called into question the existence of things that we see and touch, the nature of God, and the truth of claims like “two plus three equals five.” He resolved to affirm only beliefs that survive this skepticism. He affirmed first that he himself existed and that he had various states of consciousness, or ideas. From this, he deduced certain ideas about God, drew out the distinction between mind and body, and affirmed the existence of a physical world describable in austerely mathematical terms. Descartes’s “method of doubt” was only a strategy, a refutation of skepticism that was already held by others, which was intended to yield knowledge by uncovering its preconditions. Generally speaking, the method of deduction from first principals by pure reason may be termed rationalism.

There are four fundamental alternatives: empiricism, skepticism,

romanticism, and faith. Rationalism was most prevalent on the European continent. The British in the seventeenth century, with the exception of Hobbes, were primarily adherents to Baconian empiricism, according to which one learns the truth by experimentation. The British distrusted rationalism, a method which, in their belief, led to perilous dogmatism. Today, we think of rationalism and empiricism as complementary; but the seventeenth century was a period of religious war, in which the stakes were higher. See Alexander (2014, esp. pp. 251-253). The true skeptic finds peace of mind by suspending judgment. Descartes’s conclusions about what truths we can hold certain have been questioned by philosophers on every point. Paradoxically, the “absolute inwardness” of Romanticism, with its emphasis on feelings, could be thought of as an unintended consequence of Descartes’s more rationalistic method of selfexamination. The two points of view invoke different standards of truth; but they both imply a separation of the individual from tradition, in which orthodox beliefs and values are no longer to be taken for granted. For our purposes, what is most characteristic of Descartes’s thought is that he applied mathematical standards of proof to philosophical and metaphysical questions, thereby creating a polarization between skepticism and rational certainty that appears to reduce (if only as a strategy) the value not only of reasonable belief but also of intuition and the reality of mystery. The skeptic and the person of orthodox Christian faith disagree with each other profoundly; but they both accept that the limitation of human reasoning is a prominent feature of our cognitive landscape, which is to be celebrated rather than lamented.

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The organization and meaning of a doxology For example, the organization of a typical doxology is radically different from that of Fux’s cantus firmi; and that difference in organization arguably implies a different meaning. By far the most prominent tone in a doxology is the dominant. The tonic, if it appears at all, is one of the least important tones, other than the importance granted to it by its position at the end of the melody. The quick, perfunctory descent to the last note of a doxology is by no means an effective counterweight to the much-repeated recitation tone, which is the focus of a meditation on a text. The ending leaves the listener’s memory still dwelling on the recitation tone and its implications, which are as much spiritual as they are purely musical. The recitation tone is the heart of a doxology. The repetitions of the recitation tone, being unmeasured, do not have any predictable stopping point apart from what is determined by the text.

Having no definite length, the

repetitions can be taken as symbolically pointing toward infinity. The cadence, far from wrapping up the idea of the doxology in a definite conclusion as it would in a cantus firmus by Fux, is merely a brief and necessary – but otherwise unimportant – return to the secular world, which leaves the listener’s mind directed as far as possible toward other matters. Reverberating in memory, echoing off the walls of the sanctuary, a doxology evokes associations of physical expansiveness and large spaces. Where Fux’s cantus firmi have closure, the doxologies are more open-ended. Where the former suggest certainty, the latter suggest mystery. A cantus firmus by Fux stands with one foot in the remote past and the other foot in the Age of Reason; a doxology, firmly planted in Medieval culture, places its hopes in things unseen. As Saint Paul wrote (1 Cor. 13: 10), speaking of the end times, in words that echo Plato’s theory of knowledge, for the benefit of a Helenistic congregation: But when that which is perfect is come, then that which is in part shall be done away… For now we see through a glass darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

And in a similar vein (2 Cor. 4: 18):

While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal.

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Saint Augustine – as discussed by Treitler (2011), p. 33 – believed that all knowledge comes from God, but there are two kinds of knowledge. He distinguished what we can understand by benefit of our reason from that which we merely believe but do not understand. A contemplative member of a monastic order is presumably most concerned with knowledge of the latter kind about matters of the soul, which we can only “see through a glass darkly.”

If this is correct, then it would seem appropriate that Gregorian chant would not

necessarily be characterized by rhetorical gestures of definite closure, designed to give an impression of absolute certainty. Are the organizational differences between the cantus firmi and the doxologies merely due to the fact that the doxologies are dependent on a text and the cantus firmi are not? I think the answer is not that simple. There is more than one way to set a text to music. The symbolism of the music of the doxologies is consistent with the meaning of their text, but the influence of the text on the organization of the doxologies is not uniquely determined. The text of the doxologies could have been set in a manner much more consistent with the style of Fux’s cantus firmi. Furthermore, as we have seen, the music of the cantus firmi has a symbolic interpretation even without the presence of a text. An argument could be made that the characteristics that have just been attributed to the doxologies are nothing more than a rather abstract sort of tone-painting. If the doxology can be described as abstract tone-painting, however, the same can be said with equal force about Fux’s cantus firmi. I believe that, generally speaking, it is a mistake to draw a sharp line between so-called “pure” music and representational music. It is not self-evident that musical organization can be justified solely on its own terms, without reference to psychological or cultural values, because representation in music is not limited to the expression of feeling or the depiction of concrete things and events. It is true that music is not capable of expressing logical propositions, literally understood; but musical organization can represent cognitive processes that are devoted to finding order in the midst of disorder. These cognitive processes are part of our mental life in general. They are not limited to music; and they are, indeed, subject to the variation of personal and cultural preferences.

An indicator of melodic smoothness The distinction between smoothness and sinuosity of melodic movement has such important aesthetic consequences that it is worth our while to spend some extra time thinking

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about it. In the previous section, we examined the direction of melodic intervals without regard to their size.

Now, we will examine the sizes of intervals without regard to their direction.

Sometimes, special insights can be gained if we break up a compound idea into its component parts. It may help to begin with a little story, a fable about a Purposeful Ant and a Wayward Ant. Imagine that we see an ant climbing up a blade of grass. Suppose that the first ant we see is very purposeful. He starts climbing from the base of the blade and crawls all the way directly to the tip of the blade, without ever reversing direction, or looking back. The length of the path taken by the Purposeful Ant is exactly the length of the blade of grass. Quantitatively speaking, the ratio between the length of the path and the length of the blade of grass is equal to one. This is not a typical ant, however. Most ants meander about in all different directions in a manner that may have some hidden purpose to them but which seems almost random to a human observer. Suppose that the ant we see on the blade of grass is a more typical ant, a Wayward Ant. The ant crawls up and down the blade, frequently changing directions, only to reverse himself again and again. The Wayward ant would have started at the base of the blade of grass, and he may eventually touch the tip of the blade; but, when we first start observing him, he may be at any point on the blade of grass; and, when we stop observing him, he may be anywhere on the blade.

Perhaps, the

Wayward Ant has returned to his original starting point; or, perhaps, he has ascended higher on the blade or descended lower. The distance between the lowest and highest points on the blade of grass touched by the ant is a measure of the range of his movement. However, the total length of the path taken by the ant, which I will call the excursion, is longer than the range – perhaps, much longer – depending on how many times the ant reversed himself and how long he continued crawling on each leg of his travel. So, the ratio between the range of movement and the excursion for the Wayward Ant will be less than one. The more the ant wanders back and forth, the greater the discrepancy between the range and the total length of the path, or excursion. The ratio between the range and the excursion is an indication of the complexity of the path taken by the ants. A ratio of one indicates that the ant took a very smooth, directed path. This describes the path taken by the Purposeful Ant. A ratio less than one indicates a convoluted, undirected path, such as that taken by the Wayward Ant. The lower the ratio, the more complex the path. This little fable illustrates a way of thinking about directed motion, which suggests that the difference in linear coherence found in the comparison of the opening and closing motions of Fux’s cantus firmi can be quantified. What we have just described is perhaps the simplest such indicator – and the easiest to explain – the so-called VHF (Vertical Horizontal Filter) of Adam

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White, a stock market technical analyst. White uses this indicator to evaluate the strength of price trends, but it can just as easily be used to evaluate the strength of trends in melodic lines or the paths taken by ants (it evaluates the strength of trends, but not their directions). The VHF can be considered a formalization of Kurth’s distinction between ascending (anstiegend) or descending (abstiegend) developmental motives, on the one hand, and oscillatory (schwebend) motion, on the other (see Rothfarb, [1989] p. 57). The VHF does not make a categorical distinction between trending and non-trending motion, because the values of this indicator vary on a continuum. This is quite appropriate, because Kurth himself regards the three basic shapes as limiting cases (Grenzfälle), ideal boundaries within which countless intermediate forms arise. To compute the VHF of a melody (or section of a melody), the first thing we calculate is the range – that is, the difference between the highest and the lowest tones of the melody. Then, we calculate what I have called the excursion, which is the sum of the absolute values of all of the intervals between successive tones. The VHF is the ratio of the range divided by the excursion. (When the excursion is zero, the VHF is defined to be zero.) Since Fux’s cantus firmi are entirely diatonic, we will calculate all of the intervals in terms of the number of scale steps that they cover, not the number of half-steps. If all of the intervals are in the same direction (as in the path taken by the Purposeful Ant), the range will equal the excursion, and the value of the VHF will be 1.00. By default, if there is only one interval, the VHF can only have a value of 1.00. If there are reversals in the direction of the melody (as in the path taken by the Wayward Ant), the excursion will be larger than the range, and the VHF will be lower than 1.00. For example, consider Fux’s cantus firmus in the Dorian mode. The opening motion, as we shall see, is comparable to the path taken by the Wayward Ant of our fable. The melody first ascends from D to A, which is a perfect fifth, equal to four scale steps. From the note D, the melody ascends a third, which is equal to two scale steps. The melody then falls back a third, one step at a time. The total length of the path so far is four scale steps. Next, the melody rises a perfect fourth, for three additional scale steps, or an accumulated total thus far of seven scale steps. The melody descends again by a step, and the total has grown to eight. Finally, the melody ascends a third to the note A for a grand total of ten scale steps. The opening motion has taken ten scale steps to cover a range of four scale steps from the lowest to the highest note. The range is only forty percent of the total length of the path, or excursion.

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Contrast this with the closing motion, which descends one step at a time without changing direction all the way from A back down to D. The closing motion resembles the path taken by our Purposeful Ant, even though it is a descending motion rather than a rising motion; because, the range of the path is identical to the length of the path (the excursion), and the ratio of these lengths is exactly equal to one. The lowest possible value of the VHF occurs when the melody moves back and forth between the highest and lowest tones of the range. In that case, if there are N intervals, the excursion will be equal to N times the range; and the value of the VHF will be equal to 1/N.

Fig. 1.4. Comparison of the VHF of the opening and closing motions in Fux’s cantus firmi (descending by close).

It happens that the highest VHF of the opening motions of all six cantus firmi (0.556 for the Aeolian mode) is less than the lowest VHF of all the closing motions (0.600 for the Phrygian mode). In every case – six out of six – the close of a cantus firmus has a higher VHF than the opening of the same melody. This extraordinary fact is shown in Fig. 1.4.

The geometry of a cantus firmus The strong contrast in character between the opening and closing motions appears to be unusual; but is it, really? The answer is a bit complex. As we shall see, for computer-generated melodies similar to the cantus firmi, there is a strong positive correlation between the placement of the climactic highest note and the smoothness of the closing motions. That is, the later the climax, the more likely the closing motion is to be relatively smooth. Conversely, there is a strong negative correlation between the placement of the climax and the smoothness of the opening motions. The later the climax, the more likely the opening motion is to be convoluted. The relative smoothness of the opening and closing motions is statistically contingent on the placement

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of the climax. I will call this the smoothing effect. In other words, Fux’s practice appears to emerge statistically from the geometry of a cantus firmus, up to a point. Generally speaking, in computer simulations, the likelihood that the closing motion will be smoother than the opening motion is about even odds. It is not particularly surprising, then, that any one of Fux’s cantus firmi, taken by itself, has this property. On the other hand, it is rather unlikely that a small group of randomly selected simulations will all – in every single case without exception – be smoother in the close than the opening. The coincidence that every one of Fux’s melodies has this attribute is, if not rare, at least atypical enough to suggest that Fux’s process of composition may require a special explanation. There is an appearance of purposefulness in Fux’s choices that may not entirely result from chance.

Correlation does not prove causation; but Fux seems to have

deliberately exaggerated the natural statistical tendencies of a cantus firmus, presumably for aesthetic reasons. The resemblance of Fux’s closing motions to the Schenkerian Urlinie suggests the hypothesis that the Urlinie may emerge in a similar way. There may be a certain statistical probability that the Urlinie will emerge naturally from the geometry of a middle-ground latent line. Nevertheless, some composers comparable in aesthetic outlook to Fux may go beyond the statistical probability and deliberately enhance the smoothing effect in their treatment of the Urlinie. Conversely, some composers more comparable to Jeppesen in aesthetic outlook may on occasion deliberately mute the smoothing effect. It is a matter of interpretation whether or not any particular musical composition goes beyond the likelihood implied by its geometry. xviii In this discussion of the geometry of a cantus firmus, the question of what is cause and what is effect must be left open. Purely as a stylistic convention, I will usually frame the issues as if the broad outlines of the melodies were decided in advance and that the notes chosen to fill those outlines were decided later, with the result that the smoothness of the melodic movement – which depends on specific intervallic relationships between the tones – is secondary to the broad outlines of the melodies, rather than vice versa. In actual practice, however, there is reason to believe that Fux might have done the exact opposite, fitting the broad outlines of a cantus firmus to match a previous conception of what type of melodic movement he wanted. Perhaps, Fux was inspired to create the details and the broad outlines simultaneously, as a complete, well-formed conception of the whole. In any case, there is a correlation between the number of scale steps to be filled in a given amount of time and the character of the melodic motion that fills the available space and time. The main point to be made is not the direction of cause and effect, but simply that there is a correlation.

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For purposes of illustration, in order to simplify the issues as much as possible, let us assume for the moment that all of the motion in a melody is stepwise. Whether the melody moves up or down, the absolute value of every interval will be one scale step, without any repetitions of tones. Assume, also, that the melody is a simple arch, which begins and ends on the tonic scale degree and never descends any lower than the tonic. The broad outlines of the melody could be visualized as a triangle, where the base of the triangle is the horizontal line connecting the first and last notes. The range of the opening and closing motions will be exactly the same. Suppose, for example, that the melody goes from the tonic up to the dominant and back down to the tonic. Then, the range of both the opening and the closing motions will be four scale steps. Furthermore, the excursion in any section of the melody will always be exactly equal to the number of intervals in that section. On the other hand, if the climax occurs later than the midpoint, then, the excursion of the opening motion will be longer than the excursion of the closing motion. Suppose, for example, that it takes six intervals to go from the tonic up to the dominant; but, it only takes four intervals to go from the dominant back to the tonic. Then, the melody will pass through six scale steps to cover a distance of only four scale steps in the opening motion; but the closing motion will cover the same distance in only four steps. Somewhere in the opening motion, there will have to be a reversal in the melody, immediately followed by a counter-reversal that resumes the upward motion. The closing motion, however, will be perfectly smooth. The converse is also true. If the opening motion contains four intervals, and the closing motion contains six, then, the opening motion will be smooth, and the closing motion will be more convoluted. In other words, the geometry of an arch in this simple case implies that there is a smoothing effect, which depends on the placement of the high point of the melody. If the high point occurs late, the close will tend to be smoother than the opening, and vice versa.xix This case was deliberately oversimplified to illustrate the basic point. In actual practice, of course, the allowed range of the melody is not so restricted, and the available intervals are more varied. The distribution of intervals is asymmetrical, furthermore, since small falling intervals are preferred. For example, not all of Fux’s cantus firmi fit the description of a triangle described above, since two of them dip below the tonic. These are the melodies in the Phrygian and the Lydian modes. Each of these cantus firmi has a unique low note as well as a unique high note. The low note occurs before the high note; so, I describe both melodies as being in the Open-LowHigh-Close form. The Lydian melody does not dip as far below the tonic as it reaches above it; so, I would classify it as being both an arch and an Open-Low-High-Close form.

Strictly

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speaking, one might question whether or not the Phrygian melody should be counted as an arch, because it extends further into its low register than it does into its high register. The closing motions of the Dorian and Aeolian melodies are both perfectly smooth stepwise descents. This is possible, because the number of scale steps from the highest note to the lowest note is exactly the same as the number of notes in both cases. The Lydian cantus firmus is an exception, however. There are four scale steps from its highest note to the tonic and four notes in which to make the descent; but Fux takes an indirect route to the end, leaping downward by thirds through the tonic triad. In this case, Fux probably wanted to skip over the raised fourth scale degree because of its problematic tritone relationship to the tonic. The composer is not required to take the most direct route from the dominant to the tonic. That is only an attractive possibility. The Phrygian cantus firmus raises a different issue. There are three scale steps from the highest note of the Phrygian melody to the final, but there are four notes to complete the descent. A perfectly smooth closing motion is not possible, given these facts, because one more note is required than the number of available scale steps. Of course, Fux could have descended directly from the highest note in three steps by shortening the melody. Then, the question would be whether or not the closing motion was too short and too smooth to be consistent with the convoluted character of the opening motion. This is simply another way of stating the issues, although it reverses the framework of cause and effect. Similarly, in the closing motion of the Ionian melody, six notes must be forced into only five scale steps; so, there must be some backtracking (or vice versa, Fux wanted some backtracking; so, he forced six notes into five scale steps). In the Mixolydian melody, there are only six notes to fill seven scale steps; so, there must be at least one leap, raising the question of whether or not just one leap would seem out of place in the context of a long string of steps. Fux seems to have thought so, because he used three descending leaps where only one was strictly required by the constraints of space and time. These observations are not, of course, intended to be taken as characterizations of the stylistic constraints specific to individual modes, such as the Dorian, Phrygian, and so forth; because Fux only gave us one example of each mode. Fux’s examples are only used here as illustrations of the options that are available to a composer of cantus firmi in general.

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Computer simulation To make a thorough study of the possibilities and probabilities for melodic shape that can be created in the form of a cantus firmus, we will need to examine hundreds of examples. Fux only gave us six cantus firmi, but we can create as many as we like with a computer. The examples should be comparable in their broad outlines to Fux’s melodies, but otherwise they should be subject to as few restrictions as possible. It is not necessary that the examples be artistically convincing or compelling. For our purposes, the procedure that we use to generate the melodies needs to be as simple as possible, to avoid imposing too many preconditions. We are not trying to imitate an existing style. To the contrary, we want our procedure to wander through many possible styles, only subject to the provision that they meet the minimum requirements for a cantus firmus.xx Basically, the procedure to be followed has three steps: generation, normalization, and elimination. Generation: All of the generated melodies will be the same length – in this case, eleven notes. The intervals (measured in scale steps) will be drawn at random from a pool that lists all of the intervals actually used in Fux’s cantus firmi. This will insure that the preference for small falling intervals found in Fux will be reproduced in the generated melodies. The operational meaning of the term random is “chosen by the RANDOM function of a computer program” (where I use all upper case letters to suggest computer code). The significance of the term random, however, is that decisions are made on the basis of purely local criteria. The decisions are unpredictable, because they follow no rule having a larger scope. When the intervals are added together, they make what is called a random walk (remember the Wayward Ant?). Normalization: All of the generated melodies will begin and end on the tonic scale degree. It is statistically unlikely that a simple random walk will return to its starting point at a predetermined time.xxi To avoid throwing away an excessive amount of data, this is accomplished in our simulations by de-trending the random walks. The average interval size is calculated for each random walk, and that value is subtracted from each interval in the walk. The results are rounded to whole numbers. Normalization will distort the distribution of intervals to a small degree, but we should not expect the proportions of large and small intervals to be changed very much. Elimination: We throw away all of the normalized random walks that do not meet certain minimum criteria. First, the rounding employed by the normalization process will sometimes result in note repetitions. All of the walks having repeated notes are omitted.

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The range of the melody will be at least a fifth, but not more than an octave. The highest note will only occur once. To be an arch, the highest note must be farther above the tonic than the lowest note is below it. The last-mentioned constraint does not apply if the shape is an OpenLow-High-Close form; but, in that case, the lowest note will only occur once, and the lowest note will occur before the highest note. For the present, we will only discuss arch forms, because just one of Fux’s cantus firmi, the Phrygian, is a pure Open-Low-High-Close form. To be consistent with the cadence formulas employed by Fux, the final tonic must be approached by step from above, that is, from the supertonic scale degree. The note before that – in other words, the ante-penultimate note – must be either the mediant or the tonic scale degree, because all of Fux’s cantus firmi employ either the 3-2-1 cadence, or the 1-2-1 cadence. The former is far more common in Fux than the latter. Of the five arches by Fux, only the Lydian melody has the 1-2-1 cadence. The computer program, however, does not enforce any preference for one of the two cadence formulas over the other but leaves the choice entirely to chance. Without the restriction on the ante-penultimate note, about 40 to 60 out of a thousand normalized random walks will typically meet the minimum criteria for an arch form. With the restriction, that number is reduced to about 25 to 30. Remember, furthermore, that the normalized arches themselves represent a small proportion of what would be produced by a simple random walk without de-trending. In large samples of generated arches, the 3-2-1 cadence occurs about three times more often, on average, than the 1-2-1 cadence. Selecting minimally valid arches from thirty samples of a thousand random walks, the percentage of 3-2-1 cadences ranged from 69.8% at the twenty-fifth percentile to 78.5% at the seventy-fifth percentile. The proportion found in Fux (four out of five) is just barely outside that range. The apparent preference for the 3-2-1 cadence in the generated melodies is not the result of a predetermined rule. It is statistically emergent, resulting from the geometry of the arch form. It is not necessary to believe that Fux held an actual preference for the 3-2-1 cadence formula; because statistical emergence is sufficient to account for the proportion of such cadences found in Fux’s arch-shaped melodies, too. Generating a thousand normalized random walks at a time and selecting only the minimally valid arches, the median location of the melodic peak in the simulated arches is at the sixth note, which is the middle of the eleven notes in each melody. The frequency distribution of the climaxes is arch-shaped. The high note can be found anywhere from the second note to the ninth note out of the eleven.

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The correlation of the VHF of the opening motions with the location of the melodic peak is strongly negative, typically ranging from -0.79 to -0.88. For closing motions, the correlation is similar, but positive, ranging from about 0.73 to 0.84. In other words, the earlier the melodic peak, the smoother the opening, and the more sinuous the close; conversely, the later the melodic peak, the more sinuous the opening, and the smoother the close. Generally speaking, Fux’s preference for small falling intervals is most pronounced in his closing motions. Rising leaps are confined entirely to the opening motions in Fux’s cantus firmi.xxii Since the rising intervals include the largest intervals, the reduction in the number of rising intervals makes the absolute interval size smaller in the closes. This distinction is virtually categorical for Fux, but the change in the distribution of intervals for simulated arches is typically smoother. Plotting the percentiles of the intervals of the simulated closing motions on the Y-axis against the corresponding statistics for the simulated opening motions on the X-axis yields a curve that approximates a slightly flattened parabola. The parabola ascends relatively quickly as we step through the negative intervals, and, then, gradually approaches a maximum as we move on to intervals greater than or equal to zero.xxiii As we have seen, the five arches written by Fux are all smoother in the closing motion than in the opening motion. The question is whether or not this is likely to have occurred by chance. Ten batches of 10,000 normalized random walks were generated. The valid arches were divided into groups of five. Examples of the generated melodies will be found in Appendix A. We are interested in how many groups of five were generated in which all of the closing motions were smoother than the corresponding opening motions. Practically speaking, we cannot attribute the difference in smoothness between the opening and closing motions of one simulated arch, taken by itself, to anything but pure chance. However, the likelihood that a pre-selected group of five simulated arches would all be smoother in the close than in the opening is a different matter. Comparing the ten batches of 10,000, the percentage of the groups of five belonging to any one batch in which the VHF was higher in the close for all five cases ranged from zero to 7.84%. The average was 3.83%. On average, in other words, we can be better than 95% confident that such a result would not occur by chance. The worst case is that we can be better than 90% confident that such a result would not occur by chance. Although the numbers vary from one run to another, this suggests that there is a relatively small chance that five randomly generated arches would make as strong a distinction between their opening and closing motions as Fux did in his five arches.

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The Urlinie Although I do not always agree with Schenker, he was an acute analyst. We would not be having this discussion if that were not so. I believe that Schenker did observe something when he discovered the Urlinie, but it is possible to reframe our interpretation of what he observed. xxiv Now that we know more about the statistical properties of melodies, it is possible to re-assess the Urlinie from a completely new perspective, which was not available to previous researchers. This analysis may demystify in some small way what Pastille (p. 71) has called the “enigma,” the “sense of mystery that surrounds the concept of the Ursatz.” Presumably, composers who write many melodies will encounter some variant of the smoothing effect empirically. Whether or not these composers have a self-conscious theory of the smoothing effect, they may discover that the effect either does or does not satisfy their aesthetic values. In choosing which melodies they want to make public, they may pick either those that enhance the smoothing effect or those that mute the smoothing effect. If, like Fux, they prefer melodies that create a strong sense of certainty and closure, they may choose those with middle to late climaxes that have relatively smooth closing motions. This choice suggests a narrative in which uncertainty is overcome by certainty.

Presumably, this choice reflects a rationalistic

aesthetic, an aesthetic that prefers questions that have clear answers. Music by these composers would tend to fit Schenker’s theory of the Urlinie rather closely. On the other hand, if a composer wishes to suggest a more tragic narrative, in which some issues are never completely resolved, the approach to the final tonic can be made more indirect – as sometimes happens in Jeppesen – by taking the climax earlier, giving the closing motion more time to develop, encountering more significant deviations from expectation along the way. The analyst should not over-emphasize structural coherence in such melodies, because these melodies cast doubt on the inevitability of final closure, giving more emphasis to the struggle that one encounters on the way toward achieving closure. Since the composer is free to choose whether the peak of the arch arrives early, in the middle, or late, the difference in character between the opening and closing motions cannot be completely explained by one simple, universal law, such as the smoothing effect. The difference in smoothness between the opening and closing motions of an arch is to some degree statistically contingent and emergent, resulting from general characteristics of the vocal style, combined with geometric constraints on melodic motion. To be sure, the statistical tendencies of the arch are not strictly deterministic. The difference in smoothness also results from a strategic choice that lies in the hands of the composer.

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If Schenker was correct that the latent middle-ground lines of melodic structure have the same melodic qualities as a well-made cantus firmus, the vocal qualities that we have found in Fux ought to be equally characteristic of middle-ground lines. If it is true that the Urlinie is to some degree emergent, this would tend to eliminate certain theoretical anxieties about the Urlinie. The concept of the emergent Urlinie is attractive from both a pragmatic and a theoretical perspective. If the Urlinie is regarded as a well-defined, fixed, ontological entity, an explicit, culturally recognized schema, then, any departure from the pure type – such as a missing note, register shift, interruption, repetition, or change of direction – suggests a possible flaw in the foundations of a composition that requires a theoretical explanation (see, for example, Neumeyer [1987a, 1987b, 1989, 1990]). If the Urlinie is an emergent construct, however, its status is reversed from a cause to an effect – an effect resulting from statistical characteristics of vocal rhetoric and melodic geometry under the influence of a broadly conceived value system. In that case, departures from the pure type are only to be expected and do not require any special explanation. If the Urlinie is emergent, we can think of it as a limiting case – the idealized, sometimes unattainable, goal of a general tendency. I am skeptical about Schenker’s concept of the Urlinie. It is questionable whether or not the phenomenon is best interpreted as a categorically delimited ontological entity, comparable to an element of spoken language. It may be that, in special cases, the Urlinie is one schema among many possible schemas. In general, however, I would suggest that it might be more useful and simpler to think of the Urlinie as a construct, an idealization of an evanescent, statistically emergent, aggregate spatial property, occurring under certain conditions in a melody’s narrative arc. More research is needed, but the idea that the Urlinie is the most fundamental tonal structure is open to question. We have seen that the relative simplicity or complexity of a melodic shape ranges across a continuum of possibilities and that the smoothness of the Urlinie is simply one extreme of that range. Furthermore, we have seen that neither smoothness nor complexity is inherently more valuable than the other, because either one can be appropriate relative to the aesthetic context in which it appears. Good craftsmen know the limitations and latent possibilities of their materials and how to put these to best effect. The voice of the accomplished artist is at one with the spirit of the medium in which he or she works. In a piece of music, there is a gray area where self-expression and purely musical expression are essentially the same, where it is difficult to distinguish the unique contribution of the composer from the inherent nature of tonal relationships.

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Consequently, the status of the Urlinie as a cause or an effect is somewhat ambiguous, the Urlinie being a schema – on the one hand, emerging by chance from the time series geometry of melody, observed in arches, and possibly to be discovered in other contexts of goal-directed motion – on the other hand, subject to the supervening aesthetic taste of the composer; perhaps, in some cases arising from a subtle preference in the Classical Style for optimistic, late-climaxing, smoothclosing, dramatic arcs. In the final analysis, the question of cause and effect must be evaluated in every composition according to the logic of the individual case.

Hierarchy Lerdahl and Jackendoff (1983, 1996) suggest that the Ursatz may be an effect, not a cause (pp. 139-140). These authors argue that the Ursatz results from tonal principles such as prolongation of the tonic, the circle of fifths, and stepwise linear motion. They recognize that reductions of tonally unstable pieces at the most global levels of analysis probably will not result in a stepwise melodic descent, or even, possibly, a I-V-I progression (p. 140). They propose (pp. 288-289) that archetypal patterns in general emerge as a consequence of the preference rules of their grammar and that passages in which the preference rules reinforce each other will be heard as archetypal.

The authors identify certain dimensions of musical structure, such as timbre,

dynamics, and motivic-thematic processes, as not hierarchical in nature (p. 9). They have chosen not to treat these subjects directly in their theory, although they do take them into account as contributing to the principles that establish hierarchic structure. Nevertheless, in their advocacy of what they call the Reduction Hypothesis (and the Strong Reduction Hypothesis), they have committed themselves to the central tenet of Schenkerian analysis (p. 106).

Lerdahl and

Jackendoff believe that the function of prolongational reduction is to enable us to speak of points of relative tension and repose and the way music progresses from one to another (p. 179). Prolongation fulfills a role in their theory that corresponds to what some writers would call narrative form. Gjerdingen (1988, chapter 2) disagrees with the interpretation of archetypes by Lerdahl and Jackendoff. Gjerdingen argues that a reductive, strictly hierarchic, two-way branching tree structure – the characteristic analytical concept employed by Lerdahl and Jackendoff – is not adequate for the description of structural schemas. Our comprehension of musical structure, according to this author, results from both bottom-up and top-down perceptual processes, which are better represented using Narmour’s (1977, ch. 8) network analysis.

The Urlinie, in

Gjerdingen’s view, is one of the possible schemas; but it is not privileged, and it may coexist with

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another schema of equal importance in the same network.xxv Although this view is quite different from that of Lerdahl and Jackendoff, it has the same end result of calling into question the status of the Ursatz as a unique causative agent. Hierarchy in the theory of Lerdahl and Jackendoff emphasizes the vertical dimension of pitch relationships (p. 116). By contrast, Lerdahl’s later (2001) theory of tonal pitch space, which allows one to speak of a tension curve in the changing distance of keys from a tonic, emphasizes the horizontal dimension.xxvi The latter theory, that is, the theory of tonal pitch space, is more amenable to treatment as a dynamic system, because of its emphasis on the horizontal. Without the Ursatz as a unique causative agent, the theory of prolongation is left without a clear large-scale guiding principle. We have already seen, moreover, that hierarchic structure is not the only way to understand the complexity of melodic shapes and that reductive analysis is not the only way to understand the narrative of melodic unfolding. Nevertheless, there is an alternative side to the generative theory.

Lerdahl and

Jackendoff have greatly expanded and reformulated the basic insight of the Cooper and Meyer (1963) theory of rhythmic structure – which was conceived in terms of formal regions – to create a theory of complementary left and right branching tree structures.xxvii Not to underestimate the ground-breaking importance of Schenker’s speculations about deep structure, I would argue that a theory of rhythmic structure, however formulated, has a stronger claim than the Ursatz to be an explanation of large-scale form in tonal music; because it is based on comparatively abstract principles of tonal relationships, rather than a specific, concrete structure. One would think that the possibilities of generalizing from the concept of rhythmic structure would be limited only by the imagination. It is true that the Cooper and Meyer system of interpretation makes use of a forced-choice tree structure, which may at times seem a bit arbitrary. A more informative solution might be to weight the branches by Cope’s multi-valued SPEAC system.xxviii

Conclusions I said at the beginning that we would be concerned with what makes a musical idea distinctive. This is a large question, which will continue to occupy our minds throughout the remaining chapters of this study. For the purposes of an introductory chapter, I have chosen to concentrate on measurable properties of melodies that fall into a well-defined, iconic form. This form, the form of a cantus firmus, can be understood as representing a microcosm of more generalized patterns that one might find in much larger narrative forms. The free-form melodies of the cantus firmi led us to questions about the Urlinie, which in turn led us to larger questions

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about musical organization. What these melodies have in common with the Cooper and Meyer concept of rhythmic structure, a subject for further investigation, is that large-scale rhythmic structure implies a tension curve that is analogous in principle to the tonal tensions of a cantus firmus. The original questions raised by the cantus firmi are more limited in scope.

The

unvarying rhythm of a cantus firmus allows us to focus on intervallic relationships: specifically, the distribution of intervals, associations between them over time, and dependencies on the constraints of the form – in short, unity and variety in the selection of intervals, what is common and what is unusual, and how that varies with context. Distinctiveness, at least from a relativistic viewpoint, has to do, partly, with what is uncommon by comparison with what is more typical of a musical type; but, more generally, it has to do with freedom from constraint. The things that limit the movement of a melody tend to make it more standardized. There is more opportunity for distinctiveness, or individualization, when the melodic contour is moderately convoluted than when it is smooth, or, contrariwise, when it swings widely between extremes. Smoothness and extremity are both limiting conditions. I have chosen to emphasize arch-shaped melodies, because this is a common, though not universal, pattern. The arch represents a plurality rather than the majority of the melodic shapes in folk-song, for example; but, it is, in that sense, the most common melodic shape of the genre. In an arch-shaped melody, one can easily distinguish two parts, the opening and closing motions – that which leads up to the high point, and that which leads away from the high point. The arch is a spatial analogy of a rhetorical device called the ring-composition, a variety of chiasmus, where symmetry of form leads the auditor toward an impression of certainty. I have modeled melodic processes as constrained random walks. It is not that actual musical compositions are random; but randomness is the extreme case of localized decisionmaking, or freedom from constraint. Such a model throws into high relief the effects resulting from an opposition between local choices and larger formal limits. From the standpoint of hypothesis-testing, it is highly informative to compare stochastic processes against melodies found in the literature. It turns out that the opening and closing motions of arch-shaped melodies may have quite different properties. We found that the relationship between the unity and variety of intervallic relationships in arch-shaped melodies is highly dependent upon the location of the high point that marks the division between the opening motion and the closing motion. To begin with, the shorter the opening or closing motion, the fewer intervals there will be; and this imposes limits on the

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variety in the selection of intervals that will be possible. However, an additional peculiarity of the arch-constrained random walk is that the distribution of intervals in the closing motion, in particular, is more limited and skewed than that available to the opening motion, rising intervals being asymmetrically unavailable to closing motions. Furthermore, the shorter the opening or closing motion, the more likely that it will have a measurably smooth contour. Analysts who are inclined toward the hierarchical thinking of Schenkerian theory will be likely to interpret a smooth closing motion as structural; but it is also possible to interpret either a smooth opening motion, or a smooth closing motion, as emergent from a statistical process. The opportunity to create a distinctive pattern of intervallic successions depends to a rather large degree on a choice having been made about the overall contour and timing of an archshaped melody. I have argued that this is an aesthetic choice that is under the control of the composer. The opening motion is exploratory, comparatively free from constraints, concerned with the symbolism of possibility. The closing motion is goal-directed, seeking the relative certainty of perceptual closure, concerned with the symbolism of purpose. The balance between the two can be interpreted as expressing a judgment about the relative importance of possibility and purpose. Since the cantus firmi do not outline chord progressions and they do not modulate, however, they are only tonal in the most general sense. In this repertory, the gravitational pull of the static framework of reference pitches belonging to the tonic (or subdominant) triad generates an observable cross-current of conflicting melodic tendencies;xxix but only the ultimate attraction of the tonic scale degree has an influence on the broader gestures of melodic movement. The basic melodic impulse can be described as a meaningful, fluid, developing balance between order and disorder taking place in the field of attraction of the tonic scale degree. This takes us back to Schenker’s original concern with melodic fluency in Kontrapunkt I.

Although it is highly

instructive to analyze the cantus firmi by means of the hierarchic-reductive method that Schenker developed in his later, more mature works, we have seen that melodic fluency does not have to be understood in this way. Since the human mind presumably has difficulty conceiving a clear understanding of multiple hierarchic levels simultaneously, the mind must have some ability to conceive of complexity as such. The concept of complexity as such is a summarizing aggregate (represented here by the VHF).xxx When we compare the opening motions of the cantus firmi we see that what they have in common is essential complexity, regarded as a property in its own right, not merely as a secondary attribute of some independent method of organization. When we contrast the

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opening motions with the closing motions, we see that the two categories of melodic motion also differ in their essential complexity, or complexity as such. In a later chapter, we will investigate how the essential complexity of a melody may result from a dynamic process – the composite of an expansive exploratory process and an inhibiting process that tends toward a tonal center. The understanding of the cantus firmus that we have just described – the cantus firmus having the contour of a sinuous arch, naturally divided at the climactic peak into two parts, one leading up to the peak, and the other leading back down from the peak, the contours of the two parts being relatively more or less sinuous – is consistent with Kurth’s view of musical form, which he conceived as a shaping process that leads to layers of dynamic waves (see Rothfarb [1989], p. 191). The degree of relative sinuosity can be considered as an attribute of the whole, or any part of the whole. xxxi This concept is self-sufficient, standing on its own terms, and is not merely contributory to a reductive analysis that is somehow “more real.” In choosing between the two explanations – Schenkerian reduction and Neo-Kurthian wave analysis – we are faced with an interpretive dualism.xxxii In my view, Schenkerian linear-harmonic reductive analysis – though subject to differences of interpretation – is a fascinating and beautiful discipline, fully deserving of the intense study that it has received. Nevertheless, Neo-Kurthian wave analysis is also a valid and interesting way to think about musical organization. We are not forced to choose between the two conceptions of musical organization, because each approach reveals a different set of facts. That does not mean that every analytical technique has equal power to explain the salient facts of every musical example for every musical question, however. Each case requires us to select the most appropriate analytical technique from the repertory of available techniques. Ironically, we have come to understand the Fux of the cantus firmi as a supremely rationalistic composer, but the irregularity (the symbolic irrationality) of his opening motions is essential to the definition of his style.

Alternatively, the irregularity that is often found in

Jeppesen’s closing motions (especially in the Dorian mode) is also characteristic of his style. It would not help us to understand either of these composers if we were to attempt to explain away the irregularities found in their melodies. We need to be able to evaluate irregularity as such, as something that is real, and not merely a perturbation of something simpler.

To argue that

reductive analysis tells us that the irregularity of Fux’s opening motions and certain of Jeppesen’s closing motions is only apparent and that the linearity that implicitly lies beneath the surface of these passages represents an overriding level of truth, would only obscure the true differences between Fux and Jeppesen.

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To understand the true identity of a musical composition, we need to see it as it is, complete with all of its identifying characteristics, including its irregularities. These irregularities may very well be essential not only to the definition of the form, but also to the meaning of the composition.

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Chapter 2 Texture and Rhetoric in Debussy’s Syrinx: A Computer-Assisted Analysis

C

laude Debussy’s composition, Syrinx, for solo flute (1913), is of great theoretical interest, because its chromatic, non-triadic harmonies, emphasizing whole tone scales, foreshadow

the atonal procedures of later twentieth-century composers. The issues raised by Syrinx have broad implications, which go far beyond the original circumstances under which the composition was created. Syrinx is not merely a transitional work, however, but an important composition in its own right, arising from a particular time and place, and opening a window on the unique culture from which it came.

(A performance by Paula Robison can be viewed at

www.youtube.com/watch?v=rbRC3scRYeE.) The bulk of this chapter is concerned with computer-assisted analysis of the textures and formal procedures Debussy employed in Syrinx. The textures are defined by descriptive statistics based on nine time-series indicators, most of which are new. The procedures are conceptualized with reference to three rhetorical devices: opposition, expressiveness, and symmetry.

The

dynamic process that carries the textures from one moment to the next is described as a current of well-proportioned distinctions: juxtapositions of contrasting ideas, freely flowing in complex, judiciously channeled waves. We will be concerned not only with implicit patterns made by the textures, but also with phase relationships and divergences between independent variables. I say “computer-assisted analysis” rather than “computer analysis.” The computer makes it practical for us to think about otherwise inaccessible subjects. The computer, however, is not capable of making fundamental interpretative decisions that require domain knowledge and musical judgment. The analysis of Syrinx is a difficult challenge – almost a study in contradictions. The developmental processes of Syrinx overflow the boundaries of its ternary form. The short term waves of its melodic contour dominate its long term waves, and they are so irregular that they are better described as anti-persistent, rather than cyclical. The character of the waves changes from one area of the form to another, so it is difficult to reduce Debussy’s compositional technique to rule. For this reason, Syrinx will be viewed abstractly, as a result of dynamic processes. I will not attempt to explain everything that happens in Syrinx.

I will attempt to convey that the

complexities of the work are coordinated in a manner that supports a basically traditional but blended, not dichotomous, distinction between development and the construction of cadences – and, furthermore, that the wave properties of the melody are substantially in accord with its form.

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I will touch briefly on the deterministic version of my Limited Growth Model for exploratory behavior, which approximates the underlying shape of a simple closed-form melody (or other time series of musical interest) by an equation of motion (Eq. D1: for time t scaled to range from zero to one, and appropriately normalized data). The equation is a parameterized, generic schema, a polymorphous abstraction from theories of melodic shape by Kurth, Meyer, Narmour, Gjerdingen, Adams, and Huron.

Eq. D1.

x = atC (1 – t) – btD (1 – t) + et + f.

The equation proposes that, for musical shapes in which the deviation from the long-term linear trend has at most one distinct high point, or one distinct low point, or both, the broad outlines of the shape can be understood as the product of a single wave-like gesture rather than a succession of independent incidents – that the tendencies of such a musical process to rise and fall, expand and contract, are present throughout, and that the relative strength of these tendencies determines the outline of the shape that emerges over time. In other words, to an approximation, the unifying invariance of the form is to be found in a set of constant dynamic properties, which unfold at different rates. Debussy was, no doubt, speaking metaphorically when he wrote, “Music is a mysterious form of mathematics whose elements partake of the Infinite” (Jarocinski [1976], p. 95). Although Debussy was opposed in principle to the standard music theory and analysis of his time, I think it will be instructive to take Debussy at his word on this point – that music is somehow mathematical, in a way that mysteriously suggests perfect, Platonic ideals that are beyond our ordinary, limited understanding of the world. I will attempt to make the implicit mathematical ideals explicit. The scope of this study is rather different, however, from that of Howat (1986), who argues that some of Debussy’s musical forms are based on exact mathematical proportions, which could, conceivably, have been rationally calculated by the composer. The mathematical relationships described in this paper are only approximations to the directly observable facts, and of such a subtle nature that I can only assume that Debussy discovered them intuitively. Although the form of Syrinx is arguably complete in itself, the composition cannot be fully understood without reference to the text that originally inspired the composition. Therefore, I will begin with a few words of background.

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S

yrinx is a staple of the flute repertory, written as incidental music for the Symbolist play, Psyché, by Gabriel Mourey. The role of Syrinx in the play has been clarified by the

discovery of a manuscript of the composition in Brussels, reported by Anders Ljungar-Chapelon and Michael Stegemann in the Wiener Urtext edition of 1996. The composition was originally titled La Flûte de Pan. The name was changed when the work was published as a concert piece by Jean Jobert. The name Syrinx refers to a nymph who was the object of the Greek god Pan’s unfulfilled longing – Pan, the god of fertility, shepherds, flocks, and wild places – longing (as told by Ovid) for the chaste Naiad, Syrinx, who escaped from Pan’s pursuit to the River Ladon, where she was transformed by her Watery Sisters into tall marsh reeds. Pan cut the reeds to make the set of musical pipes with which he is identified. There is a chasm between what can be depicted by words and what can be expressed by music. Since musical textures do not have fixed, concrete meanings, it would be difficult to guess the specific story that is associated with Syrinx in the play only by hearing the piece performed in the concert hall, without accompanying words. We have to turn to the text of Mourey’s play to discover Debussy’s intention. This is discussed by Grayson (2001, pp. 132-133) and Ewell (2004). The manuscript of the composition suggests that it was originally performed as a melodrama – that is, a musical accompaniment to spoken text. The relevant section of the play is the first scene of the third act, before the depiction of Pan’s death, which consists of a dialogue between a Naiad, or river nymph, and a mountain nymph, or Oread. Ewell has translated the relevant passages, and matched the text to the music. Pan appeals to the fearful Naiad off-stage, speaking only through his music, except for the intervention of the Oread. We perceive a change of consciousness near the end of the piece, which has been explained by Ewell as representing the entranced Naiad’s final acceptance of Pan. Although I will analyze Syrinx as a continuous musical composition, as it would be performed in the concert hall, the music was interrupted by spoken text at least once, perhaps more often, when it was used as incidental music. In the manuscript copy that survives, there is an empty bar marked with a fermata between measures eight and nine of the printed score, containing a textual cue. Grayson argues that the stage directions permit the interpretation – at least as a possibility – that the music was interrupted by dialogue more frequently; and, furthermore, that the complexity of the music makes it ill-suited to serve as background music to recitation. Ewell argues for just one interruption, and discusses the text-setting in detail.

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The question of how the music was originally staged is difficult. I suspect that more than one staging was actively considered, but that, in the final analysis, it was decided to only interrupt the music once. I agree with Ewell’s interpretation, therefore, at least as far as it concerns the change of consciousness that is expressed near the end of the composition. For reasons that will be discussed later, I would only comment that the change in consciousness is temporary, and that it fades away to a state of mind that, in my view, seems rather more equivocal.

S

yrinx has been analyzed many times, by excellent theorists. An overview of Syrinx is given by Wilkins (2006), pp. 62-65, 105-6.

Curinga (2001) reviews analyses of Syrinx by

Anagnostopoulou; Austin; Baron, Borris; Cogan and Escot; Deliège, De Natale; Larson; Laske; Mahlert; Nattiez; Seraphin; Tenney and Polansky; and Whitman.

Lartillot (2009) applies

automatic extraction of motivic patterns to the analysis of Syrinx. I will try to avoid repeating much that has been said before, since my aims in this paper are rather different; but some observations must be restated, simply to set the stage. My main concern, at present, is with the textures and rhetoric of the composition, how they are unusual, and how they form a whole. Most of the formal apparatus of Syrinx is directed toward making a convincing case for the more unconventional attributes of the sounds: in particular, the musical language – which is virtually atonal – the constantly shifting textures, and the unusual ending. “Making a convincing case” is the role of rhetoric. Whether we agree with a composer’s values or not, an effective composer, like an effective speaker, employs various devices – which may collectively be referred to as rhetoric – to persuade the listener to accept his or her point of view. In this paper, we will be concerned, not only with the manner in which Debussy appeals to the listener’s emotions – an important rhetorical device in itself – but also with specific formal techniques used by Debussy, which will be explained later. Although Syrinx can be considered a harbinger of modernism – because its unconventional hierarchy of tones is a challenge to traditional tonal syntax – Syrinx is still an example of traditional rhetoric, not a contravention of traditional rhetoric. The change of consciousness at the end of Syrinx is due to what might be termed a “builtin surprise ending.” We will be concerned here with the devices by which Debussy makes this surprise ending seem convincing, and not merely arbitrary. The formal processes of Syrinx are designed to insure that the piece does not simply stop – the final close is a true ending. The ending, however, also contains – in Meyer’s terminology – a deviation from expectation. The

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“surprise” is not a mere novelty, because it does not depend on whether or not the listener has prior knowledge of how the composition comes to a close. Indeed, the surprise is only more effective as one becomes familiar with the piece, because it is an objective feature of the form, which follows from the flow of the textures.

D

ebussy used musical equivalents of three classical rhetorical devices: an appeal to the listener’s emotions, merismus (a juxtaposition of opposites that stands for the whole), and

more than one kind of formal symmetry. This is not to say that Debussy – who, though an extremely well-read composer with many friends in literary circles, was, nevertheless, by his own account, a highly intuitive composer – would have used the terms of classical rhetoric to describe his music; but, only that, in certain respects, literature and music have similar rationales. With these rhetorical devices, Debussy was able to create a rich, vivid, well-defined, and (in defiance of the reality principle) implicitly convincing world of imagination. The term merismus refers to expressions such as, “near and far,” “rich and poor,” and “young and old alike.” These expressions explicitly mention extremes, but we understand them to refer to more than just extremes. They refer not only to outer limits, but everything in between. For example, the expression “young and old alike” does not refer to a selection of people who are only young or old, but people of all ages. Exemplifying merismus, the opening two measures of Syrinx (Ex. D1a) contain a number of bold contradictions: The rhythms are both fast and slow, impulsive and hesitant. The hierarchy of accented and unaccented notes creates a tonality that fuses the intensity of the chromatic with the relaxation of the diatonic, the one embedded within the other. Small intervals descend from the bright, ethereal high register of the flute, only to be cancelled out by a large ascending leap returning to the high register. Tonal ambiguity is framed by its opposite, a closed, recirculating loop (a formal symmetry), beginning and ending on the same high B-flat above the treble clef. Paradoxically, the opening idea, though diverse, and even ambiguous in its implications, is concise and economical in its means of expression. By use of merismus, a few notes in the opening suggest more than they state. For this reason, the opening is not self-contained or complete, but ripe for development, as an opening should be. The opening of Syrinx illustrates what Tovey would call the difference between a theme and a “four-square tune.”

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Ex. D1. Incipits of the main sections of Syrinx: a. A, mm.1-2; b. A′, mm. 910; c. transition to B, mm. 14-16; d. A″, pickup to m. 26-m. 27.

As Curinga has pointed out, considerable differences are found in the way different theorists divide Syrinx into segments. The division that I am following is not unlike that made by Austin, Borris, and Cogan and Escot. Since I am primarily concerned with textures, however, and I gather data one full measure at a time, I will actually follow a somewhat different division made by Whitman. In its larger dimensions, Syrinx is in a variety of ternary form (a form much favored by Debussy, a point that is demonstrated repeatedly by DeVoto [2004]).

At a high level of

abstraction, though not in matters of detail, the ternary form, ABA, is a symmetrical mirror form, comparable to the classical rhetorical device of the chiasmus. A true mirror form, however, such as ABC…CBA, is symmetrical at every level. It is debatable, therefore, whether or not we should call the form of Syrinx a type of chiasmus. In any case, the ternary form and mirror form are both reflective symmetries, differing only in degree as to how much is reflected; and the rhetorical effects of the ternary form and the mirror form are comparable. Following the example of other theorists, I will describe the form of Syrinx as AA′BA″. The A′ section is an extension of A that serves as a transition into B. Syrinx is 35 measures long. In my division, A′ begins at measure 9, B at measure 14, and A″ at measure 26. Whitman disagrees with the other theorists about the beginning of B. I follow Whitman, because the texture changes at measure 14 (other theorists say that section B begins at measure 16). The A″ section begins with the pickup to measure 26. This pickup repeats the sustained, high B-flat that was

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introduced three beats earlier, however (at the end of section B), so the division between B and A″ is somewhat ambiguous from a textural standpoint. With respect to the broad outlines of the pitch contour, a conventional narrative arc covers B and A″. (With respect to other aspects of musical texture, development begins earlier than that, in the A′ section.) The arch in pitch contour found in B through A″ is late-peaking (the peak occurs two-thirds of the way through the arch) and, therefore – consistent with the timeseries geometry of arches – is more convoluted in the opening motion than in the closing motion. Both the opening and closing motions of this arch are convoluted; but the opening motion is more so, with a VHF equal to 0.095, less than half the VHF of the close, which is 0.235. The comparatively smoother line of the closing motion is a rhetorical device in its own right, which contributes to the sense of finality achieved by the ternary form.

T

he textures of Syrinx can be characterized by certain binary oppositions, each of which represents the poles of a merismus, the rhetorical device just mentioned that is a

summarizing opposition between extremes. The binary oppositions will be characterized by nine time-series indicators, which will be explained more fully in a moment. Five of these indicators are concerned with pitch contour: Duration-Weighted Average Pitch, Pitch Range, Interval Volatility (the average absolute size of the intervals), Extreme VHF (departures from the central value of the indicator, discussed previously, called the VHF) and Duration-Rising (the amount of time given to rising versus falling intervals). Two are concerned with temporal relationships: Rate of Attack and Metric Hierarchy (a duration-weighted average of the levels of the notes in the hierarchy of the meter, scaled by the highest hierarchic level achieved). The last two indicators are concerned with the distributions of pitch classes: Pitch-Class Concentration (a measure of how the pitch-classes are distributed around the Circle of Fifths), and Pitch-Class Dominance (a special-purpose, probabilistic indicator created to capture Debussy’s idiosyncratic treatment of pitch relationships in Syrinx). Each of the nine attributes of texture singled out for this analysis is intended to suggest either an aspect of cognitive-perceptual closure, or affective tension, or some combination of these traits. Since the composition is short, it does not provide us very many data points for purposes of statistical hypothesis testing. The observations recorded in this paper, furthermore, should not be taken as representative of any larger body of compositions, even other works by Debussy. Perhaps this study will lead to more studies in a similar vein, and a larger understanding of the

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relationship between texture and form in the works of Debussy and other composers will gradually be acquired.

For the present, however, the specific objective of this study is that it might

contribute to our conceptual understanding of the compositional technique evident in Syrinx itself. In order to accomplish this, the focus in this study will be on visual representations of descriptive statistics. To orient the reader, I will precede the detailed discussion of the indicators – which can get rather technical – with some general observations. The complexity of the constantly shifting textures in Syrinx is a consistent feature of the style. It would not, as a general rule, be possible to tell from the value of an indicator alone where in the form any particular measure is located. Nevertheless, over the course of time, the central tendencies of the indicators tend to be concentrated in identifiable channels. Each of the indicators is reducible to a long-term linear trend and a long wave. Systematic deviations from these patterns will be found where cadences are formed. Developmental processes will be seen in waves that move between cadential textures that are not typical of the piece and more typical, transitional textures. As we shall see, the flow of the textures in Syrinx runs in parallel with the ternary form, while overflowing the bounds of the form, thereby contributing to the effect of the composition as a rounded but continuous whole. For most of the indicators, the long waves are phase-shifted to lead the long wave of the Average Pitch. Exceptions: The long wave of the Duration-Rising is virtually flat, so it is not meaningful to speak of a lead time; and the long wave of the Metric Hierarchy is almost coincident with that of the Pitch. We may also calculate the divergence between the indicators, subtracting the standard scores of the detrended Pitch from the equivalent scores of the other indicators. The long wave of the divergence always peaks between the beginning of the A′ section and the middle of the B section. The lowest point after the peak that is reached by the long wave of the divergence occurs anywhere from the end of the B section to the end of the A″ section. An important consequence of the phase-shifting and divergence is that the development of the musical ideas begins as early as the A′ section. The process of development in Syrinx, therefore, results in a blurring of categories. Functionally, this serves to distinguish the end of the first part of the ternary form from the corresponding passage at the end of the third part of the form. The first part ends with transitional material; the third part ends with cadential material. Both of these sections occur in similar registers of the flute – near the bottom of descending melodic lines – but their musical functions cannot be confused. The descending line at the end of

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the first part might suggest that a cadence is approaching, if Debussy had not already begun developing his musical materials at this point. It would, of course, be fatal to the momentum of the composition if a cadence were to occur so soon. The effect of the contrast between the two endings is to avoid anticipating the final cadence prematurely.

E

ach time-series indicator will be shown in a series of charts, where vertical grid lines will represent the main divisions of the form. First, the original indicator will be compared with

its long-term linear trend. Accompanying the indicator will be a normalization of the same data, which is detrended by taking standardized residuals from a regression to the linear trend. The normalized indicator usually suggests a long wave of its own, to which a cubic polynomial will be fitted, for heuristic (that is, investigative) purposes. Since the pitch level is of special interest, it will be used as a basis of comparison for the other indicators. The data in many of these charts is smoothed to reduce the influence of outliers and make the graphs easier to understand. Smoothing is always done with centered moving averages of three measures length (the shortest time-span that will produce a centered moving average). Special-interest charts will be interjected into the main series of indicators where they are needed to make a broader point. The nine indicators considered here are not the only possibilities that one might imagine. To include more indicators might very well alter our conclusions; but, that is a matter for further research. The nine indicators that were chosen are useful, because they sample a wide variety of the attributes of musical texture, with very little overlap. A disclaimer is warranted at this point. I do not mean to imply by my use of polynomial curve fitting that I believe that the underlying process by which Syrinx was composed was actually governed by such simple deterministic mechanisms. No matter how closely the polynomials might happen to fit the data, I do not claim that the constants of the equations necessarily have direct causal interpretations. Nevertheless, I have found curve fitting helpful for the purpose of making qualitative visual comparisons between charts and for guiding my experiments in the creation of stochastic generative models.

Duration-Weighted Average Pitch The first time-series indicator we will consider is the average pitch for each measure, weighted by the relative duration of the notes (Fig. D1). Because of the intrinsic importance of pitch, I will examine this indicator more closely than any of the others.

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Pitches are assigned numbers that show the number of semitones the pitch is located above or below the final note, D-flat above Middle C. Tones below the final note are indicated with negative numbers (the lowest note is Middle C itself, which is assigned the value of minus one). The overall linear trend of the pitch level is downward. Before detrending, at major turning points, the curve makes both lower highs (in A and A″) and lower lows (in B and A″). The long wave of the detrended, normalized indicator roughly describes an Open-Low-High-Close form, a form that we have observed in Fux’s cantus firmi. To illustrate the long wave, a cubic polynomial curve of the form x = at3 + bt2 +ct +d has been fitted to the data. Curves of this form will be used extensively in this paper, as a heuristic aid to visual interpretation of the graphs. The Open-Low-High-Close form could be described as an S-curve, which can be thought of as either a ternary or binary symmetry. Considered as a ternary form, the S-curve surrounds an ascent by two descents. Considered as a binary form, the S-curve unites two major reversals in the pitch level – a trough,

 , followed by an arch,  . The binary pattern can be compared to

one of the seven frieze symmetries, the so-called glide reflection, in which a trough is transformed into an arch by a combination of inversion and translation through time. The term symmetry, however, suggests a greater regularity of form than we find here. For this reason, we will look for a more dynamic interpretation of the shapes that unfold in Syrinx. It is apparent from this chart that the general pitch level of the A′ section is transitional between A and B.

The pitch level is, presumably, arousing; and it suggests an initial

approximation of the overall climax-structure of the composition. (The concept of the climax implied by Syrinx, I should add, is subtle and goes far beyond the simple notion of going faster, louder, and higher.)

Fig. D1. Syrinx: Duration-Weighted Average Pitch, with detrended normalization.

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In a dynamic model, we shift our perspective from symmetry to process – or, rather, from observed symmetry to implied symmetry in an equation of motion. Instead of interpreting the data as a succession of contrasting, categorical entities – such as a descent followed by an ascent followed by another ascent, or a trough followed by an arch – we will view the data as an unfolding of simultaneous but opposed processes. The deterministic Limited Growth Model is shown fitted to the data in Fig. D2, using the parameters stated in Eq. D2. A more general form of the equation is stated in Eq. D1. To apply the formula, the measure numbers must be normalized to a time-scale that runs from zero to one. Also, the data stream must be rescaled, using Hurst’s method, so it begins and ends at zero. To do this, we first convert the incremental changes in the indicator to standard scores, and, then, reconstruct the indicator by accumulating the standard scores of the changes. Rescaling in this manner implies an assumption that the first and last data points are structurally important, an assumption we may not always want to make. The formula consists of a binary opposition between two binary oppositions. The term, (1 – t), is the growth limiting factor. This term ranges from one, when t equals zero, to zero, when t equals one. The term, atC, is the rising expansion factor; and the term, btD, is the falling expansion factor. These terms are equal to zero when t is zero. The formula, therefore, begins and ends with a value of zero, creating a closed form. This is why the data must be rescaled to begin and end with zero. Fitting the Limited Growth Model to the rescaled, unsmoothed Pitch, the parameters, a, b, C, and D, are estimated using a computerized numerical method of approximation (see Eq. D2). The following parameters account for 37 percent of the variance: a = 178.8, b = 157.2, C = 2.69 and D = 2.49. The degree of the equation as a whole is equal to the highest exponent plus one, C + 2.69 = 3.69. This tends to confirm that the heuristic cubic polynomial, of degree three, is a reasonable approximation to the Limited Growth Model. (Since this chapter was written, two terms have been added to the LGM, which no longer requires that the dependent variable be normalized. Statistical hypothesis testing, using Monte Carlo simulation, has also been added to the system.)

Eq. D2.

x = 178.8 t 2.69 (1 – t) – 157.2 t 2.49 (1 – t).

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The long wave established by the Limited Growth Model resembles the heuristic cubic polynomial very closely, the most notable difference being that it places the peak of the curve later. I will continue to use the heuristic polynomial throughout the rest of this study, since it does not assume that the first and last data points are structural (an assumption which could always be debated); and it appears to make a satisfactory fit in most cases.

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00 0.00

0.10

0.20

0.30

0.40

0.50

Rescaled Unsmoothed Pitch

0.60

0.70

0.80

0.90

1.00

Limited Growth Model

Fig. D2. Syrinx: Rescaled, unsmoothed Pitch vs. the Limited Growth Model.

The curve that we see in the chart is the difference between two large, late-peaking arches that are nearly identical to each other. It may be worthy of notice that the parameters of the rising expansion factor, a and C, are larger than the corresponding parameters of the falling expansion factor, b and D – that is, a > b and C > D. This is the defining characteristic of an Open-LowHigh-Close curve. In applying the Limited Growth Model to other compositions, we will want to be aware of the following: We will assume that the exponents C and D are both positive. If C and D were equal to each other (the degenerate case), the complementary relationship between the rising and falling factors would be redundant. To arrive at a unique solution in this case, the factor after the minus sign should be omitted.

The degenerate curve reaches its maximum (assuming a is

positive) when t = C/(C + 1). In the general case, where C and D are not equal, the value of the resultant curve x is zero when the time t = 0 or 1. If the resultant curve is either an Open-Low-High-Close form or an Open- High-Low- Close form, then x is zero for some value of t between 0 and 1. It can be shown in this case that t = (a/b)(1/(D-C)). The latter expression is equivalent to ln(t) = ln(a/b) / (D-C). For t in the range 0 < t < 1, ln(t) is a negative number, because ln(1) = 0. This implies that either ln(a/b) < 0, or (D-C) < 0, but not both. Since the logarithm of a negative number is not defined, the ratio

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a/b must be positive, meaning that a and b must have the same sign. From this we conclude that either a < b and C < D, or a > b and C > D. Assuming a and b are both positive, the former case produces an Open-High-Low-Close form, and the latter produces an Open-Low-High-Close form. The opposite is true for a and b negative.

Pitch Range The Pitch Range (that is, the difference in semitones between the highest and lowest pitches in a measure) also has a broad descending trend; but, the long wave of the normalized indicator is a simpler arch-form (Fig. D3a). Although the Pitch Range is presumably arousing, the peak of the Pitch Range is not directly synchronized with the peak of the Average Pitch, which occurs at the beginning of the A″ section. The peak of the Range occurs “early,” in the A′ section, and the indicator makes lower highs from that point on. De-synchronization of the indicators contributes to the variety of the musical ideas, and it insures that something interesting is happening in each section. Given that the Pitch Range makes lower highs from its peak in A′, and the fact that the Range cannot be less than zero, the variation in the indicator is much wider in the first third of the composition than it is later, where it becomes progressively narrower. This variation contributes to the improvisatory effect of the A and A′ sections, distinguishing them from the more considered character of the B and A″ sections. The progression from greater variability at the beginning to less variability toward the end is consistent with the text, which begins with a dialogue between the Oread and the Naiad, and ends with a long speech by the Naiad. Since the Pitch Range is low at or near cadences or sectional divisions, high values are also associated with regional climaxbuilding.

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Fig. D3a. Syrinx: Pitch Range.

The normalized, unsmoothed Range (Fig. D3a) is negatively correlated with the normalized, unsmoothed Pitch (R = -.0246).

Fig. D3b. Syrinx: Normalized Pitch Range vs. normalized Average Pitch, smoothed.

By visual inspection of Fig. D3b, one may see that the long wave of the normalized Pitch Range leads the long wave of the normalized Average Pitch. It is apparent that the smoothed Pitch Range moves in four regional cycles, each peaking inside one of the sections A, A′, B and A″.

Fig. D3c. Syrinx: Divergence of smoothed, normalized Pitch Range from smoothed, normalized Average Pitch.

Henceforth, the divergence between two indicators will be calculated by subtracting one from the other. For the indicators to be expressed in comparable units, it is necessary that they both be converted to standard scores, as we have done in our normalization process.

The

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examples cited here will all represent smoothed indicators, and the reference indicator will always be the Pitch. Notice in Fig. D3c that the long wave of the divergence between the smoothed, normalized Range and the smoothed, normalized Pitch peaks near the boundary between the A′ and B sections. Volatility As an indicator of Volatility, we take the average value of the absolute interval sizes for each measure (Fig. D4). The Volatility trends in the opposite direction from the Pitch. However, direct comparisons between the Volatility and the Pitch – not only for the long term trend, but also for the lead time – will be most clearly evident if we invert the standard scores of the Volatility, multiplying them by minus one. The inverted scale will be called the Continuity. The significance of the Volatility is primarily local. An arch in the Volatility curve spans the B section, and an arch of Volatility characterizes the motivic transformations of the last five measures (described by Mahlert ; see Ex. D3, D5). The very last peak of Volatility in that section contributes to the unique expressive qualities of the end of the composition. The data point for measure 33 is arguably an outlier; but, if that is the case, the data point for measure 25 (which is cadential) could also be considered an outlier. I will accept both data points and simply point out that the Volatility goes to greater and greater extremes as the piece continues. The Volatility is low at or near the ends of the three major divisions of the form (in the A′, B, and A″ sections). The peaks, which show a long-term trend toward making higher and higher highs, are located within the interiors of those divisions. The leap of an augmented sixth that is responsible for the peak of Volatility in measure 33, by the way, is not the largest leap in the composition. There is an octave in measure 17 and a ninth in measure 19, but the averages in those measures are brought down by smaller intervals.

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Fig D4a. Syrinx: Interval Volatility.

The normalized, unsmoothed Volatility (Fig. D4a), before inversion, has a very small negative correlation with the normalized, unsmoothed Pitch (R = -0.096).

Fig D4b. Syrinx: Normalized Interval Continuity (inverted Volatility) vs. normalized Average Pitch, smoothed.

The lead time of the Volatility over the Pitch is most clearly visible after the indicator is inverted (Fig. D4b).

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Fig. D4c. Syrinx: Divergence of smoothed, normalized Continuity from smoothed, normalized Average Pitch.

The divergence between the Continuity and the Pitch reaches its highest point at the beginning of the B section. The long term trend of the Volatility may be functional, even though it is dominated by short term variation. Large intervals, though arousing, do not – according to the Meyer / Narmour theory of implications – imply continued motion in the same direction. Therefore, it could be argued that – except for special cases where small intervals are confined within a narrow range – passages with lower Volatility (that is, higher Continuity), will tend to create a heightened perception of implicative momentum (a lower expectation of closure). From this point of view, the long term ascending trend in Volatility found in Syrinx suggests an overall declining trend in perceived implicative momentum. Declining implicative momentum would, presumably, be one factor contributing to a large-scale implication of closure toward the end of the composition.

Extreme VHF The VHF is more variable in the A sections than in the B section. This is shown by taking the absolute value of the deviation of the VHF from the mid-point, 0.50 (Fig. D5). The overall linear trend of this indicator, which will be called the Extreme VHF, is upward. The curvilinear trend makes an Open-High-Low-Close form. In other words, the trends are more or less the opposite of what we found for the Average Pitch, except for the fact that the Extreme VHF is highest at or near the ends of the A, A′, B, and A″ sections.

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Fig. D5a. Syrinx: Extreme VHF.

A low value of the Extreme VHF implies that the melodic contour is relatively complex and, presumably, stimulating. A high value implies that the melodic contour is relatively simple – either static or linear – and, therefore, not very arousing. So, an upward trend in the Extreme VHF is affectively consistent with a trend in the opposite, downward direction in the trend of the Average Pitch Level.

Similarly, an Open-High-Low-Close wave in the Extreme VHF is

affectively consistent with the opposite form, an Open-Low-High-Close wave in the Average Pitch. Over the short term, the relative simplicity of both stasis and linearity is appropriate for endings. That explains why we see peaks in the Extreme VHF at or near the ends of sections. Lower values of the indicator are more appropriate for development. For this reason, low values are most frequent in the B section and the first half of A″. For comparison with the Average Pitch, the Extreme VHF should be inverted. To distinguish the original form of the indicator from the inverted form, we may enclose the name in parentheses – as (Extreme VHF) – or give it a meaningful name – such as, Central VHF.

The

long wave of the normalized Extreme VHF (after inversion) leads the long wave of the normalized Average Pitch. The normalized, unsmoothed Extreme VHF (Fig. D5a), before inversion, is positively correlated with the normalized, unsmoothed Pitch (R = 0.214).

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Fig. D5b. Syrinx: Normalized Central VHF (inverted Extreme VHF) vs. normalized Average Pitch, smoothed.

Fig. D5c. Syrinx: Divergence of smoothed, normalized Central VHF from smoothed, normalized Average Pitch.

The Central VHF is a leading indicator for the Pitch (Fig. D5b). The divergence between the Central VHF and the Pitch peaks inside the B section (Fig. D5c).

Duration-Rising We are interested in the amount of time that is taken to move from one pitch level to another. The next indicator, the Duration-Rising, returns the cumulative amount of time spent in rising intervals minus the cumulative time spent in falling intervals (Fig. D6). This indicator differs from a simple count of the relative proportions of rising and falling intervals, because it compensates for passing notes: for example, it treats a descent through four sixteenth-notes the same as a descent through one quarter-note. For example, the first note of Syrinx is a dotted quarter-note, which is counted as having a duration of 0.750, that is, three-fourths of a beat. The first interval of Syrinx moves downward,

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from B-flat to A. Since it takes three-fourths of a beat to move from B-flat to A, the first entry in the calculation of the Duration-Rising is minus one times three-fourths, or -0.750. The next interval takes one thirty-second note to ascend from A to B-natural. A thirty-second note is oneeighth of a beat, so 0.125 is added back into the Duration-Rising score, giving an accumulated value of -0.625. This procedure is continued throughout the entire composition. The value that is reported is the accumulated Duration-Rising score for the last note in each measure. It is possible for trends in the Duration-Rising to diverge from the pitch contour. Before detrending, there is a strong tendency for Syrinx to spend more time in falling intervals and less time in rising intervals, a characteristic that is associated with the cantabile melodic style. True, the Duration-Rising indicator is concerned with the amount of time spent in rising and falling intervals, rather than a simple count of rising and falling intervals. Nevertheless, what we find in Syrinx is broadly consistent with the observation (discussed earlier) that folk melodies from all around the world – which we may take as representative of the traditional cantabile style in general – typically have more falling intervals than rising intervals.

Fig. D6a. Syrinx: Duration-Rising.

The normalized, unsmoothed Duration-Rising (Fig. D6a) is strongly correlated with the normalized, unsmoothed Pitch (R = 0.648). One would like to see future research investigating the possibility that the Duration-Rising may accentuate the difference in arousal potential between high and low pitch levels. The falling trend of the Duration-Rising curve for Syrinx is so pronounced that it is difficult to evaluate the embedded long wave until the data is normalized and smoothed (Fig. D6b). After detrending, we can see that the contour of the Duration-Rising curve is similar to the

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contour of the Average Pitch. The long wave of the Duration-Rising is very weak. The DurationRising, therefore, is best interpreted as a regional property of the melody, rather than a global property.

Fig. D6b. Syrinx: Normalized Duration-Rising vs. normalized Average Pitch, smoothed.

Fig. D6c. Syrinx: Divergence of smoothed, normalized Duration-Rising from smoothed, normalized Average Pitch.

The long wave of the divergence between the Duration-Rising and the Pitch (Fig. D6c) peaks in the A′ section, a manifestation of the less than perfect correlation between the two indicators. The long wave of the divergence makes an Open-High-Low-Close form, counter to the long waves of the indicators themselves. This completes the discussion of the indicators concerned with pitch contours. We now turn to indicators concerned with temporal relationships.

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Rate of Attack The Rate of Attack (Fig. D7a), presumably an arousing property, is the number of notes initiated per beat in a measure.

Fig. D7a. Syrinx: Rate of Attack.

The Rate of Attack has a long-term descending trend, like that of the Average Pitch. In spite of this fact, the normalized, unsmoothed Rate of Attack (Fig. D7a) is negatively correlated with the normalized, unsmoothed Pitch (R = -0.278).

Fig. D7b. Syrinx: Normalized Rate of Attack vs. normalized Average Pitch, smoothed.

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Fig. D7c. Syrinx: Divergence of smoothed, normalized Rate of Attack from smoothed, normalized Average Pitch.

The discrepancy is partly accounted for by the fact that the long wave of the normalized Rate of Attack leads the long wave of the normalized Average Pitch (Fig. D7b). The divergence (Fig. D7c) peaks near the beginning of Section B.

Metric Hierarchy The remaining indicators are more complex than the preceding, beginning with the Metric Hierarchy, an indication of the strength with which the metric structure is presented. Because of space limitations, the following discussion is rather terse. It is provided for the benefit of those who might wish to replicate this study. Readers who are looking for a broad overview of the analysis might want to skim through this section. Toussaint (2013, pp. 67-72) calls special attention to the metric theories of LerdahlJackendoff (1983, 1996) and Keith (1991). My treatment of the metric hierarchy – which is based on the traditional Western conception of meter, according to which certain beats are given more weight than others, according to their position in a hierarchy – begins with assumptions similar to those of Lerdahl and Jackendoff. Keith’s analysis of the combinatorics of meter and rhythm (pp. 121-139) is full of valuable information. Nevertheless, I would take issue with Keith’s theory of syncopation (pp. 133-135). One example will suffice to make the point. Keith classifies the rhythm of a dotted quarter-note followed by an eighth-note (where the dotted quarter-note falls on a strong beat) as a type of weak syncopation that he calls a hesitation. I would argue, to the contrary, that – unless the eighth-note is given more stress than the dotted quarter-note – this is an example of a strongly metric rhythm, not a syncopation.

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In this study, departures from perfect regularity will be distinguished as falling into two broad classifications: rhythms that confirm the metric hierarchy, and rhythms that counter the metric hierarchy. My understanding is that a syncopation, according to the traditional view, is a rhythm where at least one note is more strongly accentuated than would be warranted by its position in the normative metric hierarchy. There are several ways that a note can be accentuated. For example, it can be played with a louder attack than other notes or doubled by supporting instruments or voices. For present purposes, I will be exclusively concerned with another kind of accentuation: the agogic accent, in which a note is longer in duration than other notes in its immediate context. By duration, I will mean the length of time between attacks; so a half-note will be counted as having the same duration as a quarter-note followed by a quarter-note rest. To be sure, from the standpoint of performance practice, the former case – that is, a fully sustained half note – can be considered to be somewhat more accentuated than the latter – a quarter-note followed by a quarter-note rest. (The following statement is now obsolete. Treatment of rests has been added to the system since this statement was written.) It is impossible to say how much weight should be given to that factor, however, and the two rhythms can be interpreted as structurally equivalent; so, I will count them as the same. The question, then, is the degree to which the distribution of the durations of the notes in a rhythm is consistent with their levels in the metric hierarchy. For our multidimensional studies, furthermore, it is important to be able to understand the relationship between duration and position in the metric hierarchy independently from the Rate of Attack. If we were to simply assign scores to the notes based on their positions in the metric hierarchy and, then, add up the scores, the evaluation of a rhythm would be strongly affected by whether the rhythm contains few or many notes. To correct for this, we will make duration as important as position in the measure. We will look at the relative amount of time that is given to notes at different hierarchic levels.

Ex. D2. Examples of Metric Hierarchy Scores. Metrically accentuated four-four: a. 1.300; b. 1.179. Evenly divided four-four: c. 1.000; d. 1.000; e. 1.000. Syncopated fourfour: f. 0.964; g. 0.875; h. 0.700. First beat missing in four-four: i. 0.250. Metrically accentuated three-four: j. 1.111. Evenly divided three-four: k. 1.000; l. 1.000. Syncopated three-four: m. 0.800. First beat missing in three-four: n. 0.250.

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The Metric Hierarchy indicator that is proposed here distinguishes evenly divided rhythms from both metrically accentuated and syncopated rhythms (Ex. D2). Evenly divided rhythms receive a score of 1.000, regardless of how many notes are attacked in a measure. (A single note is considered to be equivalent to repeated notes of equal length.)

Metrically

accentuated rhythms receive a score greater than 1.000. Syncopations receive a score less than 1.000. The hierarchy of beats and subdivisions of beats in a duple (or quadruple) measure is, of course, straightforward: each measure divides into two parts, which are each subdivided into two parts, and so forth. Depending on where a note falls in a measure, we can assign it a Depth score. For example, in four-four, we can assign the first beat a Depth of zero. The next division of the hierarchy occurs at the third beat, which we can assign a Depth of one. Subdividing the hierarchy again, divisions occur at beats two and three, which we can assign a Depth of two. Half beats not otherwise accounted for can be assigned a Depth of three. In theory, this pattern can be extended indefinitely; but, we run into practical limitations because of the complexity of the rhythms found in Syrinx. Two-four is exactly analogous to four-four. Three-four raises a question, which we will return to in a moment. To discover the Depth scores programmatically, we construct what could be described as a hierarchically layered sieve. We test the location of a note to see whether or not it falls on a boundary of the sieve, starting with the most open layer of the sieve, and progressively testing the location against smaller and smaller divisions of the sieve. The more layers that the location passes through before it is caught, the higher the Depth score that the note receives. (The following discussion of three-four is now obsolete. In the latest version of this system, secondary accents are only considered for meters that divide evenly into two parts.) Three-four is not as symmetrical as four-four and two-four, because the measure does not divide evenly into two parts. In a three-four measure, the rhythm of a half-note followed by a quarternote is normal, but the opposite pattern would be considered a syncopation. For computational purposes, therefore, it is convenient to treat the third beat of a three-four measure as being just as normal as the third beat of a four-four measure. Putting this all together, it follows that the parser in our metric analysis program can treat a measure of two-four as an incomplete measure of four-four; and it can also treat a measure of three-four as an incomplete measure of four-four. This conclusion is surprising. It will be

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interesting to see if it holds up under future testing. As we shall see, the algorithm appears to give reasonable results when applied to the data from Syrinx. Syrinx is primarily in three-four, but it has two measures of two-four. In Syrinx, the beats can be divided into either two or three parts. The half-beats can be divided into either two, three or four parts; and there are also a number of grace-notes. In the next to last measure, five notes occur in the time of two quarter-note beats. It would be very difficult to account for all of these variations programmatically, so the hierarchy of the sieve will have a limited number of levels: shallowest (at the beginning of a measure), shallow (at two beats into the measure), deep (at either one or three beats into the measure), deeper (on the half-beats), and deepest (subdivisions of halfbeats). From the Depth, we calculate a Level score, which is equal to one-half raised to the power of the Depth. A Depth of zero corresponds to a Level of one, because any number raised to the power of zero is equal to one. A Depth of one corresponds to a Level of one-half, because any number raised to the power of one is equal to itself. A Depth of two corresponds to a Level of one-fourth; a Depth of three corresponds to the Level of one-eighth, and so on. As the Depth increases, the Level decreases. We also calculate a Weight, which is the relative duration of the note times its Level. The relative duration should not be confused with the location of the note in the measure, called the Time-In-Measure. Whereas the Time-In-Measure is a point in time given in beats past time zero, the relative duration is a length given as a fraction of the total length of the measure (strictly speaking, the total length of the notes attacked during the measure). In three-four meter, for example, the location in time of the three beats is 0.000, 1.000, and 2.000, respectively. However, the relative duration of notes held for one, two or three beats is 0.333, 0.667 or 1.000, respectively. Important differences are revealed when we compare a weighted average of the Levels against an unweighted average of the Levels. An unweighted average of the Levels of the individual notes in the measure will be called the Aggregate Level. A weighted average of the Levels will be called the Aggregate Weight. The Aggregate Weight is calculated as the sum of the individual Weights, because the multiplier for the individual Weights is a relative duration, not an absolute duration. When all of the notes in a measure are the same length, the Aggregate Weight is equal to the unweighted Aggregate Level. When the notes are different lengths, the weighted average will count more heavily than the hierarchic levels where long notes occur. A comparison of the two

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aggregates will give us information about the relative importance of the hierarchic levels in a meter. With one exception that we will address in a moment, the Metric Hierarchy score is the ratio of the Aggregate Weight divided by the Aggregate Level. In these cases, a value of 1.000 will occur when all of the notes in a measure are the same duration. A value greater than 1.000 will occur when the rhythm is metrically stressed. A value less than 1.000 will occur when the rhythm is syncopated. The rule, as described so far, is not complete, because it assumes that a note is attacked on the first beat of a measure. This is usually true in Syrinx, but there are eleven exceptions (all but two of which occur in the second half of the piece). Such cases require us to scale the Metric Hierarchy score. As long as there is an attack on the first beat of a measure, the highest hierarchic level in the measure is 1.000. If there is not an attack on the first beat, the highest hierarchic level will be less than 1.000. To adjust for this case, we will multiply the ratio between the Aggregate Weight and the Aggregate Level by the highest hierarchic level found in the measure.

This

reduces the score substantially, implying that the rhythms are highly syncopated. For example, suppose that the rhythm consists of two half-notes in common time. Each will have a relative duration of 0.500. The first note will have a location of 0.000 and a Depth of zero. Its Level is 1.000, because – as was mentioned previously – any number raised to the power of zero is equal to one. The second note will have a location of 0.500 and a Depth of one. Therefore, its Level is 0.500, because any number raised to the power of one is equal to itself. The average of these levels is 0.750. The Weight assigned to the first note is the relative duration of 0.500 times the Level of 1.000, which is 0.500. The Weight assigned to the second note is the relative duration of 0.500 times the Level of 0.500, which is 0.250. The sum of these weights gives a weighted average of 0.750, which is equal to the average Level. Therefore, the ratio of the Aggregate Weight divided by the Aggregate Level is 1.000. By similar calculations, it can be shown that four quarters in common time have an Aggregate Level and an Aggregate Weight of 0.500; so, the ratio of the two is also 1.000. A typical metrically accented rhythm would be a half-note followed by two quarter-notes. In this case, the Aggregate Level is 0.583; and the Aggregate Weight is a larger number, 0.6875. The ratio between the two is 1.179, which is greater than one. A typical syncopation would be a quarter-note followed by a half-note followed by a quarter-note. Here, the Aggregate Level is 0.500; and the Aggregate Weight is a smaller number, 0.438. The ratio between the two is 0.875, which is less than one.

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A typical end-accented rhythm (which implies the perceptual closure one might find at a cadence) would be two quarter-notes followed by a half-note. The weight-to-level ratio of this rhythm is 0.964, which is less than the ratio for an even rhythm, but greater than the score for the last-mentioned syncopation of a quarter-note followed by a half-note followed by a quarter-note. The rhythm of two quarter-notes followed by a half-note is actually a kind of syncopation, but it fits the metric hierarchy better than the previously mentioned syncopation. Similarly, a dotted half-note followed by a quarter-note gets a score of 1.300; and one quarter-note followed by a dotted half-note gets a score of 0.700. In three-four, a dotted half-note gets a score of 1.000, and so do three quarter-notes, because each of these is an even rhythm. A half-note followed by a quarter-note gets a score of 1.111; and the reverse, a quarter-note followed by a half-note, gets a score of 0.800. If there is a rest on the first beat of either four-four or three-four and the rest of the meter is filled out by one note, the highest hierarchic level is 0.250 and the score is also 0.250. The score for a dotted half-note followed by a quarter-note (1.300) is exactly balanced by the score for a quarter-note followed by a dotted half-note (0.700). Each is equidistant from the neutral score of 1.000. This is exceptional, however. The Metric Hierarchy scores are not always so symmetrical (and it is an open question whether or not the scores should be symmetrical). In both four-four and three-four meters, one can find examples of complementary rhythms that do not receive reciprocal scores. In four-four, the score for a half-note followed by two quarter-notes (1.179) is farther from 1.000 than the score for two quarter-notes followed by a half-note (0.964). In three-four, the score for a half-note followed by a quarter-note (1.111) is closer to 1.000 than the score for a quarter-note followed by a half-note (0.800). The scores, then, are useful as a rough guide to the metric structure of a rhythm. Although any deviation from the neutral value is probably stimulating, metrically accentuated rhythms (those getting scores above 1.00), will tend to imply a greater sense of continuity than syncopated rhythms. An accentuated rhythm (Ex. D2: a, b, i) will generally end with shorter notes that lead into the next measure, whereas a syncopated rhythm (Ex, D2: f, h, l) may end with a long note that suggests a braking action, or closure. On the other hand, there are grey areas in the evaluations of rhythms and their implications, such as the quarter-note followed by a half-note followed by a quarter-note (Ex. D2: g), where a braking action is followed by an accelerating action. In this case, there is an interior rhythmic progression that is not well represented by an aggregate calculated over the whole measure. One needs to take some care in interpreting the Metric Hierarchy indicator.

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The pattern of Metric Hierarchy scores in Syrinx is the familiar descending trend with an Open-Low-High-Close long wave (Fig. D8a). Before detrending, the majority of the scores are greater than the neutral value of 1.00. The rhythms, on average, are more consistently metric in the A section of the piece, least metric at the beginning of the B section, and variable on the low side thereafter. This suggests a broad tendency to move from high implicative momentum to low implicative momentum, similar to what was observed with the Volatility.

Fig. D8a. Syrinx: Metric Hierarchy.

The normalized, unsmoothed Metric Hierarchy (Fig. D8a) is negatively correlated with the normalized, unsmoothed Pitch (R = -0.101).

Fig. D8b. Syrinx: Normalized Metric Hierarchy vs. normalized Average Pitch, smoothed.

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Fig. D8c. Syrinx: Divergence of smoothed, normalized Metric Hierarchy from smoothed, normalized Average Pitch.

The long wave of the normalized Metric Hierarchy is almost coincident with the long wave of the normalized Average Pitch, leading slightly (Fig. D8b).

The long wave of the

divergence peaks in the A′ section and troughs near the beginning of the A″ section (Fig. D8c). Now we move from rhythmic indicators to indicators concerned with the distribution of pitch classes.

Pitch-Class Concentration The Pitch-Class Concentration is an indication of relative diatonicism. It is a spatial interpretation of the Herfindahl Index, which is used by social scientists to evaluate social inequality. The Pitch-Class Concentration is calculated as follows: The pitch-classes found in a measure are arranged around the circle of fifths. Each pitchclass is only counted once per measure. We count the number of fifths between pitch-classes that are adjacent to each other in the circle, going all the way around the circle to return to the place where we started counting. For example, if the two pitch-classes occurring in a measure were Dflat and G-flat, there would be two intervals between them in the circle of fifths. One of the intervals would be equal to one perfect fifth, and the other interval would be equal to eleven perfect fifths. In other words, the intervals between D-flat and G-flat would be 1/12 and 11/12 of the whole circle, respectively. To calculate the Concentration indicator, we square each of those fractions and add them together. The indicator, in this case, is equal to 0.0069 + 0.8403, or 0.8472. The highest value of the indicator is obtained for a single pitch, which has a concentration score of 1.000. The lowest possible value of the indicator is obtained when the pitch-class set

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consists of all twelve notes of the chromatic scale. In this case, the indicator is equal to 12 times the square of 1/12. Two twelves cancel out, so the minimum value is equal to 1/12, or 0.0833. The indicator is considered a measure of concentration, because it gives higher scores when the pitch-class sets are crowded together in the circle of fifths and lower scores when they are dispersed far apart in the circle of fifths. Since the seven notes of a major scale are all adjacent to one another in the circle of fifths, the Pitch-Class Concentration gives a rough approximation of the degree of diatonicism found in a pitch class set. Low values of the Pitch-Class Concentration tend to be associated with relatively chromatic passages. For Syrinx, the Pitch-Class Concentration (Fig. D9a) tends to be largest near the ends of the first, third, and fourth sections: A, B, and A″. Maximum values of the concentration occur where sustained single notes appear. The long-term trend is up, because of the distribution of cadences, which come closer together in the second half of the piece. Since this is a trend away from tension toward relaxation, it does not contradict the most common trend of the indicators from an affective standpoint. The long wave of the indicator does not fit its peaks very well, however. What we have here might be best described as a series of isolated pulses rather than a trend plus a wave. In other words, it is more like a rhythm than a melody.

Fig. D9a. Syrinx: Pitch-Class Concentration.

The normalized, unsmoothed Pitch-Class Concentration (Fig. D9a), before inversion, is positively correlated with the normalized, unsmoothed Pitch (R = 0.390).

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Fig. D9b. Syrinx: Normalized Pitch-Class Dispersion (inverted Pitch-Class Concentration) vs. normalized Average Pitch, smoothed.

Fig. D9c. Syrinx: Divergence of smoothed, normalized Pitch-Class Dispersion from smoothed, normalized Average Pitch.

We should invert the Pitch-Class Concentration before comparing it with the Average Pitch, in order to match their long-term trends. To designate the inverted indicator, it will be called the Pitch-Class Dispersion. After inversion, the long wave of the normalized Pitch-Class Concentration leads that of the normalized Average Pitch (Fig. D9b). The divergence (Fig. D9c) peaks near the beginning of the B section. Pitch-Class Dominance For an extended discussion of Debussy’s tonality in general, see DeVoto (2004). The tonality found in Syrinx can be thought of as a condensation of tendencies found in other works by

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Debussy: the importance of absolute pitch, the ambiguous modality, the avoidance of traditional cadences, and the assertion of tonal centers by emphasis rather than functional harmony. Debussy’s selection of pitch classes in Syrinx is different for relatively accented notes than it is for relatively unaccented notes. Since accentuation is distributed over several hierarchic levels, the concept of relative accentuation requires definition. The accented or unaccented status of the first note is determined manually. If a note is at a lower level in the metric hierarchy than the preceding note, it is deemed to be unaccented, and vice versa. If two notes in succession are at the same hierarchic level, the second note takes on the accented or unaccented status of its predecessor. As can be seen in the star chart that shows the frequency distribution of accented and unaccented notes, pitch-classes are not by any means equally distributed in Syrinx (Fig. D10). The most common accented pitch classes in Syrinx lie adjacent to one another in the circle of fifths, ranging from G-flat through D-flat, A-flat and E-flat to B-flat. The latter, B-flat, is the most numerous of all the accented pitch classes.

Fig. D10. Syrinx: Frequency of accented and unaccented pitch classes in the circle of fifths.

Syrinx contains allusions to the tonal framework of G-flat major (and E-flat Phrygian in measures 5-8); but its organizing principles are remote from traditional major-minor triadic tonality, being more rhythmic than harmonic in conception.

Unaccented pitch classes are

generally more common than accented pitch classes. Two of the more common unaccented pitch classes (and not the least common of the accented pitch classes) are D-flat and G-flat. The former, D-flat, is the final pitch class of the entire composition (corresponding to what would be the tonic in a traditional tonality); and the latter, G-flat, would be the subdominant in a traditional tonality

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of D-flat major. The most common of the remaining unaccented pitch classes are members of the whole-tone scale ranging through the white-note section of the circle of fifths between B and F. Except for D-flat and G-flat, there is a tendency for accented pitch classes to be black notes and unaccented pitch classes to be white notes. The metric hierarchy is used to partition the circle of fifths into rival, though somewhat overlapping, regimes. This appears to be a highly distinctive conception of tonality, but more research is needed to place it into historical perspective. Pitch-Class Dominance (Fig. D11a) is calculated by taking a duration-weighted average of the probabilities that the pitch classes found in a measure will be accented or unaccented in the composition as a whole.

Fig. D11a. Syrinx: Pitch-Class Dominance.

This is another of the charts in which the long-term trend is down and the long wave is an Open-Low-High-Close form. Before detrending, the lowest values of the indicator are found toward the end of the piece. The normalized, unsmoothed Pitch-Class Dominance is positively correlated with the normalized, unsmoothed Pitch (R = 0.438).

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Fig. D11b. Syrinx: Normalized Pitch-Class Dominance vs. normalized Average Pitch, smoothed.

Fig. D11c. Syrinx: Divergence of smoothed, normalized Pitch-Class Dominance from smoothed, normalized Average Pitch.

The long wave of the normalized Pitch-Class Dominance leads that of the normalized Average Pitch (Fig. D11b). The long wave of the divergence (Fig. D11c) peaks in the B section.

Intercorrelation analysis Next, we will examine the intercorrelations between the nine normalized indicators (that is, the detrended standard scores).

The Volatility, Extreme VHF, and the Pitch-Class

Concentration are inverted, just as they were in our analysis of phase-shifting and divergence. Every measure is compared with every other measure. The correlations are arranged in a square array (or matrix). Table D1 shows a heat map based on the correlations, smoothed to make the broad patterns easier to understand. The array is symmetrical, because the first row is the

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same as the first column, the second row is the same as the second column, and so forth. I will call the diagonal that goes from the extreme upper left of the array to the extreme lower right the main diagonal. In diagonals that move upward from left to right, the values below the main diagonal are mirrored by the values above the main diagonal. The smoothing is done as follows: Since the correlations along the main diagonal of the measure-by-measure correlation table are all equal by definition to 1.00, they do not contain any meaningful information, so we first remove all of these cells.

The measure-by-measure

correlations are then smoothed by taking a centered 3 x 3 measure moving average – the smallest block that produces a centered moving average. Low numbers are formatted with a darker gray color than high numbers.

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Bar

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Table D1. Syrinx: Smoothed intercorrelation heat map of detrended indicators. High values are represented as brighter than low values.

It is visually apparent that the correlations move in waves. A close examination of these waves will refine our understanding of how the musical form divides into sections. To understand the basic structure of the waves, we will first examine the averages of the correlations found along each row (or column) of the smoothed intercorrelation table. We shall find that predominately dark rows or columns in the heat map represent textures that tend toward the cadential, and brighter rows or columns represent more developmental textures. The distribution of the row averages is somewhat skewed (Fig. D12). Similarity is more common than dissimilarity.

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Fig. D12. Syrinx: Frequency distribution of the averages by row in the smoothed table of intercorrelations between detrended standard scores.

The row averages provide us an indication of the Typicality of each three-measure segment of Syrinx. When the row average is high, that tells us that the row is highly correlated with all of the other rows and is therefore typical of the composition as a whole. A low average tells us the opposite. As would be expected, the most typical ideas in Syrinx are developmental in character; and the most atypical ideas are cadential. Table D2 shows that there are strong positive correlations between the Typicality and the Pitch-Class Dispersion, Central VHF, Pitch Range, and Rate of Attack. These indicators are all concerned with different aspects of complexity and movement – that is, tonal complexity, complexity of melodic contour, pitch movement, and rhythmic activity, respectively. As such, they are characteristic of developmental passages, but contrary to the formation of cadences. On the other hand, the Typicality has especially strong negative correlations with the Duration Rising and the Average Pitch (corresponding to positive correlations with Atypicality). That is, with an important exception, the atypical, cadential passages tend, on balance, to be high and rising. This is only true of the internal cadences, however. The internal cadences are best understood as waiting passages rather than true closes (the concept of a “half cadence” does not quite apply in the absence of tonal harmony). The end of the piece is quite different. The Duration Rising and the Average Pitch are “switches” that help to distinguish the internal cadences from the final cadence.

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Indicator

Correlation

Duration Rising

-0.866

Average Pitch

-0.793

PC Dominance

-0.406

Interval Continuity

-0.137

Metric Hierarchy

0.062

Rate of Attack

0.485

Pitch Range

0.607

Central VHF

0.646

PC Dispersion

0.799

Table D2. Syrinx: Correlations of the normalized, smoothed indicators with the row averages of the detrended, smoothed intercorrelations (Typicality).

It is revealing to compare the row averages with the Average Pitch. There is a strong negative correlation (R = -0.793) between the row averages and a correspondingly smoothed measure of the Average Pitch (centered three-measure moving average of the Duration-Weighted Average Pitch). This is shown in Fig. D13. To make the association between the two indicators more vivid, the averages of the rows are inverted (multiplied by minus one) to change the correlation from negative to positive. The inverted curve is called Cadential, or Atypical. The two curves are on opposite sides of the zero line at the beginning and the end; but they follow each other rather closely, for the most part. At the end, they pull apart decisively – Smoothed Pitch ending below the midline, and Atypical ending above the midline. The result is that the Cadential curve has three main peaks, and the Smoothed Pitch has only two.

The three peaks in the Cadential curve correspond to the main cadences of the

composition at the ends of the A, B, and A″ sections (apart from the little cadence that rounds off the opening theme in measure three and the lesser peak in the Cadential curve near the end of the A′ section). The relative prominence of the three main peaks tends to support the interpretation that the A′ section is transitional. (Similar observations can be made about the occurrence of troughs in the Cadential curve.) In effect, furthermore, a switch is thrown at the end that changes the meaning of the three main cadences: there are two cadences at high levels of pitch, thereby marked as internal cadences; but a low level of pitch marks the last cadence as final.

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Fig. D13. Syrinx: Smoothed Pitch and the inverted averages of the rows in the smoothed intercorrelation table – low scores on the latter tending toward the typical and the developmental, high scores tending toward the atypical and the cadential.

One might expect the opening theme to be highly typical of the piece as a whole, especially since the theme provides a motivic basis for the rest of the ideas in the composition (motivic analysis is a subject for another study). From a textural standpoint, however, the first measures of the piece lie quite near the midpoint of the Typical / Atypical spectrum. In other words, from a textural standpoint, the opening theme should be thought of as central rather than typical. In the body of the piece, descending pitch is associated with increasingly developmental (non-cadential) textures, and vice versa. This pattern ceases at the end, where low pitch is associated with more atypical, cadential textures. In order to set aside the low register of the flute for the final cadence, Debussy has chosen to make the previous excursion into the low register (in the A′ and B sections) as developmental (noncadential) as possible. The indicator most clearly associated with the distinction between cadential and developmental material is the Pitch Class Dispersion. Three measure moving averages of the Weighted Average Pitch and the Pitch Class Dispersion are shown in Fig. D9b. Since these two indicators are negatively correlated, the two curves are usually on opposite sides of the zero line; but they are both above the zero line at the beginning, and both below the zero line at the end. Another way to look at the same relationship is to plot the Smoothed Pitch-Class Dispersion against the Smoothed Pitch in an XY chart (Fig. D14). Here, one can see by inspection that the timeline explores all four quadrants of the space. It begins in the upper-right quadrant,

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then explores the two adjacent quadrants, and in a large, sweeping gesture revisits all four quadrants until it comes to rest in the quadrant opposite the beginning.

Fig. D14. Syrinx: Smoothed PC Dispersion vs. Smoothed Pitch. The start is marked with an O; the end with an X.

Ex. D3. Syrinx: a. beginning, mm. 1-3; b. ending, mm 31-35.

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Fig. D15. Syrinx: The beginning versus the ending.

Except for the Duration-Rising, the smoothed indicators are all lower at the end than at the beginning (see Fig. D15).

The most typical segments, distinguished by texture as developmental As indicated earlier, if we identify the most typical segments as those having the highest average intercorrelations between measures, we find that these segments could best be described as developmental material, neither introductory nor cadential – low in pitch, tending toward falling motions, wide-ranging, having complex contours (neither linear nor static), active, and chromatically dispersed, but tending to use more recessive pitch-classes (see Ex. D4, Fig. D16a, Fig. D16b).

The most typical segments occur at mm. 3-5 (the second phrase of A); 16-18

(starting at the low point at the beginning of B); and 29-31 (starting at the time signature of 2/4 in the close of A″).

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Ex. D4. Syrinx:Tpical textures: a. mm. 3-5; b. mm 16-18; mm. 29-31.

Fig. D16a. Syrinx: Typical textures.

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Fig. D16b. Syrinx: Path from the beginning to the end through the most typical textures (omitting uncorrelated indicators).

The paths from the beginning to the end through the most typical textures show a number of crossings at the beginning, but more dramatic crossings at the end, wider and more consistent than those at the beginning (see Fig. D16b).

The Pitch, Duration-Rising, and Pitch-Class

Dominance – which are all lower than the Range, Central VHF, Rate of Attack, and Pitch-Class Dispersion in the most typical passages – switch to the opposite side at the end. I have omitted the Interval Continuity and Metric Hierarchy from the chart, because they are essentially uncorrelated with the Typicality.

The most atypical segments, distinguished by texture as cadential The two most atypical segments, having the lowest average intercorrelations between measures, are both associated with major internal cadences (see Ex. D5, Fig. D17a, Fig. D17b). They are found in measures 6-8 (the last three measures of A) and measures 23-25 (the last three measures of B). Except for the indicators that are not correlated with Typicality at all, and also excepting the Pitch-Class Dominance, which is not consistent in this context, the relative weights found for the indicators in the most atypical segments are the opposite of those found in the most typical segments. Pitch level in the major internal cadences is high and rising. These cadences use relatively small intervals and a selection of pitch-classes rather closely associated in the circle of fifths. They avoid fast notes and the middle range of scores on the VHF.

94

Ex. D5. Syrinx: Atypical textures: a. mm. 6-8; b. mm 23-25.

Fig. D17a. Syrinx: Atypical textures.

Fig. D17b. Syrinx: Path from the beginning to the end through the most atypical textures (omitting uncorrelated indicators).

Except for the Pitch and Duration-Rising, which describe arches, the paths from the beginning to the end through the most atypical textures all describe Open-Low-High-Close forms (see Fig. D17b). After the beginning, the Pitch, Duration-Rising, and Pitch-Class Dominance are always higher than the Range, Central VHF, Rate of Attack, and Pitch-Class Dispersion in this

95

selection of segments. The high-valued indicators do fall precipitously at the end, however. This reduces the divergence between the indicators.

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0 0

5

10

15

20

25

30

35

40

Divergence of PC Dispersion from Duration-Rising

Fig. D18. Syrinx: Divergence of normalized, smoothed PC Dispersion relative to normalized, smoothed Duration-Rising.

Since the normalized, smoothed Pitch-Class Dispersion is higher than the corresponding Duration-Rising in the most typical (developmental) passages, and vice versa for the most atypical (cadential) passages, we can get an estimate of the relative balance between developmental and cadential tendencies throughout the composition by subtracting the latter from the former (Fig. D18). The trend channel is ascending: it makes higher highs and higher lows. Low points mark cadences; high values are more developmental. Differences in function seem to account for much of the regional variation in the textures. There is a striking reciprocal relationship between the characteristics of the textures that are used in the major internal cadences (the atypical segments) and those that are used in the developmental material (the typical segments). Although Syrinx is written in an improvisatory, almost atonal style, which we think of in retrospect as being advanced, the concept of form is analogous to that of traditional tonality. The fact that the form is divided into three parts is confirmed by an alternation between extremes of typicality and atypicality, associated with development and closure, respectively. The division of the form based on closure does not exactly correspond to the division of the form based on thematic material and the pitch contour, however. The first point of high closure (low developmental) occurs at the end of the A section, not the A′ section. This fact

96

underlines the transitional nature of the A′ section. From a functional point of view, one could conceivably make an argument that the A′ section is part of the B section that follows. In any case, the transitional nature of the A′ section makes a striking contrast with the closing material. In terms of classical rhetoric, the A section and its reprise – though separated by intervening ideas – make a type of anaphora, that is, a pairing of statements that repeat their beginning clauses. The contrast of the endings makes a delayed antithesis – comparison (normally a juxtaposition) of contrasting ideas in parallel structures. Whereas the first statement of A leads to a transition, the reprise of A leads to a coda.

H

ere is the text that Ewell has matched to the last five measures of Syrinx:

O Pan, je n'ai plus peur de toi, je t'appartiens!

O Pan, I no longer fear you, I am yours.

Taken as a simple declarative statement, this text would appear to be unambiguously positive – except for the fact that the text speaks of fear and acquiescence together in the same sentence. It is reasonable to doubt that, under the circumstances, the Naiad’s state of mind would have been as straightforward as a literal reading of the text would imply. The complete mental image evoked by the text is psychologically ambivalent, as is the accompanying music. To understand how Syrinx ends, we have to engage Debussy as a colorist. Color in Syrinx depends on a complex interaction between subjective attributes of form, tonality, intervallic relationships, rhythm, and timbre.

The change of consciousness at the end of Syrinx is

attributable in part to a breaking up of the melodic continuity (see Ex. D3b, D5). As demonstrated by Mahlert (1986), the last three phrases (mm. 31-35) consist of transformations – each more expansive than the one before – of the cadential motive in measure two, which is developed throughout the piece, most extensively in section B. The last phrase, furthermore, contains internal contradictions – it bridges two notes that lie a dissonant, augmented sixth apart and fall into contrasting regions of the flute’s timbre. Also very important in this same passage, as described by Baron (1982), pitches initially heard in the main theme as unaccented embellishing tones are elevated to prominence, with the result that a previously subordinate form of the wholetone scale comes to the fore. The combined effect of these devices is a weakening of perceptual closure at the very moment when the larger form calls for a conclusive ending, contributing to a mysterious, seemingly revelatory sense that “what is constrained becomes released.”

97

Ex. D5. Syrinx: Contour transformations, according to Mahlert.

The long B-natural in the antepenultimate bar is the most meaningful climax in Syrinx. This is the crux of the deviation from expectation that is the major objective of the form. A recessive pitch, normally unaccented and subordinated to B-flat, the B-natural is not only approached by an isolated, volatile leap, but it is also sustained for more than three beats. This tone stands apart from its immediate surroundings in a brighter region of tonality and timbre, the non-tonal equivalent of a Picardy Third, raised a half step above the frequently repeated B-flats. The B-natural is loosely tethered by a non-diatonic, un-resolving whole-tone scale to the final, low D-flat. A significant clue to our understanding of the final tone is that this pitch is not only notated as a flat (rather than the equivalent C-sharp), but produced by a process of flatting the diatonic scale, and situated in the huskiest register of the flute.

98

The explicit and implicit symmetries of the form and developmental process invite us to hear the ending as convincing, and even inevitable. This is a paradox, however. The ending bridges the boundaries of a merismus, contrasting the bright region centered on B-natural with the dark region centered on D-flat. The B-natural and D-flat imply two spheres of influence and two loci of meaning. Without knowledge of the text, we are likely to interpret this meaning as abstract, referring to the heights and depths of human experience in general. The text suggests a more specific interpretation. One might speculate that the Naiad’s adoration of Pan – despite her overt declaration to the contrary – has not completely overcome her fear, dread or intimations of abandonment.

It is possible, therefore, to read a certain ambivalence into the music that

accompanies the last line of Mourey’s text. The ending of the composition juxtaposes bright and dark colors in close proximity, suggesting – by association with positive and negative affect – that the music can be interpreted as a nuanced, if not actually ironic, commentary on the text.

S

peaking about French music in general, but clearly talking about himself, Debussy wrote, “French music is marked by clarity, elegance and a simple and natural form of declamation;

French music desires, above all, to give pleasure . . . The French musical genius is a kind of combination of fantasy and sensibility” (Jarocinski, p. 103).

To understand Syrinx as an

accompaniment to a Symbolist play, however, Debussy’s general characterization must be put into historical context.

Debussy was working in a complex, highly politicized cultural milieu,

described in great detail by Charle (2001). A vivid capsule summary of the spirit of the times is found in Stéphane Mallarmé’s description of the sociological background of the Symbolist movement, which is reminiscent of Tocqueville and Durkheim: “In a society which lacks stability and unity it is impossible to create an art which is stable and well-defined. It is this incomplete social structure which not only creates an atmosphere of general intellectual unrest, but is also the cause of this [otherwise] inexplicable craving for individuality which is reflected directly in the literary manifestations of today” (Jarocinski, p. 31; my editorial amendation). This was a time when some composers associated with César Franck, Vincent d’Indy and the Schola Cantorum wrote what Hart (2001) calls “message-symphonies.” These composers “thought of nineteenthcentury symphonies in terms of a progression from states of darkness to light or doubt to faith, the meanings they associated with Beethoven’s Fifth and Franck’s symphony respectively.” Furthermore, they believed that symphonies could communicate frequently controversial ideological or moral convictions, with the aid of a printed exegesis, a text set in the work, or the quotation of familiar melodies. These symphonies were intentionally polemical works, invoking

99

thematic cycles and thematic transformation to depict the metamorphosis of the dark element into light. Because of the contrast of light and dark found at the end of Syrinx, it is possible to draw a certain parallel between the expressive purpose of Syrinx and that of the message-symphony, although it could be debated whether or not Syrinx expresses a simple, uncompromising ideological concept. Debussy wrote to Mourey on November 17, 1913, a few days before the performance of the play, that he had not yet found what was needed for the composition, and had made a number of false starts. In this letter, Debussy wrote about his intentions concerning the expression in the composition and the constraints under which he was working. The letter and its translation are cited from Curinga. 17

Novembre

Mon Jusqu’à

‘13

cher ce

jour

je

n’ai

pas

Mourey, encore

trouvé

ce

qu’il

faut…

Pour la raison, qu’un flute chantant sur l’horizon doit contenir tout de suite son emotion! Je veux dire, qu’on a pas le temps de s’y reprendre, à plusiere fois, et que: tout artifice devient grossier, la ligne du dessin mélodique ne pourrant compter sur aucune interruption de couleur, secourable. Dites moi, je vous prie, Très exactement, les vers après lesquels la musique intervient? Après de nombreux essasis je crois qu’il faut s’en tenir à la seule flûte de Pan, sans autre accompagnement. C’est plus difficile, mais plus (logique – crossed in the autograph) dans la nature. Affectuesement Claude Debussy Dear Mourey, so far I have not found what is needed… because a flute singing on the horizon must at once contain its emotion! That is, there is no time for repetitions, and exaggerated artificialness will coarsen the expression since the line or melodic pattern cannot rely on any interruption of color. Please tell me, very precisely, after what lines the music starts. After several attempts I think that one has to stick to the Pan flute alone without any accompaniment. This is more difficult but more (logical /crossed out in the autograph) in the nature”. Letter by Claude Debussy to Gabriel Mourey of November, 17th 1913. The original autograph is kept in the Frederick R. Koch Collection, Beinecke Rare Books and Manuscript Library, Yale University.

In other words, Syrinx was difficult to complete because the nature of the subject required a single, unaccompanied, monochromatic melodic line – without time for repetitions – which must contain all the emotion by itself, without resorting to gross artifice. I interpret this to mean that Debussy understood that his task was to balance intensity of expression with refinement and economy of means. The measure of Debussy’s success is that he was able to contain a variety of

100

textures and a high degree of chromaticism within the bounds of a thoroughly cantabile melodic style, a subtle tonal language, and a many-faceted rhetorical symmetry. What did Debussy mean by saying “exaggerated artificialness will coarsen the expression”? Overly literal tone-painting of Mourey’s text might have seemed problematic to the composer, especially in view of Debussy’s long-standing opposition to Wagnerism. Debussy might have been tempted toward a more expressionistic interpretation of Syrinx, however, if he had drawn his inspiration directly from Ovid, rather than Mourey. In stark terms, Ted Hughes (1997, pp. vii-x) compared Ovid’s Metamorphoses with Shakespeare “… in their common taste for a tortured subjectivity and catastrophic extremes of passion that border on the grotesque.” Hughes says that “The act of metamorphosis, which at some point touches each of the tales, operates as the symbolic guarantee that the passion has become mythic, has achieved the unendurable intensity that lifts the whole episode onto the supernatural or divine plane. … in every case, to a greater or lesser degree, Ovid locates and captures the peculiar frisson of that event, where the all-too-human victim stumbles out into the mythic arena and is transformed.” The transformation of catastrophic extremes of passion into the divine plane, described by Hughes, can be recognized as an intense expression of Hellenistic Dualism, a philosophy most familiar to us in Plato’s Allegory of the Cave, whereby those who are not trained in philosophy do not see the true forms of reality, but are like prisoners chained in a cave, having no view of the outside world, who see only illusory images of reality, shadows of the true forms – so to speak – cast upon the walls of the cave.

Dualistic sentiments were sometimes carved as funerary

inscriptions in the ancient Mediterranean world.

For example: “Weep not, for what use is

weeping? Rather venerate me, for I am now a divine star which shows itself at sunset” (see http://clas-pages.uncc.edu/james-tabor/hellenistic-roman-religion-philosophy/dualism/).

In the

original myth about Pan and Syrinx, the water spirit’s transformation into reeds, a medium for constructing musical pipes, is the story of a kind of death and rebirth, transporting the nymph from the tragic world of human experience into an ideal realm, the realm of melody. Pan, as depicted by Mourey, however, is not the outright rapist of Greek mythology. Mourey picked up where the original myth left off, after the water spirit’s transformation to the ideal was complete. For this reason, Mourey’s text does not evoke the same Aristotelian catharsis of pity and fear as the original myth. The mind of the Naiad in Mourey’s text is transformed, but by her own choice, not by magic. The transformation depicted by Mourey is secularized. It is not

101

for nothing that the Symbolists are sometimes characterized as decadent, because the pathological core of the original myth is presented by Mourey as normal. Mourey’s text requires a cooler, more detached, restrained, or sublimated affect than Ovid’s. This was provided by Debussy. Although Syrinx can be understood as pure music, it is not a completely abstract composition. Debussy’s handling of chromaticism in Syrinx is more indirect than Wagner’s – less strictly linear – but Debussy never fully escaped the influence of Tristan.

Technically speaking, the apparent sensuality of Syrinx is due to the fact that extreme

chromaticism, when heard in the context of a palpably vocal style, suggests a strong intensification of ordinary human feelings, the specific reference of which must be inferred from extra-musical cues.

The soft focus of Debussy’s tonality, furthermore, combines with the

impulsiveness of continuous motion by predominately small intervals to give an impression of heightened instability. (Not all atonality is alike: the priorities in Syrinx are quite different from those found in the wide-melody of Webern and his successors.) If Syrinx is more mysteriously disturbing than overtly tragic, that may be due, in part, to the subtlety of its tonal structure, which we have found to be a rhythmic rather than harmonic tonality, in which the chromatic scale is partitioned in such a way that some pitch-classes are more likely to be accentuated than others. As I have said before, there is no decisive Aristotelian catharsis of pity and fear in Mourey’s text; and the music provided by Debussy makes the ending even more ambivalent. The music, in my view, addresses an inner domain of unresolved psychological dysfunction, where the manipulative Pan is unable to win a clear-cut victory for the Naiad’s soul. As a matter of editorial policy, I generally avoid interpreting music in terms of self-expression by a composer, on the grounds that such speculations are usually subject to highly contradictory evidence. Nevertheless, the facts of Debussy’s life – the composer was a notorious womanizer – say nothing to contradict the interpretation just given. Syrinx – as much a product of bourgeois psychology as a reaction against it – domesticates the conflicts engendered by the remote gods of antiquity. The purpose of the rhetoric in Syrinx, unlike that of a message-symphony, is not to prove a straightforward ideological point, but to present a complex psychological situation in a convincing manner, as if to say, “This is the way things are.” Debussy’s setting of Mourey’s text depicts, not a role-model that we are bound to accept, but an insight into the human condition.

A

t first sight, one might think that there is a sharp dichotomy – amounting to a contradiction of purpose – between the simple, schematic, ternary form of Syrinx and its improvisatory,

nearly atonal style. As we have seen, however, analysis of the flow of the textures shows that the

102

developmental process follows an implicit symmetry in the long waves that parallels the symmetry of the ternary form. The opposition found in Syrinx between simplicity of form and complexity of content – a cognitive dissonance, in which extraordinary events are placed into the framework of an ordinary form – gives the work a mixed character of both clarity and edginess. The tension between form and content in Syrinx is a larger merismus, which can be interpreted as meaningful in itself. The combination of unresolved harmonic tension and simple form suggests the tensions of turn-of-thecentury Paris, a Pandora’s Box society riven by social instability. Pan, like the rest of the Greek gods, represents amoral forces of nature that exceed the powers of mere human beings to control. The ternary form and developmental symmetry found in Syrinx are rhetorical devices that impart an illusion of credibility to the limitless power of Pan, an anthropomorphic intermediary who invites the listener to identify with a transcendent level of reality.

103

104

Appendix A Five Groups of Five Simulated Arches

Group

Example

High Note Location

VHF Open

VHF Close

Close > Open

1

1

6

0.75

1.00

TRUE

2

8

0.63

1.00

TRUE

3

4

0.71

0.45

FALSE

4

3

1.00

0.50

FALSE

5

4

0.80

0.36

FALSE

Nbr Close > Open True

2

Close > Open

Group

Example

High Note Location

VHF Open

VHF Close

2

1

7

0.27

0.60

TRUE

2

5

1.00

0.45

FALSE

3

6

0.75

0.67

FALSE

4

7

0.50

1.00

TRUE

5

5

0.80

0.43

FALSE

Nbr Close > Open True

2

105

Group

Example

High Note Location

VHF Open

VHF Close

Close > Open

3

1

5

0.75

0.75

FALSE

2

2

1.00

0.36

FALSE

3

6

0.78

0.71

FALSE

4

4

1.00

0.78

FALSE

5

4

0.80

0.43

FALSE

Nbr Close > Open True

0

Group

Example

High Note Location

VHF Open

VHF Close

Close > Open

4

1

8

0.58

1.00

TRUE

2

6

0.88

1.00

TRUE

3

7

0.67

1.00

TRUE

4

6

0.71

0.57

FALSE

5

6

0.86

0.71

FALSE

Nbr Close > Open True

3

106

Group

Example

High Note Location

VHF Open

VHF Close

Close > Open

5

1

3

1.00

0.42

FALSE

2

3

1.00

0.64

FALSE

3

6

0.67

0.67

FALSE

4

5

0.71

0.56

FALSE

5

4

0.80

0.43

FALSE

Nbr Close > Open True

0

107

108

Appendix B Analysis of the Cantus Firmi by Jeppesen Jeppesen wrote nineteen cantus firmi for his counterpoint text: four each in the Dorian, Phrygian, Mixolydian, and Aeolian modes, and three in the Ionian mode. Because of the small amount of data, we cannot make broad generalizations about Jeppesen’s understanding of the modes. Factor analysis of the aggregate properties of these melodies, however, suggests that Jeppesen tended to treat the modes as differerent “dialects” of the cantus firmus type. The examples in the Dorian and Aeolian modes are rather similar. Surprisingly, the examples in the Phrygian and Ionian modes are also rather similar, except for the strong orientation toward the subdominant in the Phrygian mode. The Mixolydian mode is treated as a mixed type. Four statistical indicators were used to characterize the cantus firmi. The first is a measure of the relative vertical orientation of the melodies as arches, which will be called the Orientation, or Arch Strength. The number of scale steps between the lowest note and the tonic is subtracted from the number of scale steps between the tonic and the highest note. The difference between the two numbers is divided by the sum of the same two numbers. If there are no notes lower than the tonic, the difference and the sum will both be equal to the number of scale steps from the tonic up to the highest note; so, the indicator will be equal to 1.00. Such a melody will be considered to be strongly arch-shaped. Similarly, if there are no notes higher than the tonic, the indicator will be equal to -1.00. In this case, the melody will be considered to be strongly trough-shaped. Intermediate values greater than zero will be interpreted as representing melodies that are predominately arch-shaped; and, vice versa, intermediate values less than zero will be interpreted as representing melodies that are predominately trough-shaped. The second indicator is the Proportion of Rising Intervals, that is, the decimal fraction of all the intervals between successive notes that are rising. There are no repeated notes in these examples; so, any interval that is not rising is falling. For special purposes, we will reverse this indicator and call it Falling Intervals. The third indicator is the Relative High Note Time, or Late Peaks, the decimal fraction representing the location in time of the highest note. Each melody has only one highest note. Time is measured at the beginning (or attack) of each note. Each measure of the melody is counted as one unit of time. The first note is considered to occur at time zero. The indicator is calculated by dividing the time of the highest note by the time of the last note. Again, for special purposes we will also reverse this indicator and call it the Early Peaks.

109

The fourth indicator is the VHF Difference, or Smooth Close, based on changes in an indicator of melodic smoothness that was discussed in the body of the text. The VHF of a portion of a melody, it will be recalled, is the ratio between the range of the passage and its excursion, the latter being the sum of the absolute values of the successive melodic intervals occurring in the sample. For present purposes, all intervals are measured in scale steps. The VHF Difference is calculated by subtracting the VHF of the opening motion (that is, from the first note up to the highest note) from the VHF of the closing motion (from the highest note to the last note). If the VHF of the closing motion is greater than that of the opening motion, this will be interpreted as indicating that the closing motion is smoother than the opening motion, and vice versa. When this indicator is reversed, it will be called the Irregular Close. The data is summarized below: Orientation

Percent Rising

High Note Time

VHF Diff

D1

1.000

0.250

0.125

-0.500

Minor

D2

1.000

0.429

0.429

0.000

Dorian

Minor

D3

1.000

0.375

0.250

-0.444

Dorian

Minor

D4

1.000

0.286

0.429

0.333

Phrygian

Minor

P1

-0.143

0.250

0.250

-0.065

Phrygian

Minor

P2

0.500

0.625

0.625

-0.229

Phrygian

Minor

P3

-0.333

0.600

0.800

0.571

Phrygian

Minor

P4

-0.333

0.500

0.250

-0.321

Mixolydian

Major

M1

1.000

0.375

0.125

-0.500

Mixolydian

Major

M2

0.200

0.444

0.667

0.444

Mixolydian

Major

M3

1.000

0.250

0.375

0.444

Mixolydian

Major

M4

-0.600

0.571

0.857

0.545

Aeolian

Minor

A1

1.000

0.286

0.143

-0.286

Aeolian

Minor

A2

0.500

0.444

0.667

0.556

Aeolian

Minor

A3

1.000

0.333

0.167

-0.500

Aeolian

Minor

A4

1.000

0.222

0.111

-0.222

Ionian

Major

I1

0.143

0.636

0.091

-0.563

Ionian

Major

I2

1.000

0.600

0.600

0.000

Ionian

Major

I3

-0.200

0.625

0.750

0.375

Scale Type

Tonic Mode

Label

Dorian

Minor

Dorian

There is a moderately negative correlation (-0.572) between relative upward extension of the range and the percentage of rising intervals. The stronger the arch orientation, the fewer rising intervals, generally speaking. There is a moderately negative correlation (-0.523) between relative upward extension of the range and the location of the high note. The stronger the arch orientation, the earlier the

110

melodic peak, generally speaking. The arch shape seems to be associated with a strong initial upward impulse that dies out slowly. As should be expected for arch shapes, the percent rising is moderately correlated (0.614) with the location of the peak. Loosely speaking, the later the peak, the more rising intervals; but there are exceptions. There is a strong tendency (correlation equal to 0.838) for later peaks to be associated with smoother closes (closing VHF > opening VHF). This is consistent with theory. Exploratory factor analysis was performed on the correlation matrix using the Principal Component method, rotating the axes with the Varimax method. The software package employed was an Excel add-in, called statistiXL. There is only one strong factor (Eigenvalue equal to 2.61, accounting for 65% of the variance), which describes the fundamental attributes of the cantus firmi taken as a whole, regardless of mode. The strength of the upward extension of the range is positively loaded, and the other three variables are negatively loaded. The strongest loadings are negative weights given to the Relative First High Note Time and the VHF Difference.

A weak secondary factor

(Eigenvalue equal to 0.88, accounting for 22% of the variance) helps to distinguish the modes from one another. This has a strong positive loading on Vertical Arch Orientation and a strong negative loading on the Percent Rising.

Varimax Rotated Factor Loadings Variable Orientation

Factor 1

Factor 2

0.298

0.789

Percent Rising

-0.128

-0.921

First High Note Time

-0.834

-0.485

VHF Difference

-0.984

-0.106

A chart of the factor scores shows that the Dorian and Aeolian cantus firmi outline overlapping areas in the space defined by the two factors. There is also considerable overlapping between the areas outlined by the Phrygian and Ionian melodies. The Mixolydian melodies outline an area that overlaps all of the other modes. The first of the Ionian and Mixolydian melodies, all but the second of the Aeolian melodies, and all but the third of the Phrygian melodies score high on Factor 1, meaning that they have early peaks with relatively convoluted closing motions. The other melodies in these modes – the bulk of the Ionian and Mixolydian melodies, the second Aeolian melody, and the third Phrygian melody – have later peaks with relatively smoother closing motions.

111

All of the Dorian and Aeolian melodies and the first of the Phrygian melodies score high on Factor 2, meaning that they are strong arches with a high proportion of falling intervals. All but the first of the Phrygian melodies and all of the Ionian melodies score low on Factor 2 – meaning that they are not strong arches, and they tend to have a relatively high proportion of rising intervals. The factor scores were calculated by the software package using a refined method; but, if we are to understand the internal structure of the factors, we need to turn to the original data. To make each column of data comparable to the others, we first convert all of the values to standard scores, subtracting the mean of the column from the raw observations in that column and, then, dividing the difference by the standard deviation of the column.

The columns have been

regrouped and renamed, to give the values a more concrete interpretation that reflects the structure revealed by the factor analysis.

Each factor is represented by its two most heavily loaded

components. Since the components representing the second factor had loadings with opposite signs, the signs were reversed on the component having to do with the direction of the intervals in the melodies. Originally, this was a measure of the percent rising intervals. Now, it is a measure of the tendency of a melody to have falling intervals.

112

Component Standard Scores Factor 1 Factor 2 Label

Late Peak

Smooth Close

Arch Strength

Falling Intervals

D1

-1.068

-1.154

0.830

1.185

D2

0.087

0.045

0.830

-0.014

D3

-0.592

-1.021

0.830

0.345 0.945

D4

0.087

0.845

0.830

P1

-0.592

-0.110

-1.115

1.185

P2

0.834

-0.503

-0.021

-1.334

P3

1.500

1.416

-1.439

-1.166

P4

-0.592

-0.726

-1.439

-0.494

M1

-1.068

-1.154

0.830

0.345

M2

0.992

1.112

-0.531

-0.121

M3

-0.117

1.112

0.830

1.185

M4

1.717

1.354

-1.893

-0.974

A1

-1.000

-0.640

0.830

0.945

A2

0.992

1.378

-0.021

-0.121

A3

-0.909

-1.154

0.830

0.625

A4

-1.121

-0.488

0.830

1.372

I1

-1.197

-1.304

-0.629

-1.410

I2

0.739

0.045

0.830

-1.166

I3

1.309

0.945

-1.212

-1.334

Centroids were computed by taking the average values of the standard scores for each mode. The components belonging to Factor 1 were sorted by their Late Peak score. With the exception of the Phrygian mode, the modes with minor tonics have earlier peaks, on average, than the modes with major tonics. However, the range between the largest and smallest centroids is less than one standard deviation. This is much less than the range of values found in the scores for individual melodies, so it would not be possible to determine the mode of a cantus firmus from the location of the high note alone.

113

Factor 1 Component Standard Score Centroids Label

Late Peak

Smooth Close

Aeolian

-0.509

-0.226

Dorian

-0.372

-0.321

Ionian

0.284

-0.104

Phrygian

0.287

0.019

Mixolydian

0.381

0.606

Min

-0.509

-0.321

Max

0.381

0.606

Range

0.890

0.927

Here, we show a chart of the complete ranges covered by all of the examples of the modes on the two components loading heavily on Factor 1. For the sake of simplicity, each range is represented by a diagonal line running from the minimum values at lower left to the maximum values at the upper right, that is, from the point (Min(X), Min(Y)) to the point (Max(X), Max(Y)). The modes are plotted in order of their average value on the Late Peak score. The ranges overlap considerably on both scales, suggesting that Factor 1 represents a fundamental property of Jeppesen’s cantus firmi taken as a complete repertory, rather than the properties of individual modes. An unusual feature of the chart is that the values tend to cluster more closely together on the left than on the right.

114

A similar procedure is followed for the second factor. The ranges of the centroids are about twice as large. Except for the Phrygian mode, the modes with major tonics have lower arch strength than the modes with minor tonics.

Factor 1 Component Standard Score Centroids Label

Arch Strength

Falling Intervals

Phrygian

-1.003

-0.452

Ionian

-0.337

-1.303

Mixolydian

-0.191

0.109

Aeolian

0.617

0.705

Dorian

0.830

0.615

Min

-1.003

-1.303

Max

0.830

0.705

Range

1.833

2.009

The ranges of the standard scores for Factor 2, ordered by Arch Strength, appear in the next chart. Notice that the Ionian mode is off in an area by itself, and the Dorian and Aeolian modes are also off in an area by themselves. The Mixolydian range overlaps all the others. Except for its Arch Strength, which is limited at the high end, the Phrygian mode occupies a range similar to that of the Mixolydian mode. Factor 2 summarizes components that tend to distinguish the modes from one another.

115

The major-minor distinction to which we are accustomed from common-practice triadic tonality does not have a straightforward application to the modes, as we find them in Jeppesen’s practice. Jeppesen gave the modes somewhat different, though overlapping, characteristics. The Mixolydian mode does not stand out; but the Dorian and Aeolian melodies are quite distinctive, and the Ionian and Phrygian melodies also have characteristic traits. Perhaps, the Mixolydian mode, having a major mediant scale-degree on the one hand and a lowered seventh scale degree on the other, blends too much of the major and minor to have an individual identity. The Phrygian mode is a peculiar case. The lowered supertonic scale degree of this mode makes it impossible to construct a perfect triad on its dominant scale degree. Jeppesen’s Phrygian melodies reach downward to the subdominant below the tonic, which gives them an entirely different cast from the other melodies. Although the Phrygian is a minor mode, its melodies are forced to struggle upward from the subdominant back to the tonic in a manner that resembles the major Ionian mode. Traits that can clearly be assigned to our customary major and minor categories – distinguished by their third and seventh scale degrees in particular – are limited to the Ionian mode on the major side (raised third and seventh scale degrees), and the Dorian and Aeolian modes on the minor side (lowered third and seventh scale degrees).

116

Appendix C Stochastic Models for Limited Growth and Perpetual Cycles

A

ccording to the Limited Growth Model, the shape of a simple closed form can be explained as resulting from a combination of simultaneously unfolding tendencies: tendencies to rise,

fall, expand, and contract. The apparent parts of the shape can be understood, not as successions of unrelated entities, but as consequences of dynamic properties that are set at the beginning and remain relatively invariant throughout. An arch, for example, is not merely a rising motion followed by a closing motion. An Open-Low-High-Close form is not merely a falling motion followed by a rising motion followed by another rising motion, or, alternatively, a trough followed by an arch. The Limited Growth Model interprets an arch as what results when an expansive tendency to rise is countered by a contrary goal-directed tendency to contract back toward the starting point.

An Open-Low-High-Close form results when the goal-directed tendency to

contract is imposed on an opposition between simultaneously rising and falling tendencies to expand. The Limited Growth Model is abstracted from theories of melodic shape by Kurth, Meyer, Narmour, Gjerdingen, Adams, and Huron. In the stochastic form of the model, the expansive tendencies are represented by parameterized quasi-random walks. Cyclical motion, also generated stochastically, can be added to an Open-Low-High-Close form to create a more complex shape. In realistic stochastic models, state variables would be drawn from tables containing empirically observed values. For purposes of illustration, however, the demonstration models will use computed variables. Fixed parameters used by the Limited Growth generator are shown in Tables C1.1 and C1.2. All of the pre-set parameters are arbitrary. Two groups of parameters are needed, one for each of the components of the Limited Growth Model.

Parameter

Value

Range

Contraction Exponent

0.060

>0

Reversal Probability

0.500

0 to 1

Initial Direction

Up

Up or Down

Bias

1.700

>0

Initial Walk

0.000

Fixed

Table C1.1 Fixed parameters for the Limited Growth generator, Primary Curve.

117

Parameter

Value

Range

Contraction Exponent

0.200

>0

Reversal Probability

0.100

0 to 1

Initial Direction

Up

Up or Down

Bias

1.700

>0

Initial Walk

0.000

Fixed

Table C1.2 Fixed parameters for the Limited Growth generator, Contrary Curve.

The generator for the Limited Growth Model creates tables for both the Primary and Contrary curves, and each row of each table contains the following variables: 

Item Number



Time



Contraction Factor



Normal Variates



Absolute Value of the Normal Variate



Reversal Variable



Direction



Step



Walk

The calculations are performed as follows: 

The Contraction Factor is calculated by counting down the time backward from 1.000 to 0.000. Experiments were made using sequences of thirty-five points in time, a number chosen because this study supplements an analysis of Debussy’s composition for solo flute, Syrinx, which happens to be thirty-five measures long.

In these experiments, the tendency to contract imposed by a linear

countdown was too strong, so the countdown was raised to a fractional power, called the Contraction Exponent. 

The individual steps are simulated using a modified normal random variable. To approximate a normally distributed random variable, one takes the sum of twelve uniformly distributed random variables, ranging from zero to one. In the long run, the standard deviation of the sum will tend toward one.

After

subtracting six from the sum, the mean will tend toward zero.

118



We are going to skew the random variable that has just been generated; so, we take its absolute value. What we do to the absolute value depends on whether we want the current Step to ascend or descend.



We arbitrarily decide that the direction of the first Step will be up. The direction of succeeding Steps depends on another uniformly distributed random variable, the Reversal Variable. The interpretation of the Reversal Variable depends on a fixed parameter, called the Reversal Probability. If the Reversal Variable is less than the Reversal Probability, the polarity of the direction is changed to the opposite of its last state; otherwise, the direction stays the same.



Furthermore, we scale the size of the Step, referring to one more fixed parameter, called the Bias. If the direction is up, we multiply the absolute value of the normal variate by the Bias; otherwise, we multiply it by minus one and divide by the Bias. A Bias of one is neutral. Values of the Bias greater than one will tend to make the Walk ascend, and vice versa.



The first value of the curve (the Walk) is arbitrarily given a value of zero. Subsequent values are calculated by adding the most recently calculated Step to the last previously calculated value of the Walk.



This sum is multiplied by the Contraction Factor.

The last value of the

Contraction Factor will be zero; so, the last value of the Walk will also be zero, returning to its initial value.

The Resultant Curve is calculated by subtracting values of the Contrary Curve from corresponding values of the Primary Curve. More research is needed to evaluate the full range of effects of the parameters. All of the parameters are assumed to be positive. Values of the Bias greater than one tend to produce arches; lower values tend to produce troughs. The smaller the Contraction Exponent, the later the peak of an arch (or the bottom of a trough).

All other things being equal, the lower the Reversal

Probability, the larger the range. The accompanying illustrations (Fig. C1.1-C1.5) show the variety of shapes that can be generated using a fixed set of parameters. The parameters are chosen to generate late-peaking Scurves from combinations of late-peaking primary arches and early-peaking contrary arches. In every case, the Initial Direction is up; and the Bias Coefficient is 1.70. The primary curves have a small Contraction Exponent of 0.060 and a relatively large Reversal Probability of 0.500. The

119

contrary curves have a somewhat larger Contraction Exponent of 0.200 and a small Reversal Probability of 0.100. Each chart shows the first, second, and third quartile of the generated values, measure-by-measure, from a sample of twelve generated curves of each type.

Primary

Contrary

15.0

15.0

0.0

0.0

-15.0

-15.0

Q1

Q2

Q3

Q1

Q2

Q3

Resultant 15.0

10.0 5.0 0.0 -5.0 -10.0

-15.0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35

Q1

Q2

Q3

Fig. C1.1.

Primary

Contrary

15.0

15.0

0.0

0.0

-15.0

-15.0

Q1

Q2

Q3

Q1

Q2

Q3

Resultant 15.0

10.0 5.0 0.0 -5.0 -10.0

-15.0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35

Q1

Q2

Q3

Fig. C1.2.

120

Primary

Contrary

15.0

15.0

0.0

0.0

-15.0

-15.0

Q1

Q2

Q3

Q1

Q2

Q3

Resultant 15.0

10.0 5.0 0.0 -5.0 -10.0

-15.0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35

Q1

Q2

Q3

Fig. C1.3.

Primary

Contrary

15.0

15.0

0.0

0.0

-15.0

-15.0

Q1

Q2

Q3

Q1

Q2

Q3

Resultant 15.0

10.0 5.0 0.0 -5.0 -10.0

-15.0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35

Q1

Q2

Q3

Fig. C1.4.

121

Primary

Contrary

15.0

15.0

0.0

0.0

-15.0

-15.0

Q1

Q2

Q3

Q1

Q2

Q3

Resultant 15.0

10.0 5.0 0.0 -5.0 -10.0

-15.0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35

Q1

Q2

Q3

Fig. C1.5.

Stochastic Cycles The following is, loosely speaking, a stochastic interpretation of Simple Harmonic Motion. In this model, random motion is tailored to resemble the action of a body in motion under the constraint of a restoring force that tends to bring the body back toward a central value. The motion cycles about the central value indefinitely. The Perpetual Cycle generator uses its own collection of fixed parameters, shown in Table C2. Again, all of the pre-set parameters are arbitrary.

Parameter

Value

Range

Min Standard Deviation

0.000

0 to 1, < Max

Max Standard Deviation

1.000

0 to 1, > Min

Min Probability Reversal

0.000

0 to 1, < Max

Max Probability Reversal

1.000

0 to 1, > Min

Initial Direction

Up

Up or Down

Bias Base

16.000

>1

Initial Bias

1.000

>0

Amplitude

6.000

>0

Initial Walk

0.000

Fixed

Table C2. Fixed parameters for the Perpetual Cycle generator.

122

Like the Limited Growth Model, the stochastic cycle generator creates a table, but the table is somewhat more complex. Each row of the table contains the following variables: 

Item Number



Standard Deviation



Normal Variate



Absolute Value of the Normal Variate



Reversal Criterion



Reversal Variable



Direction



Bias Exponent



Bias



Step



Relative Amplitude



Interpolation Factor



Walk

We begin, as before, by generating random variables that approximate a skewed normal distribution.

As an approximation to the behavior of velocity in Simple Harmonic Motion,

however, we want to make the Walk move more quickly through the middle of a cycle than it does near the reversal points. To do this, we will control the Standard Deviation of the Normal Variate by obtaining feedback from the generated Walk. We will also control the probability that the Walk will reverse direction, in a similar manner. This refinement helps to maintain persistent motion through the middle of a cycle. 

The rows of the table are numbered consecutively. Each row represents a Step of the Walk. Each Step depends on all of the variables discussed below.



The Walk is initialized to zero.



The Relative Amplitude is the value of the Walk, divided by a pre-determined Amplitude. In the present examples, we will use an Amplitude of 6.00.



The Interpolation Factor is the absolute value of the Relative Amplitude, truncated to a maximum value of one.



The Standard Deviation is a linear interpolation between pre-set minimum and maximum values. To calculate the interpolation, we first take the product of the

123

Interpolation Factor and the difference between the minimum and maximum allowed values for the Standard Deviation. We then subtract that product from the maximum.

When the Walk is at zero, nothing is subtracted from the

maximum value of the Standard Deviation. The wider the deviation of the Walk from zero, the larger the Relative Amplitude (up to a maximum of one). The larger the Relative Amplitude, the smaller the Standard Deviation. When the Relative Amplitude is at a maximum of one, nothing is left of the Standard Deviation but the minimum allowed value.

The interpolated Standard

Deviation, therefore, will be high near the middle of a cycle and low near the extremes. In our examples, we will use a minimum Standard Deviation of zero and a maximum of one. 

Calculation of the Normal Variate begins with the second row, since it depends on a previously calculated value of the Standard Deviation. Values of the Normal Variate approximate a normal distribution. Each estimate is the sum of twelve uniformly distributed random variables in the range from zero to one. This sum will have a standard deviation of one. We subtract six from the total to center the range and multiply the difference by the Standard Deviation calculated for the previous row.



The Absolute Value of the Normal Variate will be needed later. We will divide the Normal Variate into two parts, which can be treated differently.

The

Absolute Value will serve for the calculation of either part. 

The Reversal Criterion governs the probability that the Walk will reverse direction. The Reversal Criterion is intended to be low near the middle of a cycle, but high near the extremes.

The Reversal Criterion is a linear

interpolation between a minimum and maximum value of the Reversal Probability, which we will set to zero and one, respectively. The calculation of the Reversal Criterion is similar to that of the Standard Deviation, except that they vary in opposite directions. The product of the Interpolation Factor times the difference between the maximum and minimum probabilities is added to the minimum probability. 

The Reversal Variable, as before in the Limited Growth Model, is a uniformly distributed random number between zero and one. This variable is first needed in the second row.

124



The Direction is calculated from the Reversal Variable and the previous value of the Direction, as before. A value of one signifies upward motion, and a value of minus one signifies downward motion. If the Reversal Variable is less than the Reversal Criterion, the Direction is changed to the opposite of its previous value, otherwise the Direction stays the same.

The restoring force of Simple Harmonic Motion is simulated by the Bias. This is one of the most important features of the model. It is used to skew the steps of the Walk, to bring the Walk back toward the zero line if it strays too far afield. We will calculate the Bias from a Bias Base and a Bias Exponent. In the present examples, we will use a Bias Base of 16.00. 

The Bias Exponent is simply the negative of the Relative Amplitude.



The Bias, after an initial value of 1.00, is equal to the Bias Base raised to the power of the previous Bias Exponent. The Bias is used to calculate the Step.



The Step is initialized to zero. From that point on, if the Direction is up, the Step is equal to the Absolute Value of the Normal Variate multiplied by the Bias. If the Direction is down, the Step is equal to the negative of the Absolute Value of the Normal Variate divided by the Bias. Both upward and downward motion can take place with any value of the Bias. If the Bias is equal to one, upward and downward motion will balance each other, on average. However, if the Bias is greater than one, upward motion will be augmented and downward motion will be diminished. If the Bias is less than one, the opposite will occur. Limits are imposed on the Step size. It is truncated to fall between plus or minus half the Amplitude.



The Walk is simply the sum of the Step and the previous value of the Walk, as it was in the Limited Growth Model.

The next set of examples (Fig. C2.1-C2.5) shows stochastic cycles added to curves resulting from the stochastic Limited Growth Model. The cycles are arbitrarily multiplied by three before they are added. The charts are stacked area graphs, in which cycles are built on top of Limited Growth curves.

125

Fig. C2.1

Fig. C2.2

Fig. C2.3

126

Fig. C2.4

Fig. C2.5

The results shown here are not necessarily typical. As often happens with stochastic processes, there is considerable variation between examples.

127

128

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139

Notes i

For a broader perspective on how we understand music, see Gjerdingen (1988), Zbikowski

(2002), and Huron (2006). For a thorough, sometimes provocative, analysis of the cognitive constraints on our comprehension of music, see Lerdahl (1988a). ii

See Mann (1943), p. 34, n. 1; p. 79, n. 7. According to Fux, melodies should be written for vocal

performance. Therefore, augmented, diminished, and chromatic intervals are to be avoided. The only intervals larger than a fifth to be used are the octave, and the ascending minor sixth. The range should rarely exceed the limits of the staff – that is, a ninth, assuming an appropriately chosen C clef for the voice type. As a consequence of the range limitation, Fux also has rules for the proper treatment of skips. Skips should not follow one another in the same direction – although we see occasional exceptions to this rule in Fux’s cantus firmi. One should compensate for skips subsequently – Fux’s cantus firmi occasionally leave gaps unfilled, but pitches gravitate toward a center, so skips do not result in a melody that begins in one register and ends in another. iii

The basic issue raised by what I am calling time series geometry is the behavior of a random

walk where there are reflecting barriers (a random walk is the trajectory of successive random steps). The influence of reflecting barriers on a random walk is a large subject discussed by Feller (1968), pp. 342ff. Also, see the third and fourth chapters of Berg (1993) on diffusion to capture, and diffusion with drift. In general, we are concerned here with the aggregate properties of random walks under geometric constraints. A shape such as a melodic arch can be thought of as the union of two wiggly lines, the details of which are hard to predict. We are interested in the aggregate properties of wiggly curves whose gross contours suggest lines, angles, and zig-zags. Since the smoothing effect of goal-directed motion on a random walk depends on the interaction of a number of different constraints, the most practical way to study this effect is through computer experiments, rather than formal mathematics. The purpose of these experiments is to discover the possibilities and probabilities for melodic motion in cases that are sufficiently similar to cantus firmi by Fux that there is a meaningful basis for comparison between the artificial melodies and the composed melodies. The intention is to learn what is really distinctive in Fux, and what emerges from the given logical properties of his musical materials. To coin the term time series geometry seems necessary because the most appealing alternative terminology has already been used for other subjects. The term stochastic geometry refers to the study of random spatial patterns. Antecedents for this theory can be traced to the field

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of mathematics known as geometric probability (which is concerned with such problems as Buffon’s needle – the probability that a needle dropped on lined paper will fall on one of the lines). The concerns of stochastic geometry can be distinguished from those of fuzzy geometry, which is the geometry of inexact measurements. We must distinguish time series geometry from stochastic geometry in general because time series are not directly commensurate with patterns in the two-dimensional Euclidean plane. The dimensions of a time series (in this instance, pitch and time) are not measured with comparable units. In time series analysis, we can speak of the rate of change of a variable with respect to time; and, it is permissible to compare ratios between values on the different scales, because ratios cancel out the units of measurement. It is not permissible, however, to make direct comparison between values measured in different units. For example, it would not be meaningful to compute the distance between two tones in pitch and time using the Pythagorean Theorem, which would involve taking the sum of the squares of distances measured in two different units. For that reason, the subject matter of the current study is not the same as what is normally designated by the term stochastic geometry. What the two concepts have in common is that they are both concerned with randomness and geometry. iv

Vos and Troost (1989); Huron (2006), pp. 75-77.

v

Huron (1996); Huron (2006), pp. 85-88.

vi

Generally speaking, the concepts of the climax and the dramatic arc can be traced back through

Gustav Freytag’s analysis of the five act drama to Aristotle’s tripartite theory of tragedy. In the context of Romantic music, the notion of musical form as a developing process can be interpreted as specifically Hegelian (see Schmalfeldt [2011], pp. 23-30, especially p. 269, n. 19). vii

Mann (1943), pp. vii-xiv.

viii

Jeppesen treats modes with major and minor tonics differently. 75% of Jeppesen’s cantus firmi

with minor tonics climax early, that is, in the first half. With a standard error of 12.5%, this proportion is statistically significant at the 95% level of confidence. The most common location of the melodic peak for minor tonics is in the first quarter of the melody. The distribution of peaks is more irregular for major tonics, but the most frequent area for the location of the climax is in the third quarter of the melody. In the ideal types, the affective distinction can be interpreted as follows: major tonic – bright, confident, and certain; minor tonic – dark, yielding, and uncertain. This distinction parallels what we find in Fux, although the manner in which the distinction is made differs.

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ix

Schenker (1910); Federhofer and Mann (1982); Snarrenberg (2005).

x

See Narmour (1977, ch. 11) and Gjerdingen (1988, ch. 1 and 3) for a discussion of archetype,

schema, style form, style structure, and idiostructure. These are complex subjects, but – roughly speaking – grammar is concerned with style forms (particular features, regarded as abstract forms, such as triads) and style structures (the regular combinations of features that distinguish a style, such as cadence formulas). Rhetoric, as I am using the term, is concerned with the other, more high-level concepts, particularly idiostructures. Idiostructures – again, roughly speaking – are large, relatively freeform gestures found in individual compositions, which are as much process as they are form. Style analysis, in the sense of that which is concerned with describing or defining a musical style, is primarily concerned with style forms and style structures.

This is to be

distinguished from critical analysis, which is more concerned with style structures and idiostructures. In this study, we are primarily doing critical analysis, although we are also doing style analysis in the sense that we are concerned with the differences between musical styles from a critical point of view. The critical analyst, in my view, needs special techniques to help see the forest beyond the trees. As a critical analyst, I share Zbikowski’s (2002, pp. 96-134, esp. pp. 12426, 132-34) view that a theory of music is made up of a number of conceptual models, changing in response to circumstances, which guide understanding and reasoning, provide answers to conceptual puzzles, and which simplify reality. xi

Schenker’s description of wave motion in this place uncharacteristically bears a certain

resemblance to Kurth’s conception of musical form. See Cook (2007), pp. 263-264. xii

The concept of the Ursatz is fully formed by the first two volumes of Schenker’s Das

Meisterwerk in der Music (1925-1926), but the concept is nearly complete by the fifth issue of Der Tonville (1923). See Pastille (1990), pp. 79, 81. xiii

In the current work in progress, a melodic typology adapted to the wave paradigm has been

used to describe the distribution of arch types in the cantus firmi. (The wave typology differs from – but does not contradict – Narmour’s well-known implication-realization typology [1990, 1991, 1992, 2000].

Narmour’s melodic typology is genuinely interesting, but it serves a

completely different purpose; and research on the subject is still ongoing.) In the wave typology, arches (and complementary troughs) are distinguished according to whether or not the first and last notes make a rising, level, or falling motion. They are further distinguished according to whether their top (or bottom, in the case of troughs), occurs at the mid-point, or early or late relative to the mid-point. The wave typology can be extended to include compound forms, such as

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the open-low-high-close form, or the open-high-low-close form. Applying information theory to the typology, we can evaluate the unity and variety of melodic ideas in the abstract – that is, without regard to the identification of specific motives in the traditional sense. An interesting and unexpected finding – the technical details of which are to be explained more fully in a later installment – is that there is an approximate power law relationship (which appears linear in a loglog chart) between the relative frequencies and the rank orders of the basic arch types in a composite data collection from all of Fux’s cantus firmi, segregating the opening motions from the closing motions. The exponents of the power laws for the opening and closing motions differ, reflecting the greater variety of arch types in the opening motions. This power law is comparable to Zipf’s Law (Zanette, 2006). A model of Zipf’s Law is proposed, based on the cognitive bias known as the distinction bias (Hsee and Zhang, 2004). It is hypothesized that if we always take the most preferred member (arbitrarily selecting one member where the utility is similar, even when the advantage is small, or non-existent) of a (uniformly distributed with replacement) random selection of alternatives, for a range of trial depths, the resulting distribution will approximate a power law. The exponent of the power law is related to the maximum trial depth, a larger maximum trial depth representing a more selective decision procedure. xiv

Cooke (1959, pp. 102-110) discusses examples of rising and falling melodic lines in both the

major and minor modes. xv

For a discussion of the difficulty of drawing theoretical conclusions from the overtone series,

see Lerdahl and Jackendoff (1983, 1996), pp. 290-294. xvi

In terms that are developed in the following section, the VHF of the original Dorian cantus

firmus was 0.400 for the opening motion, and 1.000 for the closing motion. The contrast in the alternative melody is theoretically nonexistent: the VHF is 0.667 in both cases. For this reason, the alternative is especially suitable to be put into retrograde motion. The VHF belongs under the umbrella of fractal geometry. I used a somewhat more complicated fractal indicator in my study of Haydn minuets (Chesnut [1996]). The ancestor of these indicators is Hurst’s rescaled-range analysis, which Hurst used to study flows of water in the River Nile. In a parallel study, Duane (2012) uses information theory to examine the nature of melody. Fractal geometry and information theory are complementary. Each has its advantages and disadvantage, depending on the context, and what one is attempting to accomplish. Fractal geometry is concerned with spatial relationships (presumably invoking the right brain) in which

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there is a measure of distance between entities (such as the intervals between pitches, or differences in loudness).

Information theory is concerned with language-like conceptual

categories (presumably invoking the left brain) in which there may not be a well-defined measure of distance between entities (it is difficult to make exact comparisons between themes or motives, the timbres of musical instruments, or the relative darkness and brightness of major and minor keys). Music consists of both spatial relationships and conceptual categories. xvii

The groundwork has not yet been laid for us to discuss narrative form in general, but a few

remarks are in order. The general plan of Fux’s cantus firmi could be called the argument and conclusion model, since it is analogous to the procedure by which one demonstrates the solution to a logical problem. As an intellectual method, the demonstration of logical conclusions traces back to Euclid and Aristotle, so there is no reason to think that pieces of music built on the argument and conclusion model are limited to the Age of Reason, as such. At an abstract level, the argument and conclusion model could be considered the basis of Aristotle’s concept of plotting in tragedy, so this is a ubiquitous plan. The smoothing effect is most pronounced when the climax of a melody occurs relatively late. It can be reduced by placing the climax earlier. The placement of the climax influences the degree of contrast between order and disorder in a melody. This has implications for narrative form, where order represents comprehensibility, stability, or fate, depending on the extra-musical context. A precedent for the general plan of Fux’s cantus firmi is found in the aria, “When I am laid in earth,” from Purcell’s Dido and Aeneas. The repetitive (theoretically unending) ground bass that accompanies the melody can be interpreted as tone-painting, representing inexorable fate. The organization of the melody as a whole is rather turbulent, containing a variety of different kinds of movement. The repeated notes at the words “Remember me” are themselves tonepainting. Like the ground bass, these repeated notes again suggest infinity, in this case the perpetuity of memory. The climax of the melody is placed very late, and the closing motion is very direct. Given the subject matter of the text, and the fact that the melody is in the minor mode, the organization of the melody can be interpreted as representing Dido’s prolonged struggle with fate, and the ultimately overwhelming victory of fate. Presumably, Purcell could have chosen to place the climax anywhere, but the choice that he actually made – a late climax which enhances the smoothing effect resulting from the constraints of the impending motion to the final tonic and, therefore, enhances the contrast between uncertainty and certainty – was the most dramatically

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effective choice that was appropriate to the text. Purcell used the smoothing effect as a rhetorical device in the service of dramaturgy. Although I have argued that the argument and conclusion model is rationalistic, the tonepainting that saturates Purcell’s opera is evidence for a more empirical point of view, since it is based on close observation of nature. On the whole, Purcell’s world view can be interpreted as being consistent with Baconian empiricism. For this reason, it is ironic that tone-painting is denigrated in the modern, scientific age, since Purcell could just as well be admired for the acuteness of his observations, just as we admire Galileo. We live in times when revolutions in mathematics and physics – such as non-Euclidean geometry, the theory of relativity and quantum mechanics – have led to radical reassessments of the meaning of “mere facts.” xviii

The project would have to address methodological issues, because Narmour (1977) and

Gjerdingen (1988) have argued forcefully that the hierarchic structure of tonal melodies is subject to multiple interpretations. xix

The possible paths between two given notes can be ordered by their lengths, but not uniquely,

because some paths will have equal lengths. The question of how many different ways a path of a given length can be partitioned into intervals of different absolute sizes is a classic problem in combinatorial number theory, which was most notably addressed by Leonhard Euler (1707-1783). xx

Highly sophisticated algorithms for analyzing and generating music – outside the scope of this

paper – are described by Loy (1989), Bidlack (1992), Cope (2001, 2005, 2009), Nierhaus (2009), and Xenakis (2001). There are several ways to influence the course of a random walk. For example, as an alternative to the algorithm described in the body of this text, arch-shaped melodies (as well as troughs) can be generated by a stochastic interpretation of a well-known difference equation, the growth-limited logistic equation.

The logistic equation has been useful in

algorithmic composition because it generates deterministic chaos when it is overdriven. See Nierhaus (2009), pp. 132-133, 146, and Bidlack (1992). The tendency of the standard deviation of a random walk to increase with the square root of time is explained in the first chapter of Berg [1993]. This result follows from the defining terms of a random walk: (1) that the sizes of the steps are fixed, (2) that the direction of the steps is not biased to one side or the other, and (3) that each step is completely independent from all the others. The rate of expansion could be other than the square root of time in a type of random walk called Fractional Brownian Motion. In order to generate a non-standard random walk, however, it is necessary to modify one or more of the defining terms. For example, instead of making a

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random choice between upward and downward motion, one might make a random choice between continuing in the same direction as the previous step, or reversing direction. If the choice is biased toward moving persistently in the same direction, the walk will tend to expand more rapidly than a standard random walk, and vice versa. xxi

See Feller (1968), pp. 76-78. In 1000 random walks based on the pool of intervals drawn from

Fux’s five arch-shaped cantus firmi (before normalization), about 7.5% return to the starting point after ten steps. In repeated trials, the range varies from approximately 5.5% to 9.5%, implying a standard error of about 1.0%. xxii

For an advanced treatment of relative statistical distributions, beyond the scope of this paper,

see Handcock and Morris (1999). xxiii

The frequency distribution of the intervals occurring in the opening motions of the generated

arches has negative kurtosis, meaning that it is flat topped, with short, fat tails. The corresponding frequency distribution for the closing motions has positive kurtosis, meaning that it is more peaked. The distribution for the closes is also more strongly skewed – due to the relative lack of ascending intervals in the close - and therefore has a long thin tail on the right. The result is that a plot of the cumulative relative frequencies of the intervals in the closing motions (on the Y-axis) against the cumulative relative frequencies of the opening motions (on the X-axis) is curved. Since repeated notes are not allowed, there are gaps in the frequency distributions at the zero level, which causes a bit of a hip in the resulting plot. xxiv

I cannot do justice here to either Schenker’s views, or the controversies they have raised. The

suggested reading list highlights a few of the key ideas. For readers who wish to make an in-depth study of these issues, there are excellent bibliographies in Cook (2007) and Pankhurst (2008). xxv

Network analysis was essential for Gjerdingen because he needed to be able to do close

analysis of the role of a particular schema – based on the progression of the scale steps, 1-7-4-3 – in a great variety of musical contexts. This is just one of a number of schema that Gjerdingen (2007) found to be characteristic of the galant style. Gjerdingen has shown by an exhaustive statistical analysis that the 1-7-4-3 schema developed gradually in the early eighteenth century, peaked in the 1770s, and then dissipated. In other words, the prevalence of the schema follows an arch. xxvi

For more on tonal tension, see Lerdahl (1996), Smith and Cuddy (2003), and Cope (2001,

2005, and 2009). Riemann (1882, [1915] 1992), George (1970), Steblin (2002), and Brower (2008) are concerned with the sharp-flat effect, which is complementary to Lerdahl’s concept of

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tonal distance. The two views of tonal pitch space can be reconciled if we recognize that there are at least two dimensions of affect, where valence (brightness) is independent of arousal (see for example, Russell (1979). For a broader overview, see Tymoczko’s (2011) geometry of tonality. The type of analysis done in the present study requires a measure of distance between musical entities; or, at least, that we be able to compare them in a rank order. As Rings (2011, p. 13) points out, not all transformational theories provide such measures. xxvii

A tree structure compares the stability, tension, or hierarchic dominance of pairs of elements.

The relationship between the two elements is diagrammed as an inverted Y, in which the dominant member of the pair is given the longer branch. Rising tension is shown by a right-branching tree, in which the tail of the Y extends to the right. Falling tension is shown in the opposite manner. xxviii

For an introduction to the subject, see Cope (2001), pp. 129-137; Cope (2005), pp. 224-243,

and Cope (2009), pp. 205-215. SPEAC is an acronym for statement, preparation, extension, antecedent, and consequent.

SPEAC analysis allows notes and chords to vary in meaning

according to their context. SPEAC identifiers follow an A-P-E-S-C kinetic order with the most unstable function to the left, and the most stable function to the right. A and P require resolution, while E, S, and C do not. xxix

For more about the tonal hierarchy, see Krumhansl and Kessler (1982) on key profiles, Lerdahl

on basic pitch space (1988b, 2001), and Thomson on the tonal frame (1999).

A detailed

discussion of the manner in which the tonal hierarchy is realized in the cantus firmi is a subject for another occasion. To summarize, evidence that a melody has a tonal framework is that some of the scale steps are privileged, in the sense that they occur more often, or they are more likely to be approached or left by leap, or they are more frequently allowed to be embellished by neighboring tones. Where the roles just described are reversed, upending these privileges, the effect tends to be expressive.

The phonological theory of markedness, which is discussed by Lerdahl and

Jackendoff (pp. 296, 313-314), has a role in explaining the reversal phenomenon. To model the motions of a melody in the tonal hierarchy would get into the difficult subject of algorithmic composition, which will not be addressed here. The motion of one free-floating note in the fields of three fixed attractors is probably best understood as chaotic; but more than that, in free composition, there is a short-term memory effect such that a note tends to harmonize with recent occurrences of itself, and those harmonizations are subject to rhythmic and syntactical constraints.

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To that has to be added tendencies toward embellishment, and motivic redundancy, not to mention the constraints of phrasing and larger form. xxx

Complexity as such can be modeled by mathematical tools drawn from information theory and

fractal geometry. The VHF, as we shall see, is a simplified version of an indicator of fractal complexity. Information theory and fractal geometry provide fundamentally different conceptions of complexity, both of which I have found lead to interesting conclusions when applied to Fux’s cantus firmi. xxxi

What I am describing here bears a certain resemblance to what Mandelbrot (1999) called a

multifractal. The degree of complexity changes with time, just as the complexity of the world’s coastlines (Mandelbrot [1967]) changes from region to region. The revolutionary impact of Mandelbrot’s world-view does have a profound impact on my work, but a formal discussion of the subject must be postponed. xxxii

The dualism issue can also be raised in the interpretation of tree structures. The tree structures

of tonal music as analyzed by Lerdahl and Jackendoff are conspicuously arch-shaped in their general outlines. These arch shapes require an aesthetically plausible explanation. A rule is needed for collapsing the trees into intensity curves (showing the flow of hierarchic depth) that are analogous in shape to a cantus firmus. See the discussion in Lerdahl and Jackendoff on p. 116 of the trade-off between the verticalization of discrete events in a hierarchy versus voice-leading. Something like the column graphs – though inverted – that Lerdahl and Jackendoff use for the analysis of metric hierarchy (Chapter 2) could be the basis for a non-hierarchical analysis of tonal process. In other words, hierarchy and process can be understood as equivalent, inter-penetrating structures. An explanatory model for the process in question is currently under investigation by this author. One might be tempted to model an arch as a segment of a sine wave plus an arbitrary number of harmonics, but a sine wave is theoretically infinite in duration. We need to model a finite closed form, with a definite beginning, middle, and end. A variation of a well-known equation, the logistic equation that is used by biologists to model the growth of a population of bacteria in a medium (using a power law xa instead of an exponential variable ax), can be used to model an arch as the confluence of two opposed non-linear monotonic processes (a driver and a damper).

The psychological motivation behind the two monotonic processes – based on

Csikszentmihalyi‘s (1990) theory of optimal experience – can be summarized under the rubric of exploratory behavior, that is, the tendency to simultaneously pursue a search for new information

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together with the tendency to achieve cognitive integration of that information over time. In other words, the basic arch of tension in tonal structure is consistent with an aesthetic that is rationalistic in its broad outlines. The model of exploratory behavior fills out a missing element of Meyer’s theory of deviation from expectation. Meyer tended to emphasize the return to a norm after a deviation, without clearly identifying the information-seeking force that tends toward divergence from a norm. Other models would be necessary to describe music based on different aesthetic principles.

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