MARKAND THAKAR - 'Counterpoint - Fundamentals Of Music Making'

COUNTERPOINT

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COUNTERPOINT ** Fundamentals of Music Making MARKAND THAKAR

Tale University Press New Haven and London

Copyright © 1990 by Yale University. All rights reserved. This book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers. Designed by Nancy Ovedovitz. Set in Galliard type by Keystone Typesetting. Printed in the United States of America by ThomsonShore, Dexter, Michigan. Library of Congress Cataloging-in-Publication Data Thakar, Markand, 1955Counterpoint : fundamentals of music making / Markand Thakar. p. cm. ISBN 0-300-04628-6 (alk. paper).-ISBN 0-300-04638-3 (pbk. : alk. paper) 1. Counterpoint. I. Title. MT55.T44 1990 781.2'86-dc20 89-70719

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The paper in this book meets the guidelines of permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources. 1

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To Sergiu Celibidache, who showed me the music in me and how to find it

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CONTENTS ^>

Acknowledgments, xi Introduction, xiii ONE I PRIMORDIAL LINE Elements of Line, 1 Stepwise Motion, Discrete Participation, Unitary Apprehension, Attributes of a, Tone, Stepwise Foundation, Addendum: The Minor Mode Cantus Firmus, 31 TWO

I TWO-PART

COUNTERPOINT

Conjunct Lines, 43 Harmonic Intervals, Addendum: Balance, Conjunction, Primordial Lines, Independence, Unitary Apprehension First Species, 62

CO N T E NT S

THREE

I

DISSONANCE

Elements of Dissonance, 67 Passing Tones, Neighbor Tones, Successive Dissonances, Cambiatas, Dissonant Suspensions

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First Species, 125 Second Species, 126 Third Species, 128

Second Species, 83

Fourth Species, 128

Third Species, 91

Fifth Species, 132

Fourth Species, 98

SIX I MIXED SPECIES

FOUR I RHYTHMIC VARIETY

Conjunct Lines in Mixed Species, 133 Consonant Agents, Multileveled Agency, Implied Harmony, Compound Melody

Characteristics of Rhythmic Variety, 105 Longer and Shorter Note Values, Diverse Note Values, Successive Short Note Values Fifth Species, 111 FIVE

I

THREE-PART

COUNTERPOINT Three Conjunct Lines, 115 Triads, Conjunction, Primordial Lines, Independence, Unitary Apprehension

Type 1, 144 Type 2, 148 Type 3, 150 Type 4, 152 SEVEN I CONSEQUENCES AND APPLICATIONS Consequences of Agency, 154

CONTENTS

ix Performance, 157 Musical Passages, 160 Preliminary Considerations, Passage A, Passage B, Functional Bass, Whole Works

Appendix / Write-Throughs, 175 Index, 305

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ACKNOWLEDGMENTS

I owe much of the content of this book to my teachers, Sergiu Celibidache and Jacques-Louis Monod. I am like a child on a bicycle: the ride is mine, but it never would have happened without a running push from each of these extraordinary men. And like a child on a bicycle, I often find it difficult to know where my own contribution begins and theirs ends. May this book be worthy of their consideration, and may it justify their efforts. My thanks go to Katie Powell and Tamar Barzel, who each worked painstakingly through the book at various stages of its evolution and provided enlightening comments, suggestions, and complaints. Thanks also go to Holly Panich of Ohio University Press for her guidance throughout the preparation of the manuscript, and to Nancy Basmajian for her meticulous proofreading. I am grateful to the Ohio University College of Fine Arts for its generous assistance with the production of the musical examples, which were created with Robert Fruehwald's ExampleKrafter program for the Macintosh computer. And I am especially grateful to my old friend Fred Kameny, who as editor has transformed much of my writing into comprehensible English. Finally, the book owes its existence to two friends,

xii Victoria Chiang and Paul H. Smith. Paul has been extremely generous with his time and attention, offering invaluable insights in every aspect of the project. It was my successful experience working on counterpoint with Paul, who understood it, that made a book thinkable.

ACKNOWLEDGMENTS And to mention Vicky's contributions is to trivialize them; they range from major considerations of substance and organization to points of grammar and logic to badly needed moral support. This book is unimaginable in this or any form without her assistance.

INTRODUCTION

Counterpoint is the conjunction of lines; the study of counterpoint is the study of line and of lines joining. The shelves of any good music library hold dozens of counterpoint manuals. Why yet another? Moreover, why was a book on counterpoint written by a conductor? Manuals of species counterpoint have been produced for two hundred fifty years, some to benefit primarily composers, others music historians, and still others theorists. Broadly stated, the aim of these manuals has been either to develop a facility with relations among tones, or to develop familiarity with a musical style. As a conductor, my function is to guide performers in bringing about the highest and most sublime experience of musical beauty. Thus I have come to face the inescapable question of why one performance is better than another-how the tones must be brought into sound so that they lead to the most valuable experience. The aim of this counterpoint manual is to aid performing musicians in their pursuit of this highest experience. This book is therefore not simply a counterpoint manual slanted toward performers; it is rather a primer for performing musicians in the shape of a manual of

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species counterpoint. It stands apart from other counterpoint manuals in two important ways. First, it recognizes an essential difference between a tone sounded and a tone written (in other words, that the same tone sounded in different ways yields essentially different experiences) . Second, whereas other manuals describe general characteristics of musical elements, this one attempts to pinpoint the essence of musical elements and treat them in their pure states. The far-reaching consequences of these two differences are discussed below. A Brief History of Species Counterpoint The discipline of species counterpoint was first put forth by Johann Joseph Fux in Gradus ad Parnassum (1725) as a tool for teaching composers. Fux found that his students made sure progress in acquiring facility in musical composition by completing a series of graded exercises, or species. Based on a set of rules derived from the music of Palestrina, the exercises began with the most elementary material and successively introduced greater complexities. Thus the student of composition was given the same advantage enjoyed by students of other disciplines: the ability to master simple elements before progressing to more complicated ones. Largely because

INTRODUCTION of this sound pedagogical principle, Fux's work became immensely popular-it served as the foundation for the teaching of composition for perhaps two centuries. The study of species counterpoint has since been adapted in many ways, for many purposes. Beethoven's teacher Albrechtsberger and others found that species counterpoint was at least as effective a tool for teaching composition if the rules were adapted from the modal system of Palestrina to the major and minor modes of the tonal system. Later scholars (Bellermann, Jeppesen) used the study of counterpoint as a tool for writing pieces in the style of Palestrina. In other words, where Fux offered graded exercises that led to facility with relations among tones, they presented graded exercises that led to pieces in the style of Palestrina. Others (Kennan; Davis, and Lybbert) likewise presented graded exercises in the major and minor modes that led to pieces in the style of J. S. Bach. These manuals of "sixteenthcentury counterpoint" and "eighteenth-century counterpoint" have been valuable tools, especially for the music historian seeking an intimate knowledge of musical styles. Still other scholars (Schenker; Salzer, and Schachter) have used the study of counterpoint not only to develop

INTRODUCTION facility with relations among tones (voice leading), but also to facilitate analysis. They have demonstrated clearly that masterworks of Western art music, stripped of ornamentation in a process that is something of the reverse of species counterpoint, are based on consecutive levels of contrapuntal structure. These manuals have proved valuable to the music theorist seeking to acquire analytical skills. Existing counterpoint manuals have been central in the musical education of generations of composers, historians, and theorists. They have served performing musicians less well, however, because none takes into account the highest experience of musical beauty-the extraordinary phenomenon that draws us to music in the first place. The Highest Experience of Musical Beauty Ordinary musical experiences are gratifying largely because they organize an aspect of human existence that generally comes to us as chaos: time. These ordinary musical experiences may also evoke pleasing memories, suggest touching characters (hope, sadness, grandeur), or set into motion a complex network of thoughts. There is also, however, an extraordinary experience

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available through music. The greatest Western art music, well performed, can lead to no less than a spiritual experience, an experience that transcends the physical parameters of time and space. It is the feeling of becoming lost-of losing oneself in the experience. It is losing the self and becoming the sounds. It is the rare and magical experience of the entire work filling a single moment. This is the ultimate, highest, experience of musical beauty. Strange and mystical? Not at all. All musicians have known this experience, either in listening or in playing. Moreover, the great majority of serious musicians take on the monumental task of becoming proficient in this difficult, insecure, ill-respected, and poorly rewarded profession precisely to pursue this seductive, sublime experience. For many musicians the pursuit of this experience ceases to be the primary motivation in doing music. Its importance often fades when music becomes a means of earning a living (it is the absence of the magical experience of music that frustrates and embitters so many professional musicians in even the finest ensembles). Others lose it as they become educated-as they feel the need to think as they listen. We cannot have the magical expe-

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rience of music if we listen for things: for historical significance, or for technical considerations of the performance, or for structural aspects of the piece, or in fact for anything. The magical, transcendent experience of music can result only from pure and open listeningfrom an open consciousness focused on the totality of the sounds. It cannot result from a listening adulterated by any adding of outside considerations to the consciousness of the sounds (historical significance, for example), nor can it result from focusing on less than the totality of the sounds (technical considerations of the performance or theoretical aspects of the piece, for example). The sublime experience of music requires more than the listener's complete and exclusive absorption with the sounds. The experience of music has three components. For the sublime experience to take place all three must allow it: the attitude of the listener, the object of the listener's attention (the sounds, the performance), and the piece itself. What Can Be Gained from This Book This book is grounded in a recognition of the essential • difference between performance and piece, and is committed to the pursuit of essence. Recognizing the differ-

INTRODUCTION

ence between the performance and the piece is critical to the creation of both. If the extraordinary experience of musical beauty depends on the performance, which in turn depends on the piece, then it is possible diat a piece permits it while a performance of the piece does not. The significance of this cannot be underestimated. For if a performance that allows the most magical experience is more valuable than one that does not allow it, it should be possible to grasp the essential conditions of a good performance. And these conditions can be grasped: a performance that can lead to a transcendent experience of musical beauty is experienced as indivisible, as unitary, as whole. Some of the specific conditions that allow a performance to be experienced as an indivisible whole are described in this book. The possibility that a piece may not allow the transcendent experience is also significant, for if a piece that allows it is more valuable than one that does not, it should be possible to grasp the essential conditions of a good piece. These conditions also can be grasped: the piece must permit a performance that can be experienced as indivisible. Some of the specific conditions under which a piece can lead to an indivisible experience of its performance are also laid forth in this book. The book yields something of value for analysis as

INTRODUCTION well. My primary purpose-to give the reader a fundamental understanding of the conditions of good music making-requires first an understanding of the essence of musical elements. For example, a conscious understanding of what is required to perform a line well requires a conscious understanding of the essence of line. And the analyst will find that such a grasp of the essence of line yields a systematic foundation for structural linear (Schenkerian) analysis. The listener who is sensitive to what constitutes a line will necessarily hear which tones do not participate in the line. Thus a grasp of the essence of line leads to a sure grasp of levels of linear activity. A Tour of the Book Music cannot be learned from a book, any more than swimming can. A book on swimming can at best preview the experience, perhaps isolate and describe the physical and intellectual components of the activity, suggest exercises that will strengthen the requisite abilities, and finally offer an idea of how to proceed once the reader is in the water. Likewise, a book on music making can at best preview the experience, isolate and describe essential conditions of the experience, suggest exercises that will sensitize the reader to the presence or

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absence of these conditions, and offer an idea of how to proceed once the reader is involved with sounds in a musical context. In terms of previewing the experience, this book has an enormous advantage. The reader of the didactic book on swimming has presumably never had the experience of swimming, and reads the book so as to gain the experience for the first time. The reader of this book, on the other hand, is likely to have a serious interest in music, presumably from having already had the indescribable experience of musical beauty, and is probably looking to repeat die experience. To preview the experience of swimming to someone who is not a swimmer may be impossible. But to preview die experience of the most sublime musical beauty one need only remind the reader of that magnificent, otherworldly, spine-tingling sensation that results when sounds come together in a single, magical moment: a single progression, or perhaps a phrase or two, or-on the rarest of occasions-an entire movement. That is the highest, most valuable experience that can be gained from music, and that is the experience we are seeking. The second task, considerably more complex, is to isolate die essential conditions of die experience and to offer exercises that will allow die reader to develop sen-

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sitivity to these conditions. Most of the book is given over to this second task. The highest, most valuable experience of music begins with the relations among experienced tones. Tones have the attributes of pitch, duration, volume, and timbre. Tones participate in lines; thus they have the attribute of linear function. Tones have a specific position within the mode; thus they have the attribute of scale degree. Tones also participate in harmonic events; thus they have the attribute of harmonic function. And tones participate in a gathering of energy (impulse] or dissipation of energy (resolution), thus they have the additional attribute of dynamic function. To study the essence of the relations among tones, we eliminate from consideration their nonessential attributes. Timbre, for instance, is essential to our experience of music, much as the color of the paint is essential to our experience of a car. But because timbre does not significantly affect the relations among the tones, we eliminate it from our study, much as an automotive engineer eliminates paint color from his study of the workings of the car. Likewise we eliminate articulation, a nonessential attribute of the relations of tones (the relation between two quarter notes is essentially the same regardless of the articulation of either). The subject of

INTRODUCTION

volume is taken up later as an aspect of performance; rhythmic considerations are postponed as well. We neutralize the effect of rhythm through the use of unchanging note values and an unchanging rate of harmonic activity. Finally we eliminate considerations of harmonic function by using unaccompanied lines. Thus the material of chapter 1 is essential indeed: unaccompanied tones in succession, differentiated by the attributes of pitch, linear function, scale degree, and dynamic function. Such a succession that can be apprehended as a unit is called a primordial line (see p. 11 for a detailed discussion of unitary apprehension). Through the study of primordial lines we will become sensitive to the nature of line and to the nature of the dynamic forces (the energy-gathering impulse and the energy-releasing resolution),1 which are critical to musical performance. Chapter 2 introduces conjunction. To a fixed primordial line, or cantus firmus, is added an accompanying primordial line, called the co-line. The conjunction of cantus and co-line imbues the tones of each with the attribute of harmonic function (first species). The cantus 1. The term "dynamic forces" is not to be confused with the common term "dynamics," which refers to inflections of volume.

INTRODUCTION

remains fixed, but with each successive species the coline becomes more complex. Dissonances are introduced in chapter 3, along with the related phenomenon of agency, in which dissonant tones are agents, or representatives, of consonant tones. We add stepwise dissonances (second species), leaping and successive dissonances (third species), and dissonant suspensions (fourth species). Chapter 4 deals with the effect of rhythmic variety on the dynamic forces of impulse and resolution (fifth species). Chapter 5 considers the conjunction of three lines. We add a second co-line to form three-part counterpoint, and begin again with exercises in uniform note values (first species). Again the complexities of one of the co-lines are increased in successive exercises, as dissonances and rhythmic variety are added (second to fifth species). Chapter 6 includes the final series of exercises, in which the fixed cantus is accompanied by two co-lines, each with differing levels of complexity (mixed species). We hear the phenomena of consonant agency, in which consonant tones represent other tones (type 1); of multileveled agency, in which agents are themselves represented by agents (type 2); of implied harmony, in which a functional consonant harmony is not explicitly

xix

sounded (type 3); and of compound melody, in which two complementary lines in varied rhythms are unfolded within a single-line texture (type 4). The performance of musical passages is examined in chapter 7. Each of two familiar passages is heard to represent a single harmonic event, prolonged through layers of ornamentation. We consider the structure of impulse and resolution required to perform each layer musically, and examine the effect of each successive layer of ornamentation on the requisite dynamic structure. This ultimately leads us to an increased sensitivity to the dynamic structure required for a musical performance of the actual musical surface. Finally we consider how such a sensitivity can be of value in the preparation of a performance. How to Use This Book Properly used, this book may leave the reader with an increased sensitivity to the most sublime experience of musical beauty, and with an increased understanding of three essential components of the musical experience: line, conjunction, and wholeness. The book attempts not to teach rules, but instead to allow the reader to experience musical principles consciously. For example, to

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promote an understanding of the essence of line, the exercises prohibit-without prejudice-all configurations of tones that do not result in a line in its essential form. Dissonant contour is an example of one such configuration.2 There is nothing inherently bad about dissonant contour (or inherently good, for that matter). Our task is simply to become sensitive to its effect. Toward that end we define it, we listen to it, we experience its effect, and we prohibit it from our exercises. This will no doubt prove confusing for the reader who is looking for rules. Such a reader might say, "But Bach uses dissonant contour all the time. It can't be wrong to use it in my exercises, especially if I'm going to be able to carry something of value from these exercises to my experience of Bach's music." We are not composing music, however, but exercises. We purposely treat these elementary exercises in complete isolation from the repertoire, so as to focus more easily on the precise points they have to teach. When we clearly hear and understand the nature of dissonant con2. A dissonant contour is a configuration in which the highest and lowest tones of a single ascending or descending motion describe a dissonant interval. See p. 9 for a discussion of dissonant contour.

INTRODUCTION

tour, for example, it will be easy to avoid in our exercises. And only then will we be sensitive to the function of dissonant contour in Bach's music or any music. The considerations of line, conjunction, and wholeness are but components of diat extraordinary but achievable experience of musical beauty, which must be paramount throughout the execution of the exercises. Although they are exercises, they must be composed with an ear toward beauty. If an exercise cannot be performed so that it is beautiful, then it will most likely have some identifiable problem. It is true that in some of the exercises the possibility of beauty is limited by the pedagogical demands at hand. For example, an experience of ultimate beauty may only rarely result from a three-part exercise in third species (four notes in the coline with each note of the cantus). But the possibility of experiencing that highest, transcendent musical beauty must still be pursued, both in the composition of the exercises and in their execution. Proper use of the book entails following these procedures: 1. Compose the exercise in your mind, away from the piano. 2. Sing one line while playing the other line or lines.

INTRODUCTION 3. Sing absolutely in tune, with precise rhythm. 4. Perform the exercises without looking at the music. 5. Listen to each exercise carefully. 1. Compose the exercise in your mind, away from the piano. You must hear the exercises you compose, and the ability to conjure sounds without actively perceiving them is critical for the serious musician. This may require going slowly at first, checking each tone or interval with the piano after composing it. The long-term benefits of this procedure are enormous, and without it the exercises in this book are of limited value. Composing the exercises away from the piano without hearing them in your mind, however, renders them utterly valueless. Notate your exercises only after having heard them.3 2. Sing one line of each exercise while playing the other line or lines. Nothing teaches as firmly or as quickly as active doing. The passenger never knows a new route as well as the driver; the best way to learn a language is to speak it. Likewise, the most efficient way for you to 3. To make the book more accessible I have notated the examples in treble and bass clefs exclusively. I strongly recommend that conductors, composers, music theorists, and music historians, who need to develop fluency with the other five clefs, notate their exercises in the clef most appropriate to the register of the line.

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learn from diese exercises is to do them actively-to create the tones yourself-by singing them. 3. Sing absolutely in tune, with precise rhythm. Singing out of tune defeats the purpose of singing, because you are no longer actively doing the intervals, you are not actively unfolding the relations. Investing the additional time to ensure that each exercise is sung perfectly in tune so that the intervals ring, and with precise rhythm, will repay itself many times over. It will help develop the hearing to a surprising degree, which will help in all other musical activities. 4. Perform the exercises without looking at the music. This is crucial in performing the exercises. And many people cannot imagine that it is serious. It is. Each exercise must be performed without the music. A performance that can lead to a sublime musical experience takes place as the unfolding of a single unitary moment-an indivisible continuum of experienced sound. If I compose an exercise and perform it from the written notation, my focused consciousness is taken up with the act of reading. My act of performance becomes one of transforming a visual representation into a sound, and not one of unfolding an indivisible single moment. When I relate a tone to its visual representation, I am relating the tone

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heard to the written suggestion, not to the other tones. Thus I separate it from its environment; I give it its own being. It is only when I am not tied to the written notation that I am free to focus exclusively on the totality of the sounds. 5. Listen to each exercise carefully. Musie is an aural phe-

INTRODUCTION

nomenon resulting from the perception of sounds. Ultimately the only way to learn about music is to listen and then to think-to listen with open, attentive ears, and afterward to think about the experience with an open, attentive mind.

CHAPTER ONEtf^s 1 nmordial Line

ELEMENTS OF LINE Counterpoint is lines joining; the study of counterpoint begins with the study of the essence of line. A line in its essential, fundamental, unadulterated form is called a primordial line. A primordial line is a succession of tones, when those tones are connected by stepwise motion and participate discretely in a unit. Simply put, the three essential attributes of a primordial line distinguishing it from any other succession of tones are: (1) the connection of tones by stepwise motion; (2) the discrete participation of each tone in that connection;1 and (3) the possibility of an indivisible unitary apprehension of the entire succession.2 STEPWISE MOTION

Let us begin by considering our experience of music. In one sense any tone exists only while it is being sounded. 1. Tones participate discretely in a line when they are part of the line in their own right-when they contribute to the linear progress individually and distinctly. See p. 5 for a detailed discussion of discrete participation. 2. By unitary apprehension I mean the mental grasping of a unitof a whole. See p. 11 for a detailed discussion of unitary apprehenSION

1

2

Once it ceases to sound it no longer exists in the now; it ceases to exist as a current perception. In another sense any tone remains in our consciousness for as long as we focus attentively on the group of sounds to which it belongs. The tone that has been present for us in the now but has ceased to sound remains present for us-is retained in our consciousness-as long as we continue to direct our attention to the group of sounds in which it participates. In other words, as long as we are concentrating our attention completely and exclusively on a given musical passage or work, any tone heard is still present. It remains present for us because it participates in shaping our experience of the tones that we hear as current perceptions. Example 1.1

3. The term "neighbor" denotes a tone adjacent to another tone (one step away from it). It does not denote the "neighbor tone" configuration, discussed on p. 70.

PRIMORDIAL

LINE

There is yet another sense in which any tone remains active only until its upper or lower neighbor is sounded, no matter when the tone ceases to exist in current perception.3 If we consider our experience of a tonal melodic line, any tone exists for us actively until a tone one step away is sounded. When a tone is quit by leap, that tone remains active until its linear neighbor is sounded. When the neighbor is sounded the original tone is still present for us, retained in our consciousness, but it loses its active, unconnected nature. For a line to exist each of the component tones must be connected to every other by stepwise motion, either directly or indirectly. Tones are connected by direct stepwise motion when they participate in an unbroken series

PRIMORDIAL

3

LINE

of half or whole steps. Tones are connected by indirect stepwise motion when they participate in a hierarchy of direct stepwise motions. • Listen to examples 1.1 a-Lie. These examples illustrate the most basic kind of pure line. Each tone in each of the three fragments is directly connected to every other by stepwise motion. When a tone is quit by leap, it remains active and unconnected. • Listen to example 1.2a. The A remains active and unconnected throughout the sounding of the C. • Now listen to example 1.2b. The A remains active and unconnected throughout

the sounding of the C, until the sounding of the B. The sounding of the B endows the C with an indirect stepwise connection to the preceding A and G. • Listen to example 1.3. The G remains active throughout the sounding of the E, the D, the C, and the B. With the sounding of the A, the G is relieved of its active and unconnected nature. Again each note is connected to all the others by step. Some are connected directly (such as E to all the tones that follow it) and some are connected indirectly (such as G to all the other tones). To participate in the line, any leap must be controlled by a stepwise motion. • Listen to example 1.4.

Example 1.2

Example 1.4

Example 1.3

PRIMORDIAL

4

The G is active and unconnected throughout the sounding of the C and the B, and remains so until the A is sounded. The stepwise motion G-A controls the leap G-C and the B that follows.4 The C and B participate in the line but are subordinate to the controlling motion G-A. They color the motion G-A; they become a characteristic of it. We can say that the stepwise motion GA is accomplished by the leap to C and the stepwise descent through B. The hierarchy of stepwise motions exists because the direct stepwise motion C-B-A is controlled by the indirect stepwise motion G-A. The C has an indirect linear connection to the G because it relates to the G in a hierarchy of stepwise motions. If a tone is not connected to the others by stepwise motion, it does not participate in the line; it interrupts the line. Example 1.5

LINE

• Listen to example 1.5. The G is active throughout the sounding of the C, until the sounding of the A. The C is itself active throughout the sounding of the A and remains active throughout the sounding of the final G. It is never relieved of its active nature. Standing unconnected, it has no linear function. It may have a harmonic function, combining with either the G or the A, but it does not promote the linear activity in any way. A tone approached by leap can participate in the line only if it is connected by stepwise motion, in other words if the leap is recovered. For a leap to be recovered all the steps between the two components of the leap must be sounded. • Listen to example 1.6. The leap G-C is recovered only when all the intermediary tones have been sounded-in this case the A and the B.5 The C is again active throughout the soundExample 1.6

4. A dash indicates tones sounded successively (as in C-E); a colon indicates tones sounded simultaneously (as in C:E), with the lowest tone given first.

5. The intermediate tones that recover a leap may be sounded in anv order.

PRIMORDIAL

5

LINE

ing of the A. With the sounding of the B it is "deactivated" and connected. The B participates in the line GA-B. The C also participates in the line, but only because of its indirect connection to the B. Both component tones of a leap cannot participate in the line until the leap is fully recovered. Summary One essential attribute of a primordial line is the stepwise connection of the tones. When a tone is quit by leap, the tone has an active and unconnected quality that remains until its upper or lower neighbor is sounded. A tone that sounds unconnected to the others cannot participate in the line; it interrupts the line. Thus a primordial line is made up only of tones that are connected by step. Exercises These exercises are for developing sensitivity to connections effected by stepwise motion.6 6. The exercises in this book are designed to help the reader experience musical phenomena, and become conscious of the phenomena experienced. The reader who has performed an exercise correctly will be conscious of the particular phenomenon it treats-in this case the

• Compose and then sing without music the following: 1. a series of four tones connected by direct stepwise motion 2. a series of four tones connected by an indirect stepwise motion 3. a series of eight tones connected directly and indirectly 4. a series of eight tones in which there is one tone unconnected 5. a series of eight tones in which there are two tones unconnected. DISCRETE PARTICIPATION

Linear progress is effected by the stepwise connection among tones. If not all members participate discretely (individually and distinctly) then this progress is interrupted.7 A tone does not participate in the linear progress as a discrete event if it fuses together with another tone or tones. When two tones fuse together they form connections effected by direct and indirect stepwise motions. For this exercise and all others the solution should first be written out on paper, then sung, in tune, without the music. 7. Although it may seem confusing at first, the concept of discrete participation will become clear over the course of this section.

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a single event, and are therefore not both able to participate discretely in the linear progress. Tones in succession are fused together when they are heard as forming harmonic intervals. It is in the nature of human perception to interpret information in the simplest way possible. The optical illusion in figure 1.1 is a good illustration. Although the object of our perception is four black shapes, each consisting of three-quarters of a solid circle, we are likely to see a single white square. It is simpler to interpret the visual image in this way, even though in physical reality there is no square at all. We similarly interpret sounds in the simplest way possible. It is simpler for us to hear a Example 1.7

Example 1.8

PRIMORDIAL

LINE

Figure 1.1

step than a leap, and when we have the option of understanding tones either as steps or as leaps we will hear them as steps, much as we will see the optical illusion of figure 1.1 as a square. • Listen to example 1.7. The G remains active throughout the sounding of the B. The B participates in the stepwise motion B-A, which is subordinate to the controlling stepwise motion G-A. • Listen to example 1.8a. This begins with the three notes of example 1.7 and follows them with a C. The additional C radically alters our perception of the B. We can hear the C as the result of a leap from the A, as illustrated in example 1.8b. Or

PRIMORDIAL

LINE

we can hear the C as the result of a step from the B, as illustrated in example 1.8c. We tend to hear the C as the goal of a step B-C, as in example 1.8c. Because it is simpler to hear a step than a leap we tend to hear two steps: B-C concurrently with G-A. When we hear concurrent lines we also hear a succession of harmonic intervals. In example 1.8 the G remains active throughout the sounding of the B, and the A remains active throughout the sounding of the C. That the G is still active while the B is sounded promotes the perception of the two tones as a single harmonic event. Likewise, that the A is still active while the C is sounded promotes the perception of the two tones as a single harmonic event. Aided by the simplifying nature of our perception, we hear these tones not as four events participating in a single line, but as two events: the third G:B followed by the third A:C. This is illustrated in example 1.8d. Two of the four tones therefore cannot participate as discrete (individual and distinct) Example 1.9

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linear events; they only support the linear activity of their mates. The flow of linear activity is interrupted. The phenomenon of concurrent stepwise motions is called compound melody. • Listen to examples 1.9a and 1.9b. These examples illustrate a compound melody in parallel thirds. In example 1.9a the G is active throughout the first B until the A is sounded; the A is active throughout the C until the second B is sounded; and the second B is active throughout the D. In example 1.9b the G is active-throughout the first B and the C-until the A is sounded; the C is active-throughout the A and ultimately the second B-until the D is sounded; and the second B is active throughout the D. Our preference for simplicity leads us to hear three steps instead of three leaps: in each example we hear the concurrent ascending lines G-A-B and B-C-D. As illustrated in example 1.9c, the six tones are fused into three thirds (G:B-A:C-B:D).

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• Listen to examples l.lOa and l.lOb. These fragments illustrate compound melody in contrary motion. Again our preference for simplicity leads us to hear three steps instead of three leaps: in each case we hear the ascending line G-A-B concurrently with the descending line D-C-B. As illustrated in example 1.1 Oc, the first four tones of both example 1.1 Oa and example l.lOb are fused into a fifth G:D and a third A:C. • Listen to examples 1.1 la and l.llb. By contrast, these examples both unfold a single line. In example 1.1 la the A has a stepwise connection with the B, as does the C. Thus we are likely to hear a single line containing a leap B-E rather than a succession of Example 1.10

Example 1.11

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two harmonic intervals (A:C-E:B). And in example l.llb the D and C do not form harmonies with any other tones, because they do not function concurrently with any other tones. The discrete participation of tones is also interrupted when three or more successive tones form a triad. • Listen to example 1.12. The G, B, and E fuse together as a single E minor triad. Such a triadic formation results in an interruption of the line, because at least one member cannot participate discretely in the linear progress. When tones in succession fuse together to form a dissonant interval, the linear progress is also interrupted.

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Leaps of dissonant intervals (larger than a perfect fourth) fuse the component tones together.8 One of the component tones of the resulting dissonant harmonic interval must resolve. A dissonant tone that is obligated to resolve to a consonant tone represents that consonant tone. The dissonance must move to the consonance, so it becomes an attribute of the consonant tone; it becomes a variant of the consonant tone.9 If a dissonant tone is a representative of another tone, it cannot participate discretely in the linear progress. • Listen to example 1.13a. The fusion of the F with the G imbues one of the two

with the obligation to resolve. Here it is the F, which resolves to E. The obligation of the F to resolve to E makes it an attribute of the E; therefore the F is not available to participate discretely in the linear progress. • Listen to example 1.13b. The fusion of F with B results in the obligation of each to resolve (respectively to E and C). As representatives of their tones of resolution, neither the F nor the B is able to participate discretely in the linear progress. A similar phenomenon results when the first and last tones of an ascending or descending motion form a dissonance. This is called dissonant contour.10 The first and

Example 1.12

Example 1.13

8. The perfect fourth is treated as a dissonance for the purposes of this book. The fourth has a strange, dual role in tonal music, however, because it may participate in a consonant harmony. Major and minor * triads are often dissonances (for instance in the cadential formula, where the 4 resolves to a 3), but they can also function as consonances. This may account for the lack of obligation to resolve a leap of a perfect fourth. See chapter 2, n. 3. 9. See p. 67 for a detailed discussion of dissonance.

10. Dissonant contour occurs between the highest and lowest tones of a particular ascending or descending motion, and not between the highest and lowest tones of the line, or of an arbitrary passage. In example 1.14 there are three motions: a two-note descending motion from B to G; a six-note ascending motion from G to F (which describes a contour of a dissonant minor seventh); and a two-note descending motion from F to E.

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last tones in an ascending or descending motion are highlighted by virtue of their position. When these two highlighted tones form a dissonance, the result is an obligation to resolve.11 • Listen to example 1.14. The G and F are highlighted by their position as the lowest and highest tones of the ascending motion. The powerful obligation to resolve fuses the F and G. The dissonant F then becomes a variant of the consonant E, its tone of resolution, and is thus unable to participate discretely in the linear progress. Summary Another essential attribute of a primordial line is that its tones participate discretely. When two (or more) stepwise motions are functionExample 1.14

11. Only a dissonance larger than a perfect fourth creates a problematic contour. See n. 8.

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ing concurrently, they are heard as a succession of harmonic intervals. A dissonant leap of a tritone or more is also heard as a harmonic interval. A harmonic interval results from the fusion of two tones into a single event. When two tones are fused into a single event they cannot both participate in the line discretely. Further, dissonant tones are obligated to their tones of resolution; they thus represent or stand in for their tones of resolution. A dissonant tone is not itself available to participate in the line as a discrete component. Therefore a primordial line cannot include any configurations of tones resulting in the perception of harmonic intervals, either consonant or dissonant. Exercises These exercises are for developing sensitivity to tones participating discretely in a line. • Compose and then sing without music the following: 1. a series of six tones resulting in compound melody in parallel thirds 2. a series of six tones resulting in compound melody in parallel sixdis 3. a series of four tones resulting in compound melody in contrary motion

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4. a series of tones including a leap of a tritone 5. a series of tones including a leap of a seventh 6. a series of tones including dissonant contour of a tritone 7. a series of tones including dissonant contour of a seventh 8. an eight-tone line in which each tone participates discretely. UNITARY APPREHENSION

Let us return to the optical illusion in figure 1.1. The tendency of our consciousness to simplify leads us to understand die four black circular shapes as a single white square. The white square does not exist in physical reality; it is the connection of the black circular shapes that exists only in our consciousness. This would also be true of a line of trees. Physically there are only trees: the line results from our perception of the trees from a particular vantage point, and therefore exists only in our consciousness. Further, the line results from die perception not of a single tree or of several individual and unrelated trees but of the totality of the trees. The same is true of a musical line. We can speak of tones in a succession as objects of die physical world.12

But a musical line is not an object of the physical world. The line is the connection of the objects, resulting only from our consciousness of them. If all die tones participate in the connection and each participates as a discrete component, then each tone must be connected to every other. Because the connection is a relationship that occurs within the consciousness, the consciousness must encompass the relation of each tone to every other. We can hear each tone in relation to every other when the entire succession exists as a unit in the consciousness. An Aspect of die Human Consciousness Let us briefly examine unitary consciousness. The human consciousness tends to constitute objects in whole units, again because of its tendency to find the simplest interpretation for perceptions. Each unit is defined by its boundaries. We tend to comprehend the largest ob-

12. An argument can be made that there are not even sounds "out there"-that the sounds do not come into existence until the rarefactions and densifications of air molecules reach the ear and are transferred into aural percepts in the brain. Let us leave this question moot, and assume as we do in the act of perception that the sounds themselves exist and are the physical objects of our perception.

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ject that has boundaries belonging to our focused consciousness. For example, imagine that we are standing one foot from a house, looking at a window. In our field of vision the window is a continuing surface with boundaries that are undefined; from this present perception we cannot constitute the entire window. We do have within our field of vision the completely defined boundaries of a pane of glass. We focus on the pane, as the largest object within our field of vision with boundaries that are completely defined. The pane is a referent to all the objects within our vision; we tend to perceive smaller objects either as part of the pane or as not part of the pane. Imagine that we step back six feet. Included in our field of vision now are the wooden borders of the window. We now have in our field of vision the completely defined boundaries of the window. We tend to focus on the window, the largest object in our vision with defined borders, rather than on the panes that it comprises. We tend to use the window as a referent, perceiving smaller objects either as part of it or as not part of it.13 Our field of vision includes only somewhat more 13. We could focus on the pane, but only if we excluded the boundaries of the window from our attentive, focused consciousness.

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than the boundaries of the single pane of glass; it includes only partly the boundaries of the entire wall. It is therefore impossible that the entire wall could be constituted from our present perception. Imagine now that we step back forty feet, so that the complete boundaries of the wall come into our field of vision. Again we tend to focus on the largest object with boundaries that are within our focused consciousness, in this case the wall. And again we tend to perceive the smaller objects, the window for instance, in reference to the larger unit. We cannot constitute the entire building from our present perception until we step back far enough to perceive its boundaries. The same phenomenon holds true in our perception of musical objects. We tend to constitute the largest musical object of which the boundaries are within our focused consciousness. If we do not have in present consciousness the completely defined boundaries of a given musical object, then that entire object cannot be a part of our present perception; we will focus instead on the next-largest unit for which we have completely defined boundaries. Musical boundaries are not given by edges, or colors, or textures, as boundaries in the visual world are. Nor are they formed by the distinction between sound and

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silence as one might imagine, for virtually every work contains moments of silence that do not signal its end. Instead, musical boundaries are given by the forces of impulse and resolution. Forces of Impulse and Resolution An impulse is an activating, energizing force. A sound or group of sounds resulting in an increase of energy yields impulse. In a succession of tones of equal rhythmic value, impulse tends to result from ascending motion. Ascending motion by leap tends to result in greater impulse than ascending motion by step. Each of the successions represented in example 1.15 tends to result in an increase of energy, and therefore in impulse. • Perform examples 1.15a-1.15d and listen attentively, so that you can hear an impulse resulting from each. You should have found that the tendency to create impulse is least in example 1.15 a and greatest in example 1.15d. A resolution is a retracting, de-energizing force. A Example 1.15

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sound or group of sounds resulting in a decrease of energy yields resolution. If the energy gathered by the tightening of a spring is impulse, the dissipation of the energy effected by the release of the spring is resolution. Because resolution is a decrease or playing out of energy, there can be no resolution if there has been no energy gathered. In a succession of tones of equal rhythmic value, resolution tends to result from descending motion. In contrast to ascending leaps, which dramatically increase the tendency to create impulse, descending leaps result in less resolution than descending steps. Each of the successions represented in example 1.16 begins with the tones of its counterpart in example 1.15. The additional tones in each succession tend to result in a playing out or decrease of the energy, and therefore in resolution. • Perform examples 1.16a-1.16d and listen attentively, so that you can hear an impulse and subsequent resolution resulting from each. Again you should have found that the tendency to

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Figure 1.2

Figure 1.3

create impulse and the subsequent tendency to resolve impulse are least in example 1.16a and greatest in example 1.16d. Figure 1.2 shows how the forces of impulse and resolution serve as boundaries in our musical experience. Imagine a motion that describes a circle (such as the motion of a person riding on a Ferris wheel). Beginning at the lowest point, the first part of the motion is an asExample 1.16

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cent. The motion continues upward along the arc on the left side until the highest point. At the highest point it has nowhere else to go but down; it returns downward along the arc on the right side, until it connects with the original point. When it connects, the circle is an indivisible unity. The whole unit is formed only when the arc of the descent matches the arc of the ascent. The boundaries are formed by the matching of the descent with the ascent. If the descent described by the right arc is too small, it will never connect with the original point. In figure 1.3 the descent is appropriate for a smaller ascent, and does not match. We can never apprehend this figure as a whole enclosed unit, because its boundaries are in-

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Figure 1.4

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Figure 1.5

complete. If the descent described by the right arc is too large, it too will fail ever to connect with the original point. In figure 1.4 the descent is appropriate for a larger ascent, and does not match. We can never apprehend this figure either as a whole enclosed unit, again because its boundaries are incomplete. Only when the descent matches the ascent, as in figure 1.5, can the connection with the original point Example 1.17

14. The tones in this line do not participate discretely in stepwise motions. This is likely to be true of any line that creates more impulse than can be resolved.

be made. And therefore it is only when the descent matches the ascent that the boundaries can be formed and the figure can be perceived as an indivisible unity. This circular motion suggests the activity of the forces at work in a musical experience. The ascending left arc represents the impulse, and the descending right arc represents the resolution. If the experience of a piece of music yields a resolution too small for the impulse the two will not connect, the boundaries will not be completed, and we cannot have that piece in consciousness as a whole, indivisible unit. • Listen to example 1.17a.14 The impulse is created by the ascent to the E; the resolution is given by the final three tones: G-A-G.

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Clearly these three tones cannot provide sufficient resolution to play out the impulse gathered by the first five. The amount of impulse generated by the first five notes demands a much larger resolution, such as that made possible by the final seven tones of example 1.17b. If the experience of a piece of music yields a resolution too large for the impulse, then again the two will not connect, the boundaries will not be completed, and we cannot have the piece in consciousness as a whole, indivisible unit. • Listen to example 1.18a. The impulse is suggested by the initial leap G-C, the resolution by the meandering stepwise return to G. Clearly the playing out of energy resulting from the sounding of the final six tones is far greater than the impulse that could be created by the first two. The resolution might match the impulse if it were significantly smaller, like that offered in example 1.18b. We can apprehend a musical line as unitary only when the boundaries are completely defined-in other words, Example 1.18

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when the impulse is followed by a consequent resolution, one of equal strength. When the resolution is not consequent to the impulse, our consciousness is led to focus on the next largest units for which the boundaries are defined: smaller groups of tones, or even individual tones. When we are focused on individual tones without having the line as a whole in our focused consciousness, then the connections between the tones are ruptured. Because the line is the connections between tones, without the connections there is no line. Summary The third essential attribute of a primordial line that distinguishes it from other successions of tones is that the entire succession must be able to exist in consciousness as a unit. A musical line exists only in the consciousness. It exists when a succession of tones is perceived as connected, and when each tone participates discretely in the connection. The connection can exist only if each tone is

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perceived in relation to every other. Therefore a line exists by virtue of the relation of each tone to every other within the consciousness. To hear each tone in relation to every other, the entire succession must be in consciousness as a unit. To have a unitary consciousness of an object, one must have its completely defined boundaries widiin the attentive, focused consciousness. Musical boundaries are defined by impulse and resolution. The musical object is completely defined when the resolution is consequent to the impulse. A line is thus experienced as unitary when its impulse is matched by a consequent resolution. Exercises These exercises are for developing sensitivity to the forces of impulse and resolution.15 • Compose and then sing without music: 1. two tones resulting in impulse 2. three tones resulting in impulse 3. four tones resulting in impulse

15. Perform these exercises with an unchanging volume. The effect of increasing or decreasing volume on the dynamic forces of impulse and resolution is discussed on p. 20.

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4. five tones resulting in impulse. • For both your four- and five-tone impulses (exercises 3 and 4 above), compose and then sing without music: 5. a resolution that is too small 6. a resolution that is too large 7. a resolution that is consequent (so that all the energy gathered by the impulse is played out). ATTRIBUTES OF A TONE

• Sound the A represented in example 1.19, and listen carefully. You heard a single, isolated tone; it had attributes of pitch, duration, volume, and timbre. Any number of repetitions will yield virtually no differences: each repeated tone will have the same attributes of pitch, duration, volume, and timbre.16 Example 1.19

16. Repetitions of the tone will differ only in the attribute of temporal placement (position in time).

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Linear Function When a tone participates in a line it also has attributes of pitch, duration, volume, and timbre, and it takes on the attribute of linear function. The linear function of a tone is given by its relation to the other members of the line in which it participates. • Listen to example 1.20a. Consider the A sounded as part of example 1.20a, and compare it with the isolated A of example 1.19. You should have found them markedly different. The two A's had the same or similar attributes of pitch, duration, volume, and timbre; but the A of example 1.20a had the additional attribute of linear function. • Now listen to both examples 1.20a and 1.20b. Compare the A of example 1.20a with the A of example 1.20b. Again you should have found them different, even though again they had the same or similar attributes of pitch, duration, volume, and timbre. Example 1.20

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This difference is due solely to linear function. The linear function of the A of example 1.20a was given by its position as the goal of the stepwise ascent G-A, which served the controlling step B-A, which itself participated in an overall stepwise descent C-B-A-G. The linear function of the A of example 1.20b was given by its position as the origin of the ascending step A-B, which was controlled by the step C-B, which itself participated in a stepwise descent from D. The linear function of a tone is given by its relations to every other member of the line. Scale Degree The attribute of linear function depends on the attribute of pitch. Pitch is the source of yet another attribute of a tone participating in a single line: the attribute of scale degree.

• Listen to examples 1.21a and 1.21b, which you per-

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form in the following manner: play the chord progression that serves to establish the key, then play the figure E-D-C. These two statements of the figure E-D-C have a markedly different quality, even though they have the same or similar attributes of pitch, duration, volume, and timbre, as well as linear function. The two statements differ only in scale degree. The E-D-C of example 1.21a functions as scale degrees 8-7-6 in E minor; the E-D-C of example 1.21b functions as scale degrees 3-2-1 in C major.

Dynamic Function When a tone is experienced as part of a primordial line it takes on yet another attribute: that of dynamic function.17 The dynamic function of a tone is given by its relation to the forces of impulse or resolution. The force of impulse is an increase of energy; a tone that contributes to the increasing of energy has the dynamic function of furthering the impulse. The force of resolution is a release of energy; a tone that contributes to the releasing of energy has the dynamic function of furthering the resolution.

Example 1.21

17. The term "dynamic function" refers to the forces of impulse and resolution, and is not to be confused with the common term "dynamics," which refers to inflections of volume. There is dynamic function attendant to any tone of any line that is heard, primordial or otherwise, because any tone in a succession creates impulse or resolution. (In fact most tones in a musical context participate in more than one level of linear activity, and may be participating in impulse on one

level and resolution on a different one. See p. 154 for a more detailed discussion.) In a heard line that is not primordial, dynamic function of a given tone is incidental and derives from its relation to a limited number of surrounding tones. In a primordial line that is apprehended as an indivisible unit, however, the dynamic function of a given tone is critical to the unitary experience and is universal: it derives from the relation of the tone to every other member of the line.

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It is critical to note that when we compose a primordial line, that line has no structure of impulse and resolution; its tones may suggest a dynamic function or they may not. The tones' attributes of dynamic functiontheir contribution to the growth or decline of energyexist only in our experience of the line. This experienced structure of dynamic function is a direct result of the sounds we experience-of the performance. A line that we compose has no structure of dynamic forces until we experience it as performed. Depending on the performance, the same written line may sound with any number of different structures of dynamic function. The performer uses many elements to create the dynamic structure, including volume, tempo, and articulation. In a limited musical context, such as within a phrase or in the primordial lines and counterpoint exercises of this book, volume is the primary generator of dynamic function. Increasing volume tends to create impulse; decreasing volume tends to promote resolution. Example 1.22

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• Perform the primordial line of example 1.22 with any number of different structures of volume (without looking at the music). Focus on the possibility of unitary apprehension. When you find the structure of impulse and resolution (determined by the inflection of volume) that allows you to experience the line as an indivisible unit, perform the line repeatedly. While focusing only on the sounds, open yourself to the extraordinary experience of musical beauty that is made available through these performances. Each of the performances of example 1.22 that did allow the experience of beauty was different. No two performances had exactly the same structure of volume. Each performance did however have the same structure of dynamic function. In each the impulse climaxed with the second note (E), the goal of an ascending leap of a major sixth. The ascent to the E was accomplished with an increase in volume, thus creating enough impulse to carry through to the end of the line. The resolution was

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effected with the aid of a gradual decrease in volume from the climactic E until the final G.18 This is true of any unitarily apprehensible musical object: no two performances will be identical, but all performances that are unitarily apprehensible (in other words, all performances in which the impulse is consequently resolved) must have the same general structure of dynamic function. The dynamic function of the tones will be the same in all unitarily apprehensible performances of the same unitarily apprehensible work. If the tones of two performances of the same work have different attributes of dynamic function, one or both performances cannot be unitarily apprehensible. If we recall the A sounded as part of example 1.22 and compare it with the A of example 1.20a, we find that they have the same or similar attributes of pitch, duration, volume, and timbre, the attribute of linear function, and (most likely) the attribute of scale degree. But the A of example 1.22 has the additional attribute of dynamic function: it is the penultimate tone of a six-

tone resolution. Note that dynamic function is complex: a tone can participate in impulse or resolution on more than one level. In example 1.22 the A in question caps a small impulse on one level, as the goal of a two-note stepwise ascent; on a broader level it participates in the six-tone resolution. This resolution begins after the height of the impulse (E) and ends with the final tonic (G);19 the small two-note impulse (G-A) participates in the larger resolution by extending it. As a result, the A has the dynamic function of participating in impulse on one level and resolution on a broader level. Whether it grows through impulse or is played out through resolution, energy does not exist in the composed tone but only in the experience of the tone in relation to the other tones. We have heard how the experience of dynamic structure is the joint responsibility of the performer and the composer. The composer suggests relations for sounds. A given sound or group of sounds suggested by the composer may or may not create dynamic force in and of itself. In a single line of

18. The resolution (and the gradual decrease in volume) is continuous from the E all the way to the final G. Even the ascent G-A toward the end of the line participates in the resolution.

19. More precisely, the impulse climaxes with the initial attack of the E. The height of any impulse is defined by the beginning of its resolution. In this case the resolution begins within the unfolding of the E itself, immediately after its attack.

22 tones, each with the same duration, a tone could create or resolve impulse on the basis of its pitch and scale degree. As we have heard, a tone with a higher pitch tends to create more energy than one with a lower pitch; thus moving from a lower-pitched tone to a higher-pitched tone tends to create impulse, and moving from a higherpitched tone to a lower-pitched tone tends to resolve impulse. • Listen to examples 1.23a and 1.23b. The ascending motion G-A of example 1.23a tends to create impulse; the descending motion A-G of example 1.23b tends to resolve impulse. Dynamic function is also affected by the attribute of scale degree. Scale degree l(or 8) creates the least tenExample 1.23

Example 1.24

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sion; among the others scale degree 5 creates the least. All the remaining scale degrees create more tension. A motion from scale degree 1 up to scale degree 2 tends to create more impulse than one from scale degree 6 up to scale degree 7. Conversely, a motion from scale degree 2 to scale degree 1 has a greater tendency to resolve impulse than one from scale degree 7 to scale degree 6. • Listen to example 1.24, performed in the following manner: first play the chord progression that establishes the key of G major, then play examples 1.24a-1.24d. The motion G-A of example 1.24a (scale degrees 1 and 2) tends to create more impulse than the motion EFtt of example 1.24b (scale degrees 6 and 7); likewise the motion A-G of example 1.24c tends to resolve more impulse than the motion Ftt-E of example 1.24d. The inherent tendencies toward dynamic function generally conflict to some degree in musical passages. • Listen to example 1.25, which is still in the key of G major.

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The inherent tendencies toward dynamic function resulting from the factors of pitch and scale degree are in conflict. The tendency of the stepwise ascent Fit-G to create impulse is in conflict with the tendency of the motion from scale degree 7 to scale degree 8 to resolve impulse. What is the performer's contribution to the dynamic forces? And in light of the possibility of conflicting inherent tendencies, how does he or she know how to structure those forces? The performer at all times must structure the dynamic forces so that the impulse created is resolved consequently. Summary A tone standing alone has the attributes of pitch, duration, volume, and timbre. In the unitary experience of a primordial line, each tone has the additional attributes of linear function, scale degree, and dynamic function. The dynamic function results both from the composiExampk 1.25

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tion and from the performance. For a line to result in the extraordinary experience of musical beauty that is possible, it must be composed so that it allows an impulse and consequent resolution, and its performance must create that impulse, which is consequently resolved.20 Exercises These exercises are for developing sensitivity to the performer's contribution to impulse and resolution. • Again perform without music each of your two im20. To allow the transcendent experience of the highest musical beauty, the performance must be experienced as unitary in every other aspect as well. More precisely, no aspect of the performance can be experienced as a multiplicity. Poor intonation is experienced as a multiplicity of intonation systems; imprecise ensemble is experienced as a multiplicity of ensemble systems. Squeaks by woodwind instruments, cracks by brass instruments, wrong notes, inappropriate or conflicting articulations or sound qualities are all experienced as multiplicities. Any of these conditions (or similar ones) prevents the indivisible experience of the totality of the sounds, and thus prevents the highest experience of beauty. A performance is unitarily apprehensible only if it is free of all these conditions that produce multiplicity. In addition, there can be no conflict of dynamic forces: the impulse that is created must be resolved consequently. I use the term "unitary apprehensibility" to refer specifi-

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pulses with consequent resolutions (exercise no. 7 on p. 17) in such a way that the resolution is consequent to the impulse. • Both these impulses with consequent resolutions can also be performed in such a way that the resolution is not consequent to the impulse. Using different inflections of volume, perform these four fragments so that the resolution is either too great or too small for the impulse. STEPWISE FOUNDATION

The diverse elements presented above come together in a remarkable way. Every primordial line is built on a foundation of direct stepwise motion. Further, the dynamic structure of every primordial line mirrors the dynamic structure of its stepwise foundation (assuming Example 1.26

cally to this latter, somewhat unfamiliar concept, and assume the more familiar requirements of intonation, ensemble, and so on.

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that both primordial line and stepwise foundation are performed so that the impulse is consequently resolved). Linear motion is stepwise; all the tones of a primordial line are connected by direct or indirect stepwise motion. Stepwise motion occurs between linear neighbor tones-tones a step apart. Direct stepwise motion occurs between two successive tones that are neighbors. Indirect stepwise motion occurs between a tone and the first neighbor tone that follows it, when the original tone is quit by leap. Indirect stepwise motion is stepwise motion because the original tone is still operative during the sounding of the intermediary tones until the sounding of the goal tone, its neighbor. The intermediary tone or tones are controlled by that step; the intermediary tones accomplish that controlling step. • Listen to the figure represented in example 1.26a.

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The G and A are connected by direct stepwise motion, as are the C and B. The A and B are connected by indirect stepwise motion: the A is still operative during the sounding of the C until the sounding of the B. There is a sense in which the C is secondary to the motion A-B that controls it; the C accomplishes the step A-B. As illustrated in examples 1.26b and 1.26c, a hierarchy of tones is established on the basis of stepwise motion. The figure of example 1.26a is built on a fundamental structure that begins with G, moves to A, and then moves on to B. • Now listen to the extended line notated in example 1.27a. Example 1.27

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The direct stepwise motion G-A is followed by E, D, and C, which are controlled by the indirect stepwise motion A-B (lower slur). The B is followed then by the E, Fit, and G, which are controlled by the indirect stepwise motion B-A (lower slur), and finally by the direct stepwise motion A-G. The stepwise descent E-D-CB is secondary to the controlling indirect step A-B; likewise the ascent E-FJt-G is secondary to the controlling indirect step A-G. Again there is a hierarchy of tones: in the indirect stepwise motions, the intervening tones E-D-C and E-Fit- G are secondary to the controlling tones A-B and A-G. As illustrated in examples 1.27b and 1.27c,

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eliminating these "secondary" tones reveals a stepwise palindromic figure that stands as the foundation of the line: G-A-B-A-G. • Perform the line of example 1.27a so that the impulse is consequently resolved. The initial ascent to E creates impulse, which is resolved consequently by the return to G. The height of the impulse comes with the sounding of the high E.21 • Now perform the stepwise foundation represented in example 1.27c, so that it is unitarily apprehensible. The stepwise ascent away from the tonic G creates impulse, the height of which comes with the motion to B; the stepwise descent back to the tonic resolves the impulse. The structure of dynamic forces in the line itself is mirrored by the structure of dynamic forces in its stepwise foundation (when both are performed as unitarily apprehensible). In the foundation, the height of the impulse comes with the stepwise motion from A to B. In the line, the height of the impulse comes with the motion to E, which accomplishes the same indirect step A-

B. After experiencing innumerable successful and unsuccessful lines, a pattern begins to emerge: the capacity of a line to be apprehended unitarily depends on the relationship of its dynamic structure to that of its stepwise foundation. The dynamic structure of any unitarily apprehensible line is mirrored by the dynamic structure of its unitarily apprehensible foundation. The height of the impulse in a primordial line comes either (a) with the tone that serves as the height of the impulse of its foundation, or more likely (b) within the indirect motion to that tone. Conversely, the dynamic structure of any line that cannot be apprehended unitarily is not mirrored by the dynamic structure of its unitarily apprehensible foundation.22 This point can be further demonstrated with another example. The line represented in example 1.28a is similar to the line of example 1.27. It begins with an indirect stepwise motion to A (which controls the E, D, and C); this is followed by a direct stepwise motion to B, by an indirect stepwise motion to A (which controls the Ftt and G), and finally by a direct stepwise motion A-G. As

21. A secondary impulse within the overall resolution climaxes with the penultimate tone A.

22. This is true of lines in which the members participate discretely in stepwise connections.

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illustrated in examples 1.28b and 1.28c, the stepwise foundation is again revealed to be G-A-B-A-G. • Try to perform the entire line of example 1.28a so that the impulse climaxes with the high E and is resolved consequently. I do not believe this can be done. When we turn to the foundation, we find it identical to the foundation of the first line: G-A-B-A-G. Again, to be unitarily apprehensible this foundation must be performed so that its impulse climaxes with the B, and is consequently resolved by the return to G. Yet in the line itself, the E acExample 1.28

27

complishes not this climactic B but the second tone, A. To perform the line so that its structure of dynamic forces mirrors that of its stepwise foundation, the impulse must reach its height with the B. • Listen to example 1.28c, performed so that its impulse is resolved consequently. You should have found that the line is unitarily apprehensible if its impulse climaxes with the B. Any primordial line must be unitarily apprehensible (its impulse must be consequently resolved). Only a line in which the dynamic structure is mirrored by that of its

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unitarily apprehensible stepwise foundation can itself be unitarily apprehensible. Clearly, any primordial line must have a dynamic structure mirrored by that of the dynamic structure of its stepwise foundation. The foundation of a primordial line is a foundation because it carries the line in a germinal state-not only its linear structure, but its structure of dynamic forces. In other words, the structure of dynamic forces of a primordial line exists-is present-in its foundation. The extraordinary consequences of this will be revealed in the succeeding chapters. Summary The dynamic structure of a primordial line mirrors the dynamic structure of its stepwise foundation. Two tones connected by indirect stepwise motion control the intervening tones, which thus become secondary. Eliminating the secondary tones of a line in which the members participate discretely in stepwise connections reveals a stepwise foundation. The foundation itself must be performed with a certain structure of dynamic forces to be unitarily apprehensible. The line itself can be unitarily apprehensible only when its dynamic structure mirrors that of its unitarily apprehensible foundation.

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Exercises These exercises are for developing sensitivity to the relationship between the dynamic structure of a line and that of its stepwise foundation. • For each of the three lines below: 1. find the structure of impulse and consequent resolution necessary for unitary apprehension 2. find the stepwise foundation 3. find the dynamic structure of the foundation necessary for unitary apprehension 4. determine how the height of the impulse of the line relates to the foundation. ADDENDUM: THE MINOR MODE The considerations for a line in the minor mode are essentially the same for one in the major mode. The minor mode introduces some additional complications, however. Before examining the minor mode it will help to reexamine the major mode. The term pitch field denotes those pitch classes that participate in a given tonality. As illustrated in example 1.29, the pitch field of the major mode consists of the tonic itself (scale degree 1); two pairs of tonic-defining neighbors-the upper and lower

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linear neighbors (scale degrees 2 and 7) and the upper and lower harmonic neighbors (scale degrees 5 and 4);23 and two tones that characterize the mode (scale degrees 3 and 6). Example 1.29

23. The harmonic neighbors of a tone lie a perfect fifth above or below it.

29

The pitch field of the minor mode is similar to that of the major. It too consists of the tonic (scale degree 1); two pairs of tonic-defining neighbors-the upper and lower linear neighbors (scale degrees 2 and 7) and the

30

upper and lower harmonic neighbors (scale degrees 4 and 5); and two tones that characterize the mode (scale degrees 3 and 6). The minor mode is achieved by two alterations from the major: the mode-characterizing scale degrees 3 and 6 are both lowered. Lowering the third is a simple matter. Wherever the third scale degree is sounded it lies a minor third above the tonic. Lowering the sixth is somewhat more complicated. For the tonality to be confirmed the seventh must sound as a leading tone: it must lie a half step below the tonic. In the minor pitch field the sixth (lying a half step above the dominant) and the seventh (lying a half step below the tonic) are therefore separated by the interval of an augmented second. The augmented second forms a powerful dissonance in which both members must resolve: the sixth downward and the seventh upward. In any linear motion that includes an augmented second, only one of the members can resolve; the other remains Example 1.30

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active and unconnected by virtue of its continuing obligation. Because the sixth and seventh in their standard positions cannot both participate in the same linear motion, an alteration is necessary both in the linear ascent from dominant to tonic and in the linear descent from tonic to dominant. Example 1.30a presents the dominant and the tonic pitches in die key of A minor. In a linear ascent from dominant to tonic the critical intermediary tone is the seventh degree, standing in its tonic-confirming position one half step below the goal (example 1.30b). To accommodate the priority of the seventh degree functioning as a leading tone, the sixth degree is raised so that it stands a whole step above the dominant and a whole step below the seventh (example 1.30c). Example 1.3la presents the tonic and the dominant pitches in the same key. In a linear descent from tonic to dominant the critical intermediary tone is the sixth de-

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gree, standing in its position one half step above the goal. The sixth scale degree in this position characterizes the minor mode. To accommodate the priority of the sixth degree in this lowered position, the seventh degree is lowered so that it stands a whole step above the sixth and a whole step below the tonic. In a primordial line in the minor mode, the linear motions that include either an altered (raised) sixth degree or an altered (lowered) seventh degree must sound explicitly and completely. In other words, the sixth may be raised if and only if it participates in a linear ascent from dominant to tonic (5-It6- 7- 8) in which each scale degree is sounded. Likewise, the seventh may be lowered if and only if it participates in a linear descent from tonic to dominant (8-t7-6-5) in which each scale degree is sounded. Example 1.31

24. More accurately, the term means fixed or unchanging song.

This sounding of the completed motion is necessary only when the sixth or seventh scale degree occurs in its altered form. When the sixth and seventh sound in their standard positions (one half step from the dominant and one half step from the tonic), they require no special consideration.

C A N T U S F I RM U S

Every species counterpoint exercise consists of one or more lines sounding in conjunction with a cantus firmus (a Latin term literally meaning firm voice).24 A cantus firmus, or cantus, is a primordial line in which all tones have the same temporal value. Cantus firmi are notated in whole notes.

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Guidelines for the Cantus A cantus firmus is a primordial line with a specific function: to serve as the foundation for species counterpoint exercises. To fulfill that function, it must meet the following conditions: 1. The cantus must begin and end on the same tonic pitch. In any musical experience, impulse is effected by movement away from the most stable referent;25 resolution is effected by returning toward that most stable referent. The tonic tone is the most stable referent in a primordial line, where the only system of reference is the relation of tones to the tonic. Thus to effect impulse, a primordial line must first establish the tonic tone. The most direct way to do so is to sound it first.26 To effect a complete resolution, the tonic must also be sounded last. A line that ends with a note other than the tonic is heard as not ending. 25. In tonal music this is most often a key; it may be a level of rhythmic density (as in many variation forms), and it may also be any number of other factors. See chapter 4, n. 1. 26. If the first tone sounded is not the tonic, the tonic tone must be confirmed as a referent before movement away can create impulse. A cantus with so few tones has none to "spare" for establishing the tonic.

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2. The cantus must end with a stepwise descent to the original tonic.27 The last tone is necessarily approached by step, because a leap to the final tone could not be recovered. The requirement of all cantus firmi to end with a stepwise descent (instead of a stepwise ascent) yields a convenient pedagogical organization: when the cantus firmi end with a stepwise descent, the colines end with a stepwise ascent.28 3. The cantus must consist of not fewer than eight tones and not more than fourteen. As we have heard, temporal extension (length in time) is tied to degree of impulse; thus the longer the cantus, the greater impulse it must have. Because impulse in a primordial line sounding alone is largely a function of range, cantus firmi longer than fourteen tones require a greater range than is convenient for these exercises. A cantus shorter than eight tones, on the other hand, does not provide enough freedom for the accompanying lines. 27. The cantus must end on the same tonic pitch as the one with which it begins, and not one an octave higher or lower. Otherwise it cannot resolve its impulse consequently. 28. More precisely, all co-lines ending on the tonic must approach it by stepwise ascent. Parallel octaves result if both co-line and cantus approach the final tonic by stepwise descent.

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Review of Common Problems A line is primordial when there are no conditions preventing its primordiality. Some of the most common conditions that prevent primordiality are given below. (Use this section to become sensitive to conditions that prevent primordiality, not as a series of "rules" to be followed. It is critical that the cantus be a heard primordial line that conforms to the three guidelines above, not a succession of tones contrived according to a set of rules.) An augmented fourth is problematic when one of its component tones is highlighted as the highest or lowest tone of a motion that contains the other component tone. The augmented fourth is a dissonance in which both members must resolve. It is particularly problematic in a linear context. When one component tone of an augmented fourth is highlighted as the highest or lowest tone of a motion that contains the other compoExample 1.32

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nent tone, the two components of the interval fuse together. • Listen to example 1.32a. The figure has an awkward, uncomfortable feeling. This is because one component tone of an augmented fourth, B, is highlighted as the highest member of an ascending motion containing the other component tone, F. The power of the augmented fourth is so great that it invests B with an obligation to move to C and F with an obligation to move to E. The failure of the B and F to resolve creates a certain discomfort. Surprisingly, the diminished fifth does not present the same problem. • Listen to example 1.32b. One member of a diminished fifth, B, is highlighted as the lowest tone of a descending motion containing its partner, F. Example 1.32b does not cause the same discomfort as example 1.32a, probably because the F is followed by the E and the B is followed by the C.

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Compound melody occurs when two (or more) stepwise motions exist concurrently. Compound melody results in the perception of harmonic events; thus not all tones can participate discretely in the line. • Listen to example 1.33a. This example illustrates compound melody. The fusion of G with D and the fusion of C with A result in the two concurrent stepwise motions G-A-B and DC-B. • Now listen to example 1.33b. This too contains a compound melody, although one slightly more complex than that of example 1.33a. We understand these tones in the simplest (most consonant) way: as a linear unfolding of the three harmonic events G:E-A:E-B:D.29 Thus the heard concurrent stepwise motions are E-E-D over G-A-B.

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• Finally, listen to example 1.33c. This example looks as though it might contain two instances of compound melody, but in fact it contains none. The D-Bt-C-F (upper bracket) is not experienced as compound melody, because the Bl> moves to its linear neighbor C, not to the F. The C-F-G-C-Bl> (lower bracket) is also not experienced as compound melody, because although the F moves to G and the C to Bt, they do not do so concurrently.30 A dissonant contour is formed when the first and last tones of an ascending or descending motion form a dissonance larger than a perfect fourth. This dissonance is perceived with its obligation to resolve, preventing the dissonant tone from participating discretely in the linear progress. • Listen to example 1.34.

Example 1.33

29. The E functions through the sounding of the A, and thus joins with it in the fifth A:E.

30. Compound melody does not occur in any configuration in which a leap is followed by one or more steps in the direction of the original tone, and then by a leap back to the original tone.

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The G serves as the highest tone of a descending motion; it joins with the lowest tone of the motion, A, to describe a dissonant contour. The seventh A:G is heard as a dissonance, requiring resolution. A harmonic embellishment is created when a leap from a tone is followed directly by a leap back to the same tone. These leaps result in a harmonic event; both component tones cannot participate discretely in the linear activity. • Listen to example 1.35. The leap E-G followed by a return to E constitutes an unfolding of a minor third E:G over three tones. The linear progress is interrupted because all of the tones cannot participate discretely in it. Example 1.34

Example 1.35

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Linear embellishment occurs when two tones of the same pitch surround an ornamental upper or lower neighbor. The ornamental tone is heard as a variant or representative of the main tone. The three tones serve to unfold one; all three are not able to participate discretely in the line. • Listen to example 1.36a. The two E's surround the F. The configuration E-FE is an extension of the tone E, which is still functioning through the ornamental F. Note that this configuration of tones does not result in linear embellishment if the two like-pitched tones function on different levels. • Listen to example 1.36b. The first E is controlled by the indirect step G-F. The

36

F is not ornamental; it participates in the descending line A-G-F-E. In this case the E-F-E configuration does not constitute a linear embellishment, because the two E's have different attributes of linear functionthus they are essentially different tones. Mixed modes occur in the minor mode when an ascending motion from dominant to tonic is not completed before a descending motion from tonic to dominant is begun, or vice versa. Example 1.36

Example 1.37

Example 1.38

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• Listen to example 1.37a, which is in C minor. This is problematic, because the descending motion of tonic to dominant C-Bl»-At is not completed with a G before the ascending form of the seventh (B'l) is sounded. • Now listen to example 1.37b, also in C minor. The ascending motion of dominant to tonic G-A-B^I is satisfactorily completed with the C before the descent back to G is begun.

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In overly conjunct motion, too many steps result in more resolution than impulse. • Listen to example 1.38. This fragment is overwhelmingly conjunct. The lack of ascending leaps that promote impulse results in a far greater resolution than impulse. In overly disjunct motion, too many leaps result in more impulse than resolution. • Listen to example 1.39. The ascending leaps G-G, D-F, and A-D create more impulse than can be resolved. Patterns occur when the same interval configurations sound at different pitch levels. Tones that are members Example 1.39

Example 1.40

37

of a pattern serve the activity of the pattern and do not participate equally in the line. • Listen to example 1.40. Example 1.40 contains successive statements of a pattern: ascending step, ascending third, descending step. Although the entire passage is controlled by the ascending stepwise motion D-E-F-G-A-B, the tendency of tones in a pattern to fuse together breaks the passage into two sections, one beginning on D and the other on G. (Note: the more tones in the pattern, the greater the tendency for the component tones to fuse together.) A repetition is created by the sounding more than once in succession of the same group of two or more

38

tones. Tones in a repeated group tend to fuse together, and thus are unable to participate discretely in the linear progress. • Listen to example 1.41. The two statements of the figure Bl>-C-D cause the tones to fuse together, as do the repeated figures E\>- D. There are too many steps when successive steps in the same direction fuse together into a harmonic interval. Generally, more than four successive tones tend to result in such a fusion. Example 1.41

Example 1.42

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• Listen to example 1.42a. The six successive ascending tones C-D-E-F-G-A fuse together into the harmonic interval C:A, and the five successive descending tones G-F-E-D-C fuse together into the harmonic interval G:C. The entire example outlines the tones C-A-D-G-C, as illustrated in example 1.42b. • Now listen to example 1.43a. The five successive ascending tones D-E-F-G-A serve to recover the leap Bl> - D. In this case thefivesuc-

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cessive ascending tones do not seem to fuse together to form a single interval; most likely this is because the overall linear progress of the line is furthered-the successive steps serve the controlling motion Bl>-A, as illustrated in example 1.43b. Thus five successive tones do not fuse together into a fifth where they serve to recover a leap of a sixth. A triad is formed when successive leaps outline an arpeggiated triadic figure, which interrupts the linear progress. • Listen to example 1.44.

Example 1.43

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The leap from B down to D is followed by a leap up to G. The B, D, and G fuse together to form a triad; not all its components are free to participate discretely in the linear progress. An unrecovereA leaf is a leap not followed directly by its intervening tones. This must result in at least one tone disconnected from the line. • Listen to examples 1.45a and 1.45b. Both contain unrecovered leaps. In example 1.45a the leap D-G is unrecovered: it is not followed by all the intervening tones. That the F is missing separates the G from the rest of the tones and prevents it from participating in the line. In example 1.45b the leap D-A is unrecovered. That the E is missing separates the C and the D from the rest of the tones. Cantus Write-Through

Example 1.44

"Write-throughs" for each type of counterpoint exercise are given in the appendix. The write-throughs constitute the guided composition of counterpoint exercises. Each write-through illustrates the steps one might go through in composing a counterpoint exercise, from the beginning to the final counterpoint. These are not intended as precise representations of how counterpoint exercises

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are composed. Rather, they offer a taste of the types of considerations involved and demonstrate some ways of proceeding. The write-through sections are offered as practical aids, and can be used at the discretion of the reader. In a given chapter the reader may choose to complete the write-through section before proceeding to the exercises, or to skip the write-through section entirely. Then if the exercises prove troublesome, the reader may wish to return to the write-through section for the practical information it offers. The process of composing each of three cantus firmi is illustrated and described in the appendix, beginning on page 175.

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Exercises • Negative examples. Compose the following and identify the problems: 1. one cantus with a dissonant contour, an unrecovered leap, a linear embellishment, and a triad 2. one cantus (minor mode) with a compound melody, a three-note pattern, and mked modes 3. one cantus with too many steps, a problematic augmented fourth, and a repetition 4. one cantus with the sole problem of not enough impulse (overly conjunct) 5. one cantus with the sole problem of not enough resolution (overly disjunct). • Identify the problems in each of the following cantus firmi:31

Example 1.45

TCf

31. Answers to the exercises for identifying problems: Exercise 6 (eight problems): three unrecovered leaps (Cft-Bt, A-A,

A-E), two dissonant contours (A up to Bl>, Bl> down to A), mked modes (B& sounded before the lowered seventh degree C'l progresses

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6. eight problems

7. six problems

8. four problems

9. three problems

down to A through B!>), dissonant leap (Cd-Bt), and overly disjunct (not enough resolution). • Exercise 7 (six problems): three linear embellishments (D-E-D, EF-E, and again D-E-D), a three-note pattern (C-D-E, D-E-F), a two-note repetition (D-E), and overly conjunct (not enough impulse).

• Exercise 8 (four problems): compound melody (E-C-D-B), unrecovered leap (D-B), augmented fourth (Ftl down to C'l), and triad (D-B-Ftt). • Exercise 9 (three problems): three-note pattern (C-E-D, F-A-G), too many steps (A-G-F-E-D-C), and overly conjunct (not enough resolution).

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• Compose and sing without music:32 10. two good cantus firmi in sharp major keys (G major, D major, and so on)

32. You will undoubtedly find composing counterpoint exercises difficult at first, but the more exercises you do the more comfortable you will become. In composing your cantus firmi (and your counterpoints in later chapters) you may get bogged down if you focus on avoiding problematic conditions. Instead you might begin by composing your cantus (or counterpoint) so that it pleases you, and then listen for any problematic conditions.

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11. two good cantus firmi in flat major keys (F major, Bl> major, and so on) 12. two good cantus firmi in minor keys.

CHAPTER TWO

Two-Part Counterpoint

CONJUNCT LINES

If counterpoint is lines joining and the study of counterpoint begins with the study of the essence of line, then it continues with the study of the essence of conjunction. In two-part counterpoint the cantus is joined with a second primordial line, termed a co-line.l The conjunction of the cantus with the co-line results in a larger whole, the counterpoint. To repeat, the counterpoint is a unit resulting from the conjunction of two primordial lines. The lines must be primordial, they must join yet remain independent, and the totality must be apprehensible as an indivisible unit.2 1. The line accompanying the cantus is often referred to itself as a counterpoint. The term "counterpoint" is already in use to signify both a technique of combining lines and a specific example of combined lines. To avoid confusion I use the term "co-line" to indicate the line that accompanies the cantus. 2. The co-line by itself must display two of the three aspects of primordiality: (a) its members must be related by step, and (b) all members must participate discretely in the line. The third aspect, unitary apprehensibility, arises only in conjunction with the cantus: the counterpoint exercise resulting from the conjunction with the cantus must be unitarily apprehensible. Whether the co-line by itself is or is not unitarily apprehensible is irrelevant.

43

44 HARMONIC INTERVALS

The sounding together of two lines creates harmonic intervals. There are three categories of harmonic interval: stable consonances, unstable consonances, and dissonances (single and double). Illustrated in examples 2.la-2.Id, these categories result from the degree of tendency to motion. Stable consonances have no particular tendency to motion. Example 2. la illustrates the stable consonances: octaves and perfect fifths. Unstable

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consonances have a tendency to motion. Example 2.1b illustrates the unstable consonances: major and minor sixths and thirds. Dissonances have an obligation to motion. Single dissonances have one component tone with an obligation to motion. Both component tones of double dissonances have an obligation to motion. Example 2.1c illustrates the single dissonances: major and minor sevenths, augmented fifths, perfect fourths,3 diminished fourths, and major and minor seconds. Example 2. Id

Example 2.1

3. The perfect fourth is often considered a "swing interval." When it is formed between the tones of the lower and middle lines, or between the tones of the lower and upper lines, it is dissonant. But when it is formed between the tones of the middle and upper lines it may be consonant. For the purpose of this counterpoint study, the harmonic fourth is considered exclusively dissonant. The fourth is not heard as a dissonance when formed between two upper lines, because of our tendency to hear intervals in relation to the bass. When we hear a 3 triad

we do not hear the fourth formed between the two upper lines as dissonant, because we do not hear a fourth-we hear a sixth and a third. We hear upper-line tones as dissonant with each other only when they stand in conflict with each other. For example, a s chord is dissonant even though both the major sixth and perfect fifth are individually consonant with the bass, because the two tones belong to conflicting triads. Likewise an augmented 3 chord is dissonant even though both the major third and the minor sixth are individually consonant with

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illustrates the double dissonances available within the major and minor pitch fields: tritones,4 diminished sevenths, and augmented seconds. Ordinarily the perfect fifth and the sixth (major or minor) are consonant intervals. Under certain circumstances either can become dissonant. If a sixth represents a perfect fifth, it has an obligation to move and is by definition dissonant. Examples 2.2a and 2.2b illustrate a dissonant sixth and a dissonant fifth.5 Example 2.2

the bass, because the two tones have conflicting tendencies (the chord occurs as the III* in the minor mode: thus the chord's major third is the seventh scale degree in its ascending position, and its minor sixth is the modal third scale degree that is lowered from the major). The tendency of the major third (ascending seventh scale degree) to ascend conflicts with that of the minor sixth (lowered third scale degree) to descend.

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• Listen to example 2.2a. The A of the sixth C:A in bar 2 represents the consonant G that follows;6 it is obligated to move to the consonant G. Through its obligation to resolve, the sixth assumes the characteristics of a dissonance. • Now listen to example 2.2b. The G of the fifth C:G in bar 2 is obligated to move to the consonant A.7 Through its obligation to move, the fifth assumes the characteristics of a dissonance. Compound intervals (those larger than an octave) have the same tendency to motion as their simple counterparts do. It is an interesting phenomenon that the greater the distance between tones, the weaker the relationship between them. Therefore, simple stable consonances (an octave and a perfect fifth) are more stable than their compound counterparts. And at the same

4. I use the term "trirone" to mean both augmented fourth and diminished fifth. 5. This is assuming a harmonic rhythm that changes by the bar. 6. The G must be consonant because it is quit by leap. In this context, only a consonance may be quit by leap. See p. 78. 7. See n. 5.

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time, dissonances of less than an octave are more dissonant than their compound counterparts. Harmonic Function In chapter 1 an isolated A (example 1.19) was heard to have the attributes of pitch, duration, volume, and timbre. When the same A was heard as part of a line (example 1.20), it took on the attributes of linear function and scale degree. And when the same A occurred as part of a primordial line (example 1.21), it took on the attribute of dynamic function. Similarly, a tone that participates in a harmonic event takes on the attribute of harmonic function. Just as the linear function of a tone is given by its relations to the other members of the line in which it participates, the harmonic function of a tone is given by its relations to the other members of the harmonic event in which it participates. • Perform examples 2.3a-2.3d, sounding each chord Example 2.3

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separately but in a similar way. Listen carefully to the quality of the A in each of the four different contexts. The four A's sounded quite different from each other; yet they had the same or similar attributes of pitch, duration, volume, and timbre. They had neither linear function nor dynamic function. The sole difference among the four was the attribute of harmonic function: each stood in a different relation to the other members of the harmonic event in which it participated. ADDENDUM: BALANCE We have heard that tones in succession must be sounded with a structure of volume so that the totality can be experienced as an indivisible whole. The same is true of tones sounding concurrently: they must be sounded with a structure of volume so that the totality can be experienced as an indivisible whole. In other words, they must be in balance. Tones sounding concurrently are not

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COUNTERPOINT

necessarily in balance when they can be distinguished from each other, or when they can all be heard, or when they are sounded with precisely or approximately the same intensity (volume). On the contrary, tones sounding concurrently are in balance when they combine to form a new sound-a single, indivisible sound in which all the sounding components participate. For this to occur, lower-pitched tones must be sounded with less intensity than higher-pitched tones. • As an experiment, perform the C major chord illustrated in example 2.4a so that each successive tone enters at an intensity great enough for it to join with the tone or tones sounding, but not great enough for it to dominate. You should have found that each successive tone had to be softer than the tone or tones already sounding. If Example 2.4

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you perform the chord members simultaneously (as illustrated in example 2.4b) with a structure of intensity similar to that of example 2.4a, you will hear the four blend into a single indivisible whole-a new sound resulting from the blending of the four component parts. If lower-pitched tones are louder than higher-pitched tones, then at best the experience will be one of individual tones sounding concurrently, and at worst the higher-pitched tones will be obliterated by the lower ones. • Listen again to the C major chord illustrated in example 2.4a. This time perform it so that each successive tone is at least as loud as the preceding one. If you sound the four chord members simultaneously (as in example 2.4b) with this structure of intensity, you will hear the low C most prominent, followed by the upper C. Depending on how much louder the low C is, you may not hear the G or the E at all-or possibly not even the upper C. If a lower tone obliterates an upper tone, the tones are certainly blended into one sound, but the oneness results from the obliteration of tones, and not from their relations. The examples and exercises below involve concurrently sounding tones. It is important to perform them

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with a structure of intensity that allows the sounds to blend into a single, indivisible new sound resulting from the combination of the two (or more) concurrent events. CONJUNCTION In a two-part counterpoint, the cantus must be joined with a co-line. Two conditions preventing such a conjunction are (a) too great a distance between the lines, and (b) conflicting modal inflections. Distance. The greater the distance between two lines, the harder it is to hear them as joined into a single entity. Lines separated by a distance of greater than a thirteenth are difficult to perceive as joined together; lines separated by a thirteenth or less are easier to perceive as joined.

TWO-PART

COUNTERPOINT

• Listen to examples 2.5a and 2.5b. Example 2.5a, in which the two lines are separated by a seventeenth, is more likely to be heard as two distinct lines and less as a single entity than is example 2.5b, in which the two lines are separated by a tenth. Two lines tend to be perceived as distinct and separate when separated by more than a thirteenth. Mixed modes. In exercises in minor keys, the two lines cannot join if their modal inflections conflict. Modal inflections conflict when an ascending motion from dominant to tonic is begun before a descending motion from tonic to dominant has been completed, or vice versa. Thus if an altered sixth or seventh is sounded, no unaltered sixth or seventh may be sounded in either line before the altered tone completes its motion between tonic and dominant. • Listen to example 2.6a, which is problematic. The upper line sounds an altered descending seventh scale degree B!>, which participates in a descending motion from tonic to dominant. Before the descending motion is completed, however, the lower line sounds the altered ascending sixth scale degree A^l, which participates in an ascending motion from dominant to tonic.

TWO-PART

COUNTERPOINT

• Now listen to example 2.6b, which is not problematic. The ascending B^ in the lower line is sounded as the upper line descending motion is completed, and not before it. Thus lines in minor keys can join only if their modal inflections do not conflict. Positive effects of conjunction. Certain problematic configurations in a single line can be eliminated solely through the conjunction of that line with another. For instance, repetitions or linear embellishments are prob-

49

lematic in a single line. In either case the sounding of two like tones or group of tones interrupts the linear progress. When as a result of conjunction with another line the two like tones or groups of tones participate in different harmonic intervals, the problem is eliminated. The two tones or groups of tones then have differing attributes of harmonic function; thus they are essentially different and do not impede the linear progress. • Listen to example 2.7a.

Example 2.6

Example 2.7

50

This example contains a repetition (two statements of the successive tones E-D), which breaks the linear progress. • Now listen to example 2.7b. The line is joined with another line above it. Here the first E and D participate respectively in a sixth and a third, whereas the second E and D participate in a third and a fifth. Because their attributes of harmonic function are different, these two statements of E and D are essentially different, and are not problematic. • Listen to example 2.8a. The linear progress is interrupted by the linear embellishment of G. • Now listen to example 2.8b. As a result of the conjunction with the lower line, the

Example 2.8

TWO-PART

COUNTERPOINT

first G participates in a third and the second G participates in a sixdi. Because the two G's differ in the attribute of harmonic function, they are no longer the same G; thus the G-F-G is not problematic. PRIMORDIAL LINES

In a two-part counterpoint the primordial cantus is joined with a primordial co-line. When one primordial line is joined with another, die conjunction itself-the way that the two come together-can prevent a tone from participating discretely in its own line. If a tone does not participate discretely in its line, diat line cannot be primordial.8 There are a number of ways that the conjunction of otherwise primordial individual lines can render them not primordial. This happens when the joining of lines results in harmonic intervals diat are not consonant, or in parallel stable consonances (perfect fifths or octaves). Consonant harmonic intervals. For the conjunction of die lines to take place, each tone of die cantus must join widi die corresponding tone of die co-line to form a consonant harmonic interval. If die conjunction of the 8. See p. 5.

TWO-PART

COUNTERPOINT

lines results in a dissonant harmonic interval, at least one line cannot be primordial. In a dissonant harmonic interval, at least one of the tones is obligated to resolve; this tone represents its consonant resolution. Because the dissonant tone represents its tone of resolution, it is effectively no tone at all. This is illustrated in examples 2.9a and 2.9b. In example 2.9a the G forms a dissonant fourth with the lowest tone, D. Because the G must resolve to the Ftt, 9 it substitutes for the Ftt-it stands in for the Ft, or represents it. Thus the G is sounding but die Ft is functioning; die G is an agent of the Fl. Likewise, in example 2.9b the F forms a dissonant second with the G. Because the F must resolve to the E, it substitutes for the E-it stands in for or represents the E. The F is sounding but die E Example 2.9

9. Depending on the key, of course, the G might be obligated to resolve to an F*l.

51

is functioning; the F is an agent of the E. If the conjunction of lines causes a tone to become dissonant, it effectively eliminates the tone; the tone can no longer participate discretely in its line. Thus each tone of the cantus must participate in a consonant harmonic interval. Parallel stable consonances. A primordial line is rendered not primordial if two or more successive tones join with the accompanying line to form the same stable consonant intervals (in either simple or compound form). We can make an analogy between linear progress and climbing stairs. The linear flow is analogous to a smooth ascent of the stairs, one foot after the other. A stable consonance is equivalent to having both feet on the same step. And moving directly between two like

52

stable consonances is analogous to moving directly from one step to another with both feet simultaneously-it cannot be done without interrupting the flow. Parallel fifths or octaves likewise interrupt the flow.10 Tones participating in parallel fifths or octaves do not continue their lines but instead effect a new start, severing their connection to the previous tones and preventing primordiality. • Listen to examples 2.10a and 2.10b.

TWO-PART

COUNTERPOINT

In example 2.10a the lines are interrupted by the motion between the octaves B:B and C:C. In example 2.1 Ob the lines are interrupted by the motion between the fifths F:C and E:B. Exercises These exercises are for developing sensitivity to the conjunction of two lines. • In conjunction with the three-note stepwise descend-

Example2.10

10. More precisely, a succession of parallel fifths and octaves has one of two effects on our experience of the counterpoint. We may hear the succession of intervals as stopping and then starting again, effecting an interruption of the lines (as described in the paragraph above). Or we may hear the two lines combining into a single linear progression, pre-

venting the lines from remaining independent (see p. 53). Given an extended string of parallel consonances, we are more likely to hear the two lines combine into a single linear progression and lose their independence. Given a succession of only two parallel consonances, however, we are more likely to hear an interruption of the lines.

TWO-PART

COUNTERPOINT

53

ing "cantus" represented in figure 2.1, compose and perform (without looking at the music) each of the "colines" described below (each harmonic interval formed must be consonant):

5. 6. 7. 8.

Figure 2.1

Figure 2.2

1. eight different "co-lines" above (at a higher pitch), each of which is a three-note figure connected by direct stepwise motion 2. seven different "co-lines" below (at a lower pitch), each of which is a three-note figure connected by direct stepwise motion 3. one "co-line" above that contains an indirect stepwise motion 4. one "co-line" below that contains an indirect stepwise motion. • In conjunction with the six-note "cantus" represented in figure 2.2, compose and perform (without looking at the music) each of the "co-lines" described below:

INDEPENDENCE

one "co-line" above with parallel octaves one "co-line" below with parallel octaves one "co-line" above with parallel fifths one "co-line" below with parallel fifths.

Counterpoint results from the conjunction of lines. There can be no joining of objects if they are not discrete-independent from each other. There are two different conditions that prevent independence of lines. The first prevails when one line is taken over by the other. If the co-line becomes subsumed into the cantus or the cantus into the co-line, there is no conjunction because there are no longer distinct lines to join together. The second condition preventing independence of lines prevails when the two lines become confused with each other-when tones of one line are heard to participate in the other.

54

One line can be taken over by'another when too many like intervals occur in succession. Crossed voices, some unisons, and some parallel leaps cause confusion of lines. Too many like intervals. When two lines join to form four successive similar intervals (thirds or sixths in their

TWO-PART

COUNTERPOINT

simple or compound forms) one line is subsumed under the other. Lines tend to remain distinct through a succession of three like intervals, but not four.11 • Listen to examples 2.1 la and 2.lib. In example 2.1 la the two lines join to form a succession of four thirds. With the sounding of the fourth

Example 2.11

11. This fusion occurs whether the interval is in simple or compound form. In other words, any combination of four successive

thirds or tenths will fuse the two lines, as will any combination of four successive sixths or thirteenths.

TWO-PART

COUNTERPOINT

third, the perception of two distinct lines is dissolved. In example 2.lib, however, the lines remain distinct throughout the succession of three thirds. • Now listen to examples 2.lie and 2.lid. In example 2.1 Ic the two lines lose their distinction as a result of the succession of four sixths being formed. In example 2.lid, however, the two lines remain distinct throughout the succession of three sixths. Crossed voices and unisons. Crossed voices and unisons obscure the independence of the lines.12 The configuraExample 2.11

12. In addition to obscuring the independence of the lines, unisons interrupt the linear progress. This is primarily because of their striking contrast with the other intervals in the succession, each of which has two different component tones.

55

tion of tones called crossed voices occurs when tones of the upper line are lower in pitch than the concurrent tones of the lower line. • Listen to example 2.12a. When the second tone of each line is sounded, resulting in a third G:B, we perceive the upper-line tone to be the B. Thus we perceive an upper line of D-B-C-B and a lower line of D-G-A-G, as illustrated in example 2.lib. The lines are no longer independent. When the conjunction of two lines results in a unison,

56

the lines become indistinct. They lose their distinction after the unison, when it becomes unclear which tones participate in which line. • Listen to example 2.13a. After the unison C it is unclear whether the upper line continues with the higher-pitched tones or the lower-pitched tones. An exercise may begin or end on a unison, however, because the two lines do not lose their distinction from each other.

TWO-PART

COUNTERPOINT

• Listen to examples 2.13b and 2.13c, which represent the beginning and end of an exercise. In neither case does the unison confuse the apprehension of distinct lines. Parallel leaps. Parallel leaps occur when both lines leap in the same direction simultaneously. Some parallel leaps cause confusion between the lines; some do not. Parallel leaps cause confusion if one line leaps to within a step of the preceding tone of the other line.

Example 2.13

Example 2.14

TWO-PART

• Listen to examples 2.14a-2.14c. In example 2.14a the lower line leaps to Bl>, which is within a step of the preceding tone (C) of the other line. Because of the simplifying nature of our perception, we tend to hear a step C-Bt, with the linear progress moving from the upper line to the lower line, instead of the more complex parallel leaps. This is especially true where one line leaps to the same tone as did the preceding tone in the other line, as in example 2.14b, or to a Example 2.15

57

COUNTERPOINT

tone past the preceding tone of the other line, as in example 2.14c. With one exception, no problems are presented by parallel leaps in which the two lines remain independent (in other words, leaps in which one line does not leap to within a second of the preceding tone of the other). Nor are problems presented by simultaneous leaps by contrary motion. • Listen to examples 2.15a and 2.15b. Neither the parallel leaps of example 2.15a nor the simultaneous leaps by contrary motion of example 2.15b prevent independence of lines. The single exception involves parallel leaps to stable consonances, either perfect fifths or octaves. Because of the contrast between the stable quality of the fifth or octave and the unstable quality of other intervals, leaping to the stable consonance creates an interruption. • Listen to example 2.16a.

Example 2.16

58

The linear progress is interrupted in both lines by the parallel leap from the tenth G:B up to the octave E:E. • Listen also to example 2.16b. The linear progress is interrupted in both lines by the parallel leap from the sixth C:A up to thefifthF:C. Exercises These exercises are for developing sensitivity to how lines lose their independence. • Compose and then perform without music (playing one line and singing the other): 1. a two-part counterpoint with too many consecutive thirds 2. a two-part counterpoint with as many consecutive thirds as possible, without losing the independence of the lines 3. a two-part counterpoint with too many consecutive sixths 4. a two-part counterpoint with as many consecutive sixths as possible, without losing the independence of the lines 5. a two-part counterpoint with crossed voices 6. a two-part counterpoint with a problematic unison

TWO-PART

COUNTERPOINT

7. a two-part counterpoint with parallel leaps that confuse the lines 8. a two-part counterpoint with parallel leaps to a stable consonance 9. a two-part counterpoint with parallel leaps that present no problems. UNITARY APPREHENSION

We have explored the conjunction of two primordial lines: lines joining, yet remaining primordial and independent. The final requirement for our counterpoint exercises is that each be apprehensible as a unit. As was heard in the study of unitary apprehension above,13 a single line must have a consequent resolution to be apprehended as unitary. This section explores the apprehension of a unit comprising two lines. Impulse and resolution. To be apprehended as a unit, two conjunct lines must be unfolded with dynamic forces structured so that the impulse is resolved consequently. • Perform examples 2.17a-2.17e and listen attentively, so that you can hear an impulse resulting from each. 13. Seep. 11.

TWO-PART

COUNTERPOINT

You should have found the tendency to create impulse least in example 2.17a and greatest in example 2.17e. In each of these examples the succession of intervals tends to result in an increase of energy, and therefore in impulse. Ascent of the upper line by leap yields greater tendency to create impulse than ascent by step; and tendency to create impulse generally results from expansion-when contrary motion results in a larger interval. • Perform examples 2.18a-2.18e and listen attentively, so that you can hear an impulse and subsequent resolution resulting from each. Again you should have found that the tendency to

59

create and resolve impulse is least in example 2.18a and greatest in example 2.18e.14 Each of these examples begins widi its counterpart from examples 2.17a-2.17e that suggests a gathering of energy, or impulse. In each of the progressions in examples 2.18a-2.18e, the succession of intervals that follows the impulse tends to result in a playing out or decrease of that energy, and therefore in resolution. Descending motion in the upper line generally suggests resolution. Descent by leap yields no particular tendency to resolve; but tendency to resolve generally results from contrary motion to a smaller interval. Generally, but by no means always, the direction in

Example 2.17 14. Because there are more elements involved than in a single line and because these fragments are so short, their tendency to create impulse is more loosely defined, and thus a greater portion of the re-

sponsibility for dynamic function lies with the performance. The tendency to effect dynamic function is much more clearly defined in a complete exercise.

60

TWO-PART

which the upper line moves determines the dynamic tendency. There is a tendency to create impulse when the upper line ascends, and a tendency to resolve impulse when the upper line descends. This can be heard Example 2.18

Example 2.19

COUNTERPOINT

in each of the impulses of examples 2.17a-2.17e. It is however possible for an upper-line descent to create impulse and for an upper-line ascent to resolve impulse. • Listen to example 2.19.

TWO-PART

For this fragment to be apprehended as unitary, the height of the impulse must come with the third F:A; the expansion back out to the octave C:C must effect the consequent resolution. The structure of impulse and resolution required for a unitary experience of a single line may change dramatically when the line is joined with another. Earlier, example 1.22 was performed so that it could be apprehended as an indivisible unit.15 The energy was gathered by the ascent to the second tone, E, accomplished with a growth in volume. This impulse was resolved by the succeeding six tones with the help of an overall decrease in volume. In example 2.20 the same line serves as the lower line of a two-part counterpoint. Example 2.20

61

COUNTERPOINT

• Listen to example 2.20, performed so that the entire unit has the structure of dynamic function of the lower line when performed alone: greatest volume with the lower line E, decreasing gradually to the final G. This exercise has an incomplete quality, resulting from too little impulse. • Now perform the example again, focusing on the totality of the sounds. Keeping as a goal the extraordinary experience of beauty that results from the apprehension of the counterpoint as an indivisible unit, try different ways of structuring the impulse and resolution until you find the way to create an impulse that is consequently resolved. You should have found that the resolution can be consequent to the impulse if the height of the impulse is carried by the sixth D:B in the upper line. Exercises

15. Sec p. 20.

These exercises are for developing sensitivity to impulse and resolution within two-part counterpoints. • Compose and perform without music (playing one line and singing the other): 1. three harmonic intervals resulting in impulse 2. four harmonic intervals resulting in impulse

62

TWO-PART

3. five harmonic intervals resulting in impulse. • For your four- and five-interval impulses (exercises 2 and 3 above), compose and then perform without music: 4. a resolution that is too small 5. a resolution that is too large 6. a resolution that is consequent. The following exercises are for developing an awareness of the performer's contribution to impulse and resolution. • Both of your impulses with consequent resolutions (exercise 6 above) can also be performed so that the resolution is not consequent to the impulse. Perform these four fragments again, varying the inflections in volume in such a way that the resolution is either too great or too small for the impulse. FIRST

SPECIES

Two-part counterpoint exercises involve the composition of a co-line sounding in conjunction with the cantus. First species exercises are note against note: each

COUNTERPOINT

tone of the cantus is accompanied by one tone in the coline. First species co-lines are notated in whole notes. Guidelines for All Two-Part Counterpoints The co-line is a primordial line, which accompanies the cantus so that the complete counterpoint can be apprehended unitarily. It must therefore be composed according to certain guidelines: 1. Beginning: Co-lines sounding above the cantus may begin on the tonic or the fifth; co-lines sounding below the cantus must begin on the tonic. The counterpoint must begin by establishing the tonic scale degree as the stable referent. This is most directly accomplished if the counterpoint begins with a stable harmonic consonance built on the tonic scale degree (octave, perfect fifth), or with a unison. Thus if the co-line sounds above the cantus it must begin with either the tonic or the fifth. If the co-line sounds below the cantus it must begin with the tonic. 2. Ending: All co-lines must end with a stepwise ascent to the tonic. All two-part co-lines must end with the most stable consonant interval (an octave), or with a unison. Because the cantus ends with a stepwise de-

TWO-PART

63

COUNTERPOINT

scent, the co-line must end with a stepwise ascent to prevent parallel octaves. 3. Each tone of the cantus must participate in one consonant harmonic event.16 The co-line must join with the cantus so that each tone of the cantus participates in not more and not less than one consonant harmonic event. This eliminates the complication of varying harmonic rhythm. Note that the co-line differs from the cantus in two ways. First, it may begin on the fifth scale degree (if it sounds above the cantus). If the co-line begins on the fifth scale degree its foundation will unfold a stepwise motion to the tonic (almost always descending).17 Second, the co-line need not be unitarily apprehensible on its own. The structure of impulse and resolution in a two-part counterpoint is given by the totality-by the experience of the two lines sounding together. When two lines are experienced in conjunction of each other, 16. In first species counterpoints the consonant harmonic event is always an interval. In second species to fifth species the consonant harmonic event will be triadic if the co-line leaps within the bar. 17. This is true only of co-lines in two-part counterpoint. Co-lines sounding in the middle or upper line in three-part counterpoint may end on the third or fifth as well.

neither has any structure of impulse or resolution of its own; any impulse or resolution that might be experienced if either line were to sound alone is irrelevant. Characteristics of First Species Counterpoints 1. Each tone of the co-line forms a consonant interval with the corresponding tone of the cantus. There are no dissonances between cantus and co-line. 2. On one occasion in the exercise a tone in the co-line may be tied over two successive tones of the cantus. Both tied tones may not participate in the same harmonic interval. The repetition of a tone affects the perception of dissonant contour, because it halts the motion (ascending or descending). Thus the first statement of the repeated tone serves as the end of the previous motion and the second statement of the tone begins another motion-even if it proceeds in the same direction. • Look at example 2.21, which seems to contain a dissonant contour of a diminished fifth, from the highest tone of the descending motion (F) to the lowest tone (B). Now listen to the example carefully. You should have found that you do not hear a dis-

TWO-PART

64

sonant contour. The descending motion is divided by the restatement of C; thus there is one descending motion from F to C, and a second from C to B. Review of Common Problems Crossed voices occur when the tones of the upper line are lower in pitch than the tones of the lower line. This destroys the independence of lines. • Listen to example 2.22. The upper line is heard as A-G-F-G over a lower lineofC-E-D-E. Distance between two lines is problematic when it is so great that the lines do not sound in conjunction (in general a thirteenth is the outside limit). • Listen to example 2.23. The seventeenth C:E strains the perception of the two lines as joined. Parallel fifths or octaves occur when two or more successive tones of the cantus participate in the same stable consonant harmonic interval, either a fifth or an octave. Consecutive stable consonances interrupt the linear continuum. • Listen to example 2.24. The perfect fifth D:A is heard not as a progression

COUNTERPOINT

from the fifth E:B but instead as a new beginning. This separates the tones before the parallel fifths from those after, preventing primordiality.18 Example 2.21

Example 2.22

Example 2.23 18. No problem results if an octave is followed by a perfect fifth, or conversely if a fifth is followed by an octave.

TWO-PART

Parallel leaps are problematic when a line leaps to within a step of the original tone of the other line, or to the same pitch as the original tone of the other line, or beyond the original tone of the other line. • Listen to example 2.25a. These parallel leaps cause confusion between the lines and prevent their independence. The parallel leaps from the sixth BkG to the fifth F:C are problematic because the lower line leaps to F, one step from the preceding

Example 2.24 Example 2.25

65

COUNTERPOINT

tone G of the upper line. Because we tend to hear a step G-F instead of two leaps, the lines lose their independence. • Now listen to example 2.25b. The parallel leaps from the tenth F:A to the octave C:C interrupt the linear progress. • Finally, listen to example 2.25c. The parallel leaps from the octave F:F to the sixth C:A are not problematic at all: they cause no confusion between the lines, nor do they interrupt the linear progress. Too many like intervals occur when more than three successive tones of the cantus are accompanied by the same interval (either a third or a sixth, in simple or compound form). After the third consecutive parallel interval the two lines lose their independence: one is subsumed under the other. • Listen to example 2.26. c

66

TWO-PART

You should have heard the two independent lines becoming reduced to one with the fourth consecutive thirteenth. A unison between a tone of the cantus and the corresponding tone of the co-line prevents independence if it occurs other than at the beginning or end of the exercise. Such a unison reduces the two lines to one. • Listen to example 2.27. The unison G formed by the two lines reduces the two to one. Example 2.26

Example 2.27

COUNTERPOINT

Counterpoint Write-Through The write-throughs illustrating the composition of twopart counterpoints in first species above and below begin on page 181. Exercises • Using one of your successful cantus firmi from chapter 1, compose and perform (without the music) one first species counterpoint in which the co-line sounds above the cantus.19 • Using another of your successful cantus firmi from chapter 1, compose and perform (without the music) one first species counterpoint in which the co-line sounds below the cantus. 19. You may try composing more than one counterpoint. But composing one good counterpoint is sufficient, and more valuable than composing any number of problematic ones. As with cantus firmi, I recommend that you begin by composing your counterpoint so that it pleases you, and then listen for any problematic conditions. When you encounter difficulties, you will have to retrace your steps and try different options. Keep in mind that the attributes of any tone depend on its relation to every other tone. Changing any tone of the co-line changes the attributes of every other tone as well. And with any change, tones that had been problematic may become acceptable, and tones that had been acceptable may become problematic.

CHAPTER

Dissonance

THREE

ELEMENTS

OF

DISSONANCE

There are three essential attributes of dissonance in tonal music. First, dissonance is defined by obligation to motion: a dissonance is a dissonance because it must move. Second, a dissonant tone stands for, or represents, a specific, identifiable tone other than itself. A dissonance represents a consonance-a dissonant tone is a representative or agent for a consonant client tone. Third, dissonance is a linear phenomenon. Dissonant agent tones have a stepwise relationship with their consonant clients. First species counterpoint exercises do not make use of any dissonant harmonic intervals; these exercises consist of successions of consonant intervals. Second, third, and fourth species exercises also consist in essence ot successions of consonant intervals; each species introduces a more complex type of ornamental dissonance.: 1. In first species exercises linear progress is tied to harmonic change; the two happen at the same rate. Dissonances sever the bond between the linear progress and the rate of harmonic change. In the case of dissonant passing tones and neighbor tones, the linear activity progresses while the harmonic event remains constant. In the case of dissonant suspensions, the linear progress is delayed while the harmonic event changes.

67

D I S S O NA N C E

68

The dissonant harmonic intervals formed between tones of the major and minor pitch fields are as follows: major, minor, and augmented seconds and their inversions, diminished and augmented fourths and their inversions, and perfect fourths.2 PASSING TONES

In tonal music there are essentially only three kinds of dissonances: passing tones, neighbor tones, and suspensions. Passing tones are dissonances that are approached by step and quit by step in the same direction. • Listen to example 3.1. The dissonant D is a passing tone; it forms a dissonant fourth with the A sounding below it. The dissoExample 3.1

nant D is approached by step from the consonant C, and continues in the same direction by step to the consonant E. The passing tone creates a direct stepwise connection between two consonant tones separated by a third; it ornaments the linear progression of a third, C-E. In ornamenting the linear progression C-E, the passing tone D also ornaments the progression of harmonic intervals A:C-G:E. The D is a dissonance; it ornaments but does not materially affect the underlying harmonic progression. The consonant harmonic interval A:C then remains in effect-remains functional-until another consonant interval is sounded. And if the harmonic event A:C remains functional then its component tones must in some way remain functional. The A remains functional because it continues to sound, and the C remains functional because the passing tone D represents it. The D is sounding, but the C is still functioning. The dissonant passing tone D is an agent of its consonant client tone C. Exercises

2. These intervals are dissonant in their simple or compound form.

These exercises are for developing sensitivity to the ornamental nature of dissonant passing tones.

69

D I S S O NA N C E Figures 3.1 and 3.2 are models. The whole notes DC represent a two-note "cantus"; the lines above the whole notes D-C (figure 3.1) and below them (figure 3.2) represent "co-lines." The "co-lines" unfold ascending motions of a third above and below the two-note

Figure 3.1

Figure 3.2

3. In each exercise the two functional harmonic intervals must be consonant. Take care that the passing tone does not create parallel

"cantus." In each figure example a represents the underlying whole-note motion of the "co-line" and example b represents the "co-line" ornamented by passing half notes. The model exercises should be performed by playing the "cantus" and singing the "co-lines": first the unadorned leap of a third, then the three-note stepwise figure resulting from the addition of the ornamental passing tone. • As in the model exercises (figures 3.1 and 3.2), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows:3 1. an ascending or descending leap of a third above the "cantus," first unadorned, then adorned with a passing tone 2. an ascending or descending leap of a third below the "cantus," first unadorned, then adorned with a passing tone.

fifths. Remember to perform your exercises without looking at the music.

DI SSON A N CE

70

NEIGHBOR TONES

Neighbor tones are also approached by step, but are quit by step in the opposite direction; thus they are preceded and followed by the same pitch. • Listen to example 3.2. The dissonant D is a neighbor tone; it forms a dissonant seventh with the E sounding below it. The dissonant D is approached by step from the first consonant C and is followed by a step in the opposite direction, before returning to the second consonant C. Neighbor tones effect a stepwise embellishment of a single tone; in this case the D effects a stepwise embellishment of the C. Because the neighbor tone D ornaments the consonant C sounded before and after, it again ornaments also the progression to which the consonant C's belong: E:C-F:C. And again, although it ornaments the pro-

Example 3.2

gression it does not affect it significantly. The consonant harmonic interval E:C remains functional until the sounding of the new consonant harmonic interval F:C. Thus although the neighbor tone D is sounding, the C is functioning. The D represents the C; the D is a dissonant agent of its consonant client tone C. Exercises These exercises are for developing sensitivity to the embellishing nature of dissonant neighbor tones. Figures 3.3 and 3.4 are models. The whole notes D-

Figure 3.3

Figure 3.4

D I S S ON A N CE

C again represent a two-note "cantus"; the lines above the whole notes D-C (figure 3.3) and below them (figure 3.4) represent "co-lines." In each figure the "co-line" appears in its fundamental, unadorned whole-note form in example a, and as adorned by a half-note neighbor motion in example b. The model exercises should be performed by playing the "cantus" and singing the "colines": first the unadorned repeated tones, then the three-note stepwise figure resulting from the addition of the embellishing neighbor tone. • As in the model exercises (figures 3.3 and 3.4), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows:4 1. a repeated tone above the "cantus," first unadorned, then adorned with a neighbor tone 2. a repeated tone below the "cantus," first unadorned, then adorned with a neighbor tone.

4. In each exercise the two functional harmonic intervals must again be consonant. Again take care that the neighbor tone does not

71 SUCCESSIVE DISSONANCES

A consonant tone can be represented by two or more dissonant tones in succession: either two (or three) dissonant passing tones in succession, or two neighbor tones in succession. Successive passing tones occur when a consonant tone is represented by more than one successive dissonant passing tone. • Listen to example 3.3a. The upper line unfolds a stepwise ascent of a perfect Example 3.3

create parallel fifths.

D IS S O N A N C E

72

fifth from the G of bar 1 to the D of bar 2. The consonant G is followed by the successive passing tones A and B, and then the consonant C. The consonant G is represented by its dissonant agents A and B. Successive passing tones in a lower line are somewhat more complicated. • Listen to example 3.3b. The lower line unfolds a stepwise descent of a perfect fifth from the A of bar 1 to the D of bar 2. The consonant A is followed by the dissonant passing tones G and F. Paradoxically, although the E joins with the upperline E to form a consonant octave, it cannot be considered a consonance in species counterpoint. In conjunction with the functioning harmonic event (the fifth A:E), the E creates a dissonant second-inversion triad.5 In effect the consonant A is represented by three successive dissonant passing tone agents, G, F, and E. 5. This is essentially a technicality in this context, but it will be critical in others. See p. 86. 6. The successive neighbor figure may seem to imply a hierarchy: the second neighbor G functions as a dissonance that ornaments the consonant first neighbor Fit, which itself ornaments the consonant E. If the first neighbor (the Ftt) is not consonant in any way, then there can be no suggestion of hierarchy; the figure can be understood only

Successive neighbor tones occur when a single tone is ornamented by successive neighbor tones. • Listen to example 3.4. The consonant E is ornamented by a dissonant neighbor tone, Ftt, and again by its dissonant neighbor tone G. The successive dissonances Ftt-G-Ftt effect a stepwise ornamentation of the consonant tone E. The E is represented by its dissonant agents Ftt and G.6 Example 3.4

as two successive neighbor tones that function on the same level of dissonance. In species counterpoint each tone of the cantus may be accompanied by only one consonant harmonic event. Thus in example 3.4 the E and the Ftt cannot both be consonant. As the first tone of the bar, the E must be consonant; therefore the Ftt must be dissonant. Both Ftt and G are then experienced as simple dissonances on the same level. For a more extended discussion of hierarchies of dissonance and consonance see p. 133.

D 1 SSO NAN C E Exercises These exercises are for developing sensitivity to the ornamental nature of successive passing tones and successive neighbor tones. figure 3.5

73

Figures 3.5 and 3.6 are models that illustrate successive passing tones. The whole notes E-F represent a two-note "cantus"; the lines above the whole notes E-F (figure 3.5) and below them (figure 3.6) represent "colines." The "co-lines" unfold ascending and descending motions of a fifth above and below the two-note "cantus." In each figure example a represents the underlying whole-note motion of the "co-line," and example b represents the "co-line" ornamented by successive quarter-note passing tones. The model exercises should be performed by playing the "cantus" and singing the "co-lines": first the unadorned leap of a fifth, then the five-note stepwise figure resulting from the addition of the successive passing tones. • As in the model exercises (figures 3.5 and 3.6), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows: 1. an ascending or descending leap of a fifth above the "cantus," first unadorned, then adorned by successive quarter-note passing tones

D I SS ON A N CE

74

2. an ascending or descending leap of a fifth below the "cantus," first unadorned, then adorned by successive quarter-note passing tones. Figures 3.7 and 3.8 are models that illustrate successive neighbor tones. The whole notes G-F# represent a two-note "cantus"; the lines above the whole notes GFtt (figure 3.7) and below them (figure 3.8) represent "co-lines." In each figure the "co-line" appears in its fundamental, unadorned whole-note form in example a, Figure 3.7

and as adorned by successive quarter-note neighbor tones in example b. The model exercises should be performed by playing the "cantus" and singing the "colines": first the unadorned repeated tones, then the fivenote stepwise figure resulting from the addition of the successive embellishing neighbor tones. • As in the model exercises (figures 3.7 and 3.8), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows: 3. a repeated tone above the "cantus," first unadorned, then adorned by successive quarter-note neighbor tones 4. a repeated tone below the "cantus," first unadorned, then adorned by successive quarter-note neighbor tones. CAMBIATAS

Figure 3.8

A cambiata results from the elision of two successive neighbor-tone or passing-tone motions. There are two different types of cambiatas, the neighbor cambiata and the passing cambiata,. The neighbor cambiata results from the elision of two successive neighbor-tone motions, one upper and one lower.

75

DISSONANCE

• Listen to example 3.5a. The figure A-B-A-G-A constitutes an embellishment of a consonant A with first the dissonant upper neighbor B, and then the dissonant lower neighbor G. The two dissonant neighbor tones are approached and quit by step. • Now listen to example 3.5b. The middle A is elided to form a neighbor cambiata. The dissonances B and G are both heard as stepwise embellishments of the consonant A's, although on a local level each dissonance is either quit or approached by leap.

Neighbor cambiatas accompany a single tone of the cantus, and are essentially static. They begin and end with the same consonant tone, of which both component dissonant neighbors are agents. The passing cambiata results from the elision of two successive passing-tone motions. • Listen to example 3.6a. The successive passing-tone motion from the consonant A up to the consonant D is followed by another passing-tone motion from the consonant D down to a consonant B. • Now listen to example 3.6b.

Example 3.6

Example 3.6

DISSONANCE

76

The first dissonant passing tone C is omitted. As a result of the elision of the C, the dissonant B is quit by leap. The obligation of the dissonant B to progress by step is met by the sounding of the second C. In a passing cambiata, the critical element of the second passing motion is that it recovers the leap created by the earlier elision. It may happen that the second dissonant motion is a neighbor-tone motion. • Listen to example 3.7a. The initial passing-tone motion from the consonant A up to the consonant D is followed by a dissonant lower neighbor C. • Now listen to example 3.7b. The dissonant passing tone C is again elided. Again the dissonant B is quit by leap as a result of the elision, and again the stepwise obligation of the B is met by the sounding of the lower-neighbor tone C. Example 3.7

Exercises These exercises are for developing sensitivity to the nature of neighbor cambiatas and passing cambiatas. Figures 3.9 and 3.10 are models illustrating neighbor cambiatas. The whole notes B-A represent a two-note "cantus"; the lines above the whole notes B-A (figure 3.9) and below them (figure 3.10) represent "co-lines." In each figure example a represents the underlying Figure 3.9

Example 3.6

DISSONANCE

whole-note stepwise motion of the "co-line," and example b represents the same motion ornamented by a neighbor cambiata. The model exercises should be performed by playing the "cantus" and singing the "colines": first the unadorned stepwise motion, then the five-note embellishing figure resulting from the addition of the neighbor cambiata. • As in the model exercises (figures 3.9 and 3.10), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows: 1. first an ascending or descending stepwise motion in whole notes above the "cantus," then the same motion adorned with a neighbor cambiata 2. first an ascending or descending stepwise motion in whole notes below the "cantus," then the same motion adorned with a neighbor cambiata. Figures 3.11 and 3.12 are models that illustrate passing cambiatas. Note that in this counterpoint study passing cambiatas can occur only in co-lines above the

7. See p. 94.

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cantus.7 The whole notes G-A (figure 3.11) and G-Ftt (figure 3.12) represent a two-note "cantus"; the lines above represent "co-lines." In each figure the "co-line" Ft/jure 3.11

Example 3.6

DISSONANCE

78

appears as an unadorned whole note in example a, and as adorned by a quarter-note passing cambiata in example b. Figure 3.11 represents the first type of passing cambiata, which is built with an elided successive passing-tone motion followed by another passing-tone motion. The first type of passing cambiata ornaments a stepwise motion (ascending or descending) from one bar to the next. Figure 3.12 represents the second type of passing cambiata, which is built with an elided successive passing-tone motion followed by a neighbortone motion. The second type of passing cambiata ornaments a leap of a fourth (ascending or descending) from one bar to the next. The model exercises should be performed by playing the "cantus" and singing the "colines": first the unadorned whole-note motion, then the five-note embellishing figure resulting from the addition of the passing cambiata. • As in the model exercises (figures 3.11 and 3.12), compose and perform two "co-lines" for each of the four two-note "cantus firmi" below, as follows:

3. an ascending or descending stepwise motion above the "cantus," first unadorned, then adorned with the first type of passing cambiata 4. an ascending or descending leap of a fourth above the "cantus," first unadorned, then adorned with the second type of passing cambiata. DISSONANT SUSPENSIONS

If two tones form a consonance and one is held while the other changes, the held tone may become dissonant. This held tone is called a suspension. In general, dissonant suspensions resolve downward by step. Dissonant suspensions represent the consonances to which they resolve. The consonant intervals are major and minor thirds, perfect fifths, major and minor sixths, and octaves (and their compound counterparts). When dissonant suspensions occur in the upper line, dissonant fourths resolve down to the consonant thirds they represent; dissonant sixths resolve down to the consonant fifths they repre-

DI SSON A N CE

79

sent; dissonant sevenths resolve down to the consonant sixths they represent; and dissonant ninths resolve down to the octaves they represent. • Listen to examples 3.8-3.11, playing one line and singing the other. In each of the four examples, a presents a progression

of two consonant intervals and b presents the same two intervals with the first tone of the upper line transformed by delay into a dissonant suspension, followed by its resolution. Example 3.8 illustrates a third C:E transformed into a 4-3 suspension. Example 3.9 illustrates a fifth C:G transformed into a 6-5 suspension.

Example 3.8

Example 3.10

Example 3.9

Figure 3.11

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DISSONANCE

Example 3.10 illustrates a sixth B:G transformed into a 7-6 suspension. Finally, example 3.11 illustrates an octave C:C transformed into a 9-8 suspension. When suspensions occur in the lower line, dissonant seconds resolve down to the consonant thirds they represent; dissonant fourths resolve down to the consonant fifths they represent; dissonant fifths resolve down to the consonant sixths they represent; and dissonant sevenths resolve down to the octaves they represent.

• Listen to examples 3.12-3.15, playing one line and singing the other. In each of these four examples, a again presents a progression of two consonant intervals and b presents the same two intervals with the first tone of the lower line transformed by delay into a dissonant suspension, followed by its resolution. Example 3.12 illustrates a third A:C transformed into a 2-3 suspension. Example 3.13 illustrates a fifth A:E transformed into a 4-5 suspen-

Example 3.12

Example 3.14

Example 3.13

Example 3.15

DISSONANCE sion. Example 3.14 illustrates a sixth G:E transformed into a 5-6 suspension. And example 3.15 illustrates an octave G:G transformed into a 7-8 suspension. The tendency to downward resolution can be related to the physical law requiring all objects to return to a state of stable equilibrium. In a musical experience, stable equilibrium occurs when die force of impulse is matched by a consequent force of resolution. The musical equivalent of stable equilibrium is silence, either before a musical experience when no impulse has been created, or at the end of an exquisite musical experience when all the impulse has been resolved consequently. While the work is sounding there is tension, which carries with it the obligation to return to a state of stable equilibrium-silence. The sounds can resist this obligation and create more tension or impulse, resulting in an even greater obligation to return to silence. Or they can yield to the obligation and resolve the tension. We have heard that the musical experience is unsatisfactory if the performance creates more impulse than it resolves, or if it continues to sound after all the impulse is resolved. Higher-pitched tones have greater tension than lower-pitched tones, because of the greater frequency of vibrations that creates them.8 Simply to be higher pitched, a tone must carry with it greater activity and re-

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sultant greater tension than a lower-pitched tone. When a consonant tone becomes dissonant it becomes infused widi tension, in part because of the conflict between it and the tone functioning. This functioning tone, the client tone of the suspension, has yet to be sounded. We experience the suspension as seeking its tone of resolution, seeking it as part of its obligation to rid itself of tension, and seeking it in a downward direction, away from tension. There are two dissonant suspensions that do not resolve downward by step: (1) a dissonant fifth resolving upward to a consonant sixth in the upper line, and (2) a dissonant sixth resolving upward to a consonant fifth in the lower line. In both situations, the tendency to downward resolution conflicts with the stronger tendency of our consciousness to apprehend percepts in the simplest possible way. The simplest way to understand a dissonant suspension is as an agent of a client tone one step away; this is one of the essential attributes of dissonance. When the suspension resolves to the consonant 8. This is true of tones outside any musical context. In a musical context higher-pitched tones may produce less impulse than lowerpitched tones, or vice versa, as a result of their attributes of scale degree, volume, harmonic function, linear function, and so on.

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D I SS ON A N CE

not represent the F lying one step below, because the F is itself dissonant. The F is dissonant because it creates a seventh with the E in the upper line. Instead the dissonant sixth G represents the consonant fifth A lying one step above, as represented in example 3.17b.

tone one step down, the two tendencies support each other. When the step below the suspension is dissonant and the step above it is consonant, then the stronger tendency of our consciousness to understand dissonances as clients of tones one step away overcomes the weaker tendency to downward resolution. • Listen to example 3.16a. The dissonant fifth A in bar 2 of the upper line could not represent the G lying one step below because the G is itself dissonant, for it creates a fourth with the D in the lower line. Instead the dissonant fifth A represents the consonant sixth B lying one step above, as represented in example 3.16b. • Now listen to example 3.17a. The dissonant sixth G in bar 2 of the lower line could

These exercises are for developing sensitivity to the nature of dissonant suspensions. In figures 3.13 and 3.14 the whole notes E-D-C represent a three-note "cantus"; the tones C-B-A sounding above (figure 3.13) and below (figure 3.14) represent "co-lines." In each figure, example a gives the "co-line" in its functional form, sounding concurrently with the cantus. Example b gives the same line delayed

Example 3.16

Example 3.17

Exercises

83

DISSONANCE figure 3.13

compose and perform two "co-lines" for each of the two three-note "cantus firmi" below, as follows: 1. a descending three-note "co-line" above 2. a different descending three-note "co-line" above 3. a descending three-note "co-line" below 4. a different descending three-note "co-line" below.

Figure 3.14

SECOND

by a half note, which creates suspensions. The model exercises should be performed by playing the "cantus" and singing the "co-lines": first the tones of the "co-line" concurrently with those of the "cantus," then delayed by a half note, so that dissonant suspensions are formed. • As in the model exercises (figures 3.13 and 3.14),

SPECIES

Second species exercises are two notes against one: each tone of the cantus is accompanied by two tones in the co-line. Second species co-lines are notated in half notes. Second species exercises introduce dissonant passing tones and neighbor tones.9 9. Traditionally, counterpoint manuals do not introduce the neighbor tone until the third species. For pedagogical reasons I prefer to introduce in the same species passing tones as well as neighbor tones, both ot which are approached and quit by step.

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Characteristics of Second Species Counterpoints 1. Each tone of the cantus is accompanied by two tones in the co-line. 2. The first tone of each bar must be consonant with the cantus tone. The second may be consonant, or it may be dissonant. 3. All dissonances must be approached and quit by step. Therefore passing tones and neighbor tones are the only allowable dissonances. • Listen to example 3.18. The dissonant D is a passing tone, effecting a stepwise motion from the consonant C to the consonant E. The dissonant F is an upper neighbor tone, embellishing the two E's. To repeat: the only dissonances allowed are pass-

D I S S O N A N CE

ing tones or neighbor tones; dissonances may occur only in the second half of a bar. If the second tone of a bar is dissonant then the first tone of the next bar must be approached by step, because a dissonance must be quit by step. 4. Consonances on the second part of a bar must be approached by leap. Leaps may also occur over the bar line. If the second tone of a bar is consonant, it must be approached by leap (remember that a fifth and sixth occurring in the same bar cannot both be consonant).10 In general, leaps will occur from the first tone in the bar to the second; leaps over the bar line will occur less frequently.': 5. For convenience, the second tone of a bar in the coline may be tied to the first tone of the next bar. This may occur no more than once in any exercise. As illustrated in example 3.19, the earlier of the two

Example3.18 10. When the co-line leaps between consonances within a bar, the cantus tone is still accompanied by a single harmonic event. Instead of a single interval, however, that event is a triad (complete or incomplete). For a more detailed discussion of triads see p. 115. 11. A leap that participates in the recovery of a larger leap may occur over the bar line. Also, if the first bar of the co-line begins with a rest then the first tone of the co-line may leap freely over the bar line.

DISSONANCE

tied tones (occurring in the second half of a bar) must be consonant. The latter (occurring on the first half of the following bar) is a suspension; it may be consonant, or it may be dissonant. If it is consonant it is treated like every other consonant first tone. If it is dissonant it must resolve by step to the consonant tone it represents. This is the only Example 3.19

Example 3.20

12. Sec the discussion of dissonant suspensions on p. 78.

occasion in second species counterpoint in which the first tone of the bar is not consonant, and it is the only occasion in which a dissonance is not approached by step.12 6. Second species co-lines permit more than four consecutive steps. In such a succession in the second species, consonant tones alternate with dissonant tones. This weakens the tendency for the succession of steps to unfold a single harmonic interval. An ascending or descending motion of more than four consecutive steps unfolds a harmonic interval between the first and last tones of the motion when each step is consonant. • Listen to example 3.20a. The upper line participates in a first species tex-

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ture of note against note that renders each upperline tone consonant. The stepwise ascent from C to A unfolds a harmonic sixth C:A. • Now listen to example 3.2Ob. The upper line is identical, but it accompanies a cantus in a second species texture of two against one. Because of the alternation between dissonant and consonant tones, it does not unfold a harmonic sixth C:A. 7. Linear augmented fourths present no problems if at least one of the component tones is dissonant. • Listen to example 3.2la. The Bl> in the upper line is highlighted as the lowest tone of a descending motion that contains an E; Bl> and E together form an augmented Example 3.21

D I S S ON A N C E

fourth. This presents no problems, however, because the E is dissonant. • Now listen to example 3.21b. This example is problematic because both E and Bl> are consonant. If both component tones of an augmented fourth are consonant, each is perceived along with its unfulfilled obligation to resolve. • Finally, listen to example 3.21c. If the highest and lowest tones of an ascending or descending motion form an augmented fourth, the result is a problematic dissonant contour, even if one of the tones is dissonant. A co-line sounding below the cantus may not leap a fourth or a fifth within a bar. • Listen to example 3.22a.

DISSONANCE

In the first bar the upper-line A of the cantus is consonant by definition, as is the first tone of the co-line, F. The lower-line C too must be consonant, for it is approached by leap. If F, C, and A are all consonant, then either the C creates a dissonant fourth in the context of the consonant F that precedes it, or the harmonic motion progresses by the half note. Because neither is possible in counterpoint exercises, descending leaps of a fourth (or ascending leaps of a fifth) are not possible in co-lines below. In the third bar the upper-line D of the cantus is consonant by definition, as is the first tone of the bar in the co-line, D. The lower-line G too must be consonant, for it is approached by leap. If both D and G are consonant, then either the D Example 3.22

87

creates a dissonant fourth in the context of the functional consonant harmonic event G:D, or the harmonic motion progresses by the half note. Again neither is possible in counterpoint exercises, and therefore descending leaps of a fifth (or ascending leaps of a fourth) are not possible in co-lines below. No such prohibition governs leaps between bars, because the two tones describing a fourth or fifth do not combine with the same cantus tone to form a single harmonic event. • Listen to example 3.22b. The co-line F of bar 1 leaps down a fourth to C in bar 2. The F combines with the cantus F to form the unison F:F; the C combines with the cantus G to form the consonant perfect fifth C:G.

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9. As in the first species, parallel perfect fifths or octaves between tones of successive bars interrupt the line. • Listen to examples 3.23a-3.23c. All octaves in successive bars interrupt the line and are problematic, as illustrated in example 3.23a. Also problematic are all parallel fifths between tones sounding in the first half of successive bars, as in example 3.23b, and all parallel fifths between successive tones, as in example 3.23c. Parallel fifths present no problems if they do not occur between successive tones and at least one is dissonant. • Listen to example 3.23d.

Example 3.23

DISSONANCE The G of the first bar is a dissonant passing tone, representing the F before it. Thus the fifth G:D does not combine with the consonant fifth F:C of the following bar to interrupt the linear progress. 10. Four successive like intervals can still fuse the lines. But if the succession of four like intervals is interrupted by the sounding of any other consonant interval, such a fusion does not occur. • Listen to example 3.24a. Bars 1-4 unfold a succession of uninterrupted tenths and thirds from the Fit: A of bar 1, through the E:G of bar 2 and the D:Ftt of bar 3, to the Cti:Eofbar4. • Now listen to example 3.24b.

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DISSONANCE The sounding of the fifth B:Ftt in bar 3 interrupts the succession of thirds, and thus prevents any fusion of the lines. 11. Unisons may occur, but only in the second part of the bar. • Listen to examples 3.25a and 3.25b. Unisons in the first part of the bar, as illustrated in example 3.25a, have the same deleterious effect as unisons in first species do: they weaken the inde-

pendence of the lines. Unisons in the second part of the bar, as illustrated in example 3.25b, do no such thing, because there is no question that the tone sounding on the second part of the bar participates in the co-line. There is still no restriction on unisons beginning or ending the counterpoint, for they do not compromise the independence of the lines. 12. The metric consideration of the first or second half

Example 3.24

Example 3.25

DISSONANCE

90

of the bar can influence the tendency to create or resolve impulse. Second species co-lines have two notes in each bar. The tendency to create impulse is greater when the height of the impulse occurs on the second half of the bar (note that ascending leaps, which suggest impulse, generally approach tones on the second half of the bar). 13. Beginning. The first bar of a second species co-line may contain two half notes, or a half rest followed by a half note. In either case the first tone of colines above the cantus may be the tonic or fifth, and the first tone of co-lines below the cantus must be the tonic. 14. Ending. The last bar of the co-line sounds the tonic as a single whole note. The penultimate bar may also contain a whole note, or it may contain two half notes.13 Review of Common Problems There are two harmonies in a bar when the tones in a single bar of the co-line form the intervals of a fifth and a 13. The penultimate bar often presents the syncopation figure characteristic of fourth species. Thus it would contain two half notes, the first of which is tied to the preceding bar.

sixth with the cantus, where both intervals are necessarily consonant. This prevents the cantus tone from participating in a single, consonant harmonic interval, as is required. • Listen to example 3.26a. This example illustrates two harmonies in a bar. The upper-line E is necessarily consonant, because only consonances may be quit by leap. The D must also be consonant, because the first tone of the bar must be consonant in second species co-lines. The two tones participate in conflicting harmonic events, creating a harmonic motion by the half note. Harmonic motion is however tied to the linear progress of the cantus: each tone of the cantus must represent but a single harmonic interval. • Now listen to examples 3.26b and 3.26c.

Example 3.26

91

DISSONANCE

Neither example is problematic. In example 3.26b the D is consonant; the E, which is approached and quit by step, functions as a dissonant passing tone. Thus the cantus G participates in the single harmonic interval G:D. In example 3.26c the E is consonant; the D, approached and quit by step, functions as a dissonant passing tone. Thus the cantus G participates in the single harmonic interval G:E. Counterpoint Write-Through The write-throughs illustrating the composition of twopart counterpoints in second species above and below begin on page 194. Exercises • Using one of your cantus firmi, compose one second species exercise in which the co-line sounds above the cantus. • Using one of your cantus firmi, compose one second 14. Two different co-line tones in the same bar may form consonances with the cantus tone (except if one is a fifth and the other a sixth). As in second species co-lines, the cantus tone participates in a single harmonic event-a triad instead of a single interval.

species exercise in which the co-line sounds below the cantus. THIRD

SPECIES

Third species exercises are four against one: each tone of the cantus is accompanied by four in the co-line. Third species co-lines are notated in quarter notes. Third species exercises introduce successive stepwise dissonances and dissonances approached or quit by leap. Characteristics of Third Species Counterpoints 1. Each tone of the cantus is accompanied by four tones in the co-line. 2. The first tone of each bar must be consonant with the cantus. The other three may be consonant or dissonant.14 3. The passing cambiata figure may begin on the first, second, or third quarter note; the neighbor cambiata figure may begin on the first or second quarter note.15 15. In other words, the first note of the passing cambiata may sound on the first, second, or third quarter note; the first note of the neighbor cambiata may sound on the first or second quarter note.

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• Listen to example 3.27a. The passing cambiata figure begins with the D on the third quarter note of the first bar; the E participates in an incomplete passing motion and is followed by the leap to the consonant G of the second bar. The passing tone Fit of the second bar recovers the leap from the dissonant E. • Now listen to example 3.27b. The passing cambiata begins on the second quarExample 3.27

D I S S ON A N C E

ter. The consonant F is followed by the passing tone G, which leaps to the consonant fit. The consonant A of the following bar recovers the leap from the dissonant G. Neighbor cambiatas can begin on the second quarter, but not on the third. • Listen to example 3.27c. A neighbor cambiata begins with the consonant F on the second quarter note. The elided upper and

93

DISSONANCE lower neighbor motions are completed by the sounding of the consonant F in the beginning of the next bar. • Now listen to example 3.27d. This example illustrates a problematic neighbor cambiata, which begins with the third quarter note, D. The dissonant lower neighbor Ctt is followed by a leap to what should be a dissonant upper neighbor, E; instead it is necessarily consonant, for it falls on the first quarter of the bar. And what should be a return to the consonant D is a dissonant lower neighbor. No cambiatas, neighbor or passing, may begin on the fourth quarter note of a bar. A key feature of both passing and neighbor cambiatas is the leap from the dissonant second tone. If the cambiata began on the fourth quarter note, the second tone of the cambiata would fall on the first quarter note of the following bar, and would necessarily be consonant. 4. All dissonances must be approached and quit by step, either actually or functionally. Although cambiata figures involve dissonances either approached or quit by leap on an immediate level, functionally

all such dissonances have a stepwise relation to the tones before or after. In passing cambiatas the dissonant second tone is approached by direct stepwise motion and quit by indirect stepwise motion. In neighbor cambiatas the first neighbor tone is approached by direct stepwise motion and quit by indirect stepwise motion; the second neighbor tone is approached by indirect stepwise motion and quit by direct stepwise motion. • Listen to example 3.28. The upper line presents a passing cambiata figure in which the third tone, the dissonant Bt>, is not approached by stepwise motion, direct or indirect. Although there is nothing inherently "wrong" or unmusical about such an uncovered dissonance, it creates a significant surge of impulse, and invests Example 3.28

DISSONANCE

94

the B!> with a linear attribute different from that of any of the other tones. This prevents the Bl> from participating discretely in the line. 5. Passing cambiatas can occur only in co-lines above the cantus. • Listen to example 3.29. The co-line below the cantus presents a passing cambiata. The cantus Bl> is consonant by definition, as is the first quarter note, G. According to the definition of a passing cambiata, the D approached by leap must be consonant. But the D creates a dissonance with the consonant harmonic event, which includes die consonant G. Therefore the passing cambiata cannot occur in co-lines below the cantus. No such restriction applies to the neighbor cambiata. 6. Leaps over the bar line may have a tendency to inExample 3.29

terrupt the linear progress in third species co-lines. The last tone of a bar is metrically the weakest; the first tone of the bar is the strongest, by virtue of its sounding together with the cantus tone. A leap effects a separation of tones. When the leap is from die weakest to the strongest, the separation may become great enough to threaten die linear connection of the tones. If the co-line leap over the bar line accompanies a leap in the cantus, the tendency to result in interruption is increased. • Listen to example 3.30a. The lower line leaps over the bar line from the consonant F up to a consonant Bl>. Because it accompanies a step in the cantus, the leap over the bar line does not seem to threaten die linear progress of the co-line. • Now listen to example 3.30b. Example 3.30

D I S S O N A N CE

The co-line leaps over the bar line up a sixth to a consonant D. Because it accompanies a leap in the cantus up to an A, this time the leap over the bar line does seem to effect an interruption of the linear progress of the co-line. 7. Octave unfoldings are appropriate to third species co-lines. The stepwise unfolding of an octave tends to result in the perception not of stepwise activity, but of a single harmonic event. Third species colines have an abundance of tones; it is easier to maneuver the mass of individual tones if they participate in larger harmonic structures.16 8. Some octaves or perfect fifths occurring in two consecutive bars interrupt the linear progress, and some do not. Where the third species co-line forms the same perfect, stable consonance with the cantus in consecutive bars, the disruptive phenomenon of parallel octaves or fifths does not occur if (a) neither stable consonance occurs on the first quarter, 16. The stepwise unfolding of an octave in third species extends a single tone over eight quarter notes, or two tones of the cantus. The same stepwise unfolding of an octave in second species may be problematic, because it extends a single tone over eight half notes, or four tones of the cantus.

95

and (b) both of the co-line components of the stable consonances are approached and quit by step. • Listen to example 3.3la. Octaves occur between a co-line and cantus in two successive bars (C:C in bar 1, B:B in bar 2), but they do not disrupt the progress of either line. The octaves are not disruptive because both are incidental to the functional harmonic activity; neither octave occurs on the first beat, and both are apExmnple 3.31

DISSONANCE

96

preached and quit by step in the co-line. The same is true when perfect fifths occur in similar circumstances, as in example 3.31b. When one of the two stable consonances in consecutive bars participates as a fundamental part of the functional harmonic activity, then the parallel octaves or fifths do interrupt the linear progress. • Listen to examples 3.31c and 3.31d. In example 3.31c the tones G, Ftt, and E of the

co-line in bars 1-3 all participate as fundamental members of the functional harmonic activity. The G is approached by leap, the Ftt occurs on the first beat of the bar, and the E is quit by leap. Each forms an octave with the tone of the cantus. Any two of these bars occurring in succession would interrupt the linear progress. Likewise, in example 3.3 Id the G of the second bar is quit by leap, and assumes a position within the functional harmonic

DISSONANCE activity. The fifth BkF of the first bar followed by the fifth C:G of the second bar disrupts the linear progress. This would not be a problem if one of the fifths were dissonant. • Listen to example 3.31 e. The dissonant fifth G:D does not join with the consonant E:B of the following bar to disrupt the linear progress. On the other hand, fifths between consecutive quarter notes are problematic regardless of their degree of consonance. • Listen to example 3.3If. The dissonant fifth F:C does join with the fifth E:B to interrupt the linear progress. 9. A third species co-line may form a unison with the cantus under certain conditions. A unison may occur as die first interval in a bar only in the first and the last bars of the exercise; it may occur within a bar if approached by leap. • Listen to example 3.32a. The upper line descends by step to a unison B on the third quarter note. Unisons approached by step confuse the independence of the lines.

97

• Now listen to example 3.32b. The unison on the second quarter note is approached by leap, and no such confusion exists. 10. Metric considerations can affect the dynamic function in third species co-lines. Third species co-lines have four notes in each bar. The tendency to create impulse is greater when the height of the impulse occurs after the first quarter of the bar. 11. Beginning. The first bar of a third species co-line may contain either four quarter notes, or a quarter rest followed by three quarter notes. 12. Ending. The last bar of the co-line consists of a single whole note. The penultimate bar consists of four quarter notes.

Example 3.32

D I S S ON A N CE

98

Counterpoint Write-Through The write-throughs illustrating the composition of twopart counterpoints in third species above and below begin on page 207. Exercises • Using one of your cantus firmi, compose one third species exercise in which the co-line sounds above the cantus. • Using one of your cantus firmi, compose one third species exercise in which the co-line sounds below the cantus. F OUR TH

SPECIES

Fourth species exercises are syncopated note against note: each note of the cantus is accompanied by one note, sounding on the second half of the bar, in the coline. Fourth species co-lines are notated in half notes. The fourth species introduces the suspension. Characteristics of Fourth Species Counterpoints 1. Each tone of the cantus is accompanied by one tone in the co-line. The co-line is syncopated, so that the

functional consonant tone accompanying the cantus generally sounds in the second half of the bar. Fourth species co-lines are notated in half notes; the second half note of each bar is tied to the first half note of the following bar. 2. The syncopated rhythm creates suspensions, both dissonant and consonant. In a dissonant suspension the dissonance occurs when a consonant tone of the co-line is held, or suspended, while the cantus changes.17 This results in a dissonance on the first half of the bar. The second tone of the bar resolves the dissonance; the second tone of each bar then forms a consonance with the cantus tone. The suspended tone on the first half of the bar does not always form a dissonance with the new cantus tone; often it is consonant. When it is consonant the first tone of the bar must leap to another consonance, with one exception (see number 4, below). To repeat: in fourth species counterpoint dissonant suspensions occur on the first half of a bar and resolve to a consonance on the second half. A dissonance on the second half cannot be tied over the 17. See the discussion of dissonant suspensions on p. 78.

99

DISSONANCE

bar; it must move by step to a consonance on the first half of the next bar. 3. Fourth species is essentially a rhythmic variant of first species; in both species the cantus is accompanied by a single consonant interval. • Listen to examples 3.33a and 3.33b. A series of first species sixths (example 3.33a) is transformed rhythmically into a string of 7-6 suspensions in fourth species (example 3.33b). • Now listen to examples 3.34a and 3.34b. A series of first species thirds (example 3.34a) is transformed rhythmically into a string of 2-3 suspensions in fourth species (example 3.34b). Fourth species counterpoints lend themselves to such strings of consonant sixths or thirds: 7-6 and 4-3

in co-lines above, 5-6 and 2-3 in co-lines below. With two exceptions, all dissonant suspensions in the fourth species resolve down. The two exceptions are the 5-6 suspension in co-lines above and the 6-5 suspension in co-lines below. 4. Fourth species is quite restrictive. For convenience, the tie may be omitted on occasion. If the tie is omitted the second tone of a bar is not tied to the next bar; each of the two untied half notes sounds a different tone. Thus the second of the untied half notes always functions as it would in second species: it occurs on the first part of the bar and must be consonant. The first of the untied half notes functions in one of two ways, depending on the tied note that pre-

Example 3.33

Example 3.34

100

cedes it. If the half note preceding it is dissonant, the first untied tone must resolve downward by step. • Listen to example 3.35a. The G in bar 2 is not suspended into the next bar. The A preceding the first untied half note is a dissonant fourth that must resolve to the consonant third, G. If the half note preceding the first untied tone is consonant, that first untied half note functions as it would in second species: it may be a consonance approached by leap, or it may be a dissonant passing or neighbor tone. • Listen to example 3.35b. The G in bar 2 is again not suspended into the

Example 3.35

DISSONANCE next bar. The B preceding the first untied half note G participates in a consonant perfect fifth. Thus it is free to progress to a dissonant passing or neighbor tone, or to leap to a consonance (as it does to the consonant G). The first tied tone after the two untied ones resumes the characteristics of fourth species: it must be consonant (and must therefore be approached by leap). These brief excursions into second species may occur more than once, but should not occur so often that they compromise the essential syncopated character of the exercise. 5. The dissonant suspension is an agent of its consonant resolution. Thus while the dissonant suspension is sounding, its consonant tone of resolution is

DISSONANCE functioning as part of the functional harmonic interval. 6. Octaves between the cantus and co-line in two successive bars interrupt the linear progress, as do consonant perfect fifths. • Listen to example 3.36a. The octaves in successive bars interrupt the linear progress. • Now listen to example 3.36b. Example 3.36

101

It may look as if the perfect fifths in successive bars occur between consonant sixths on the first half of each bar, and are thus benign. The sixths are clearly dissonant, however, because the fifths are suspended, and only consonances can be suspended. In each bar the dissonant sixth resolves to the consonant perfect fifths. The succession of perfect fifths interrupts the linear progress. • Listen to examples 3.36c, 3.36d, and 3.36e.

DISSONANCE

102

Example 3.36c illustrates a problematic succession of octaves. Examples 3.36d and 3.36e illustrate two problematic successions of consonant perfect fifths: the first a string of suspensions from a dissonant fourth to a consonant fifth, the second a string of suspensions from a dissonant sixth to a consonant fifth. • Finally, listen to examples 3.36f and 3.36g. These examples illustrate problematic octaves approached by leap in successive bars. Example 3.36 (continued)

7. Unisons can occur on the second half of a bar if approached by a leap. • Listen to examples 3.37a and 3.37b. These examples illustrate unisons in fourth species co-lines above and below. In example 3.37a the co-line F effects a unison on the second half of the first bar; the suspended F participates in a consonant third when the cantus moves to D. In example 3.37b the co-line Ctt effects a unison on the second half of the second bar; die suspended Ctt participates in a consonant perfect fifth when the cantus moves to Ftt. Unisons are problematic if they occur on the second half of the bar approached by step, or if they occur on the first half of the bar. • Listen to example 3.37c. The unison D in the second bar is approached by step; it results from a 2- 1 suspension, which repre-

DISSONANCE sents a first species unison, and thus confuses the independence of the lines. • Finally, listen to example 3.37d. The unison A in the first half of bar 3 also confuses the independence of the lines. 8. The metric considerations of the fourth species affect the tendency to create impulse in an even more powerful way than those of second or third species.

Example 3.37

103

Impulse is created when the tone that functions with the cantus is delayed; and a dissonance on the first half of the bar itself propels the impulse. 9. Beginning. The first bar of a fourth species co-line may contain either two half notes or a half rest followed by a half note. 10. Ending. The last bar contains a single whole note. The penultimate bar may also contain a whole note,

104

D I S S O N A N CE

or it may contain a half note tied to the preceding bar, followed by a single half note. Counterpoint Write-Through The write-throughs illustrating the composition of twopart counterpoints in fourth species above and below begin on page 218.

Exercises • Using one of your cantus firmi, compose one fourth species exercise in which the co-line sounds above the cantus. • Using one of your cantus firmi, compose one fourth species exercise in which the co-line sounds below the cantus.

CHAPTER FOUR tf^s

Rhythmic Variety

CHARACTERISTICS OF RHYTHMIC VARIETY Constancy or rhythmic values is an essential characteristic of co-lines in the first four species. We have heard that metrical considerations affect the forces of impulse and resolution. In these successions of unchanging note values, however, the element of rhythmic variety is not present. Variety of rhythmic activity exerts a powerful influence on the forces of impulse and resolution. To reiterate, the composer's contribution to impulse and resolution is given primarily by the activity of the pitches,1 but variety of rhythmic values has a strong influence. Rhythmic variety affects the dynamic forces principally in three ways: (1) shorter note values tend to create more impulse than longer note values; (2) greater diversity between the longest and shortest note values tends to create more impulse; and (3) more successive short notes tend to create more impulse. 1. In Western music dynamic forces are often generated by factors other than the activity of pitches. This can happen, however, only if there is no pitch activity, or if the activity of the pitches is in some way constant. For example, in the first movement of Beethoven's Symphony no. 7 just before the recapitulation (bar 274), resolution is gen705

RHYTHMIC

106 LONGER AND SHORTER NOTE VALUES

Shorter note values tend to create more impulse or less resolution than longer note values do. • Listen to the succession of indefinitely pitched half notes represented in example 4.la. These are fairly neutral with regard to tendency toward impulse or resolution. • Now listen to the succession of quarter notes represented in example 4.1b.

VARIETY

In relation to the half notes, the quarter notes create somewhat more tension. This is borne out when we hear these rhythmic values applied to a configuration of pitches that itself suggests impulse. The ascending scale of example 4.2 tends to create impulse. • Listen first to example 4.2a, dien to example 4.2b. You should have found the tendency to create impulse greater in example 4.2b than in example 4.2a. • Listen to example 4.3a, keeping in mind the tendency

Example 4.1

Example 4.2

crated largely by the change in articulation from staccato to legato. In the three-bar xylophone solo that opens the third movement of Bartok's Music for String Instruments, Percussion, and Celeste, the increase in rhythmic density and volume creates impulse, which is resolved by decreased rhythmic density and volume. In most variation forms (in tonal music) the pitch framework for each variation remains

constant; impulse is generated on the broadest level by the increasing rhythmic density in successive variations. In all these examples, impulse or resolution is effected by one or more factors other than the activity of pitches. In every case, though, either there are no definite pitches or the pitch activity is in some way constant.

RHYTHMIC

VARIETY

to create impulse in the ascending octave and resolution in the descending octave. You should have found that the tendency to resolve is about equal to the tendency to create impulse. • Now listen to example 4.3b. The tendency to resolve in example 4.3b is less than in example 4.3a. In example 4.3b you should in fact have found the resolution insufficient to the impulse, regardless of the inflection of volume with which you performed it. Example 4.3

107

• Now listen to example 4.3c. You should have found the resolution too great for the impulse. The dynamic tendency is dramatically affected by changing note values. Moving from longer to shorter note values tends to create impulse; moving from shorter to longer note values tends to create resolution. • Listen to the succession of indefinitely pitched note values represented in example 4.4a. You should have heard a tendency to create impulse.

108

• Now listen to the succession represented in example 4.4b. You should have heard a tendency to resolve impulse. When occurring in conjunction with configurations of pitches, rhythmic variety does not generally suggest impulse or resolution itself but acts on the tendencies to impulse or resolution suggested by the pitches. • Listen to the ascending scale of example 4.5a and then to example 4.5b, where the note values change from longer to shorter. You should have found the tendency to create impulse significantly greater in example 4.5b than in example 4.5a. Now listen to example 4.5c, where the note values Example 4.4

Example 4.5

RHYTHMIC

VARIETY

change from shorter to longer. You should have heard that the tendency to create impulse is substantially weakened. Changing note values likewise affect the tendency to resolve impulse. Moving from shorter note values to longer ones heightens the tendency to resolve; moving from longer note values to shorter ones reduces it. • Listen again to the ascending and descending scales of example 4.6a and then to example 4.6b. You should have found the tendency to resolve in the descending octave significantly greater in example 4.6b than in example 4.6a. • Now listen to example 4.6c.

RHYTHMIC

VARIETY

You should have heard that the tendency to resolve in the descending octave is substantially weakened. DIVERSE NOTE VALUES

Range of note values also affects the tendency to create impulse: the greater the difference between the longest Example 4.6

Example 4.7

109

and shortest note values, the greater tendency to create impulse. • Listen to the succession of indefinitely pitched whole notes, half notes, and eighth-notes represented in example 4.7a. Then listen to example 4.7b, which contains only half notes and quarter notes.

HO

You should have heard greater tendency to create impulse in example 4.7a.. This is also borne out when we hear these rhythmic values applied to a configuration of pitches. • Listen to examples 4.8a and 4.8b. The duration of both passages is virtually the same. Further, in example 4.8b the tendency toward impulse created by the ascending scale is intensified by the motion to shorter note values, whereas in example 4.8a the end of the passage moves from shorter note values to longer ones. Nevertheless, because of its far greater range of note values you should have found the tendency to create impulse greater in example 4.8a. Example 4.8

Example 4.9

RHYTHMIC

VARIETY

SUCCESSIVE SHORT NOTE VALUES

The more successive short notes, the greater the tendency to create impulse. • Listen to the succession of indefinitely pitched notes represented in example 4.9a. The motion of the shorter eighth-notes gives the succession a tendency to create impulse. • Now listen to the succession of indefinitely pitched notes represented in example 4.9b. Although the succession of example 4.9b has the same duration as example 4.9a, it contains more eighthnotes and therefore has a greater tendency to create impulse.

RHYTHMIC

• Finally, listen to the succession of indefinitely pitched notes represented in example 4.9c. This succession has the same duration as the two preceding examples. Here there is even greater tendency to create impulse, for this succession contains even more eighth-notes. We can hear the same phenomenon borne out in the context of a configuration of pitches that itself tends to create impulse. • Listen to example 4.10a, which traverses an ascending octave. The passage finishes with two eighth-notes; it tends toward impulse because of the ascending motion combined with the move from longer to shorter note values. • Now listen to example 4.1 Ob, which traverses the same octave but finishes with seven eighth-notes instead of three.

Example 4.10

III

VARIETY

You should have found the tendency to create impulse greater in example 4.10b than in example 4.10a. Exercises These exercises are for developing sensitivity to rhythmic variety as it affects dynamic forces. • Compose and perform the following (using a meter of two half notes in each bar): 1. a four-bar passage, using rhythms but not pitches, that creates impulse 2. an eight-bar passage using rhythms only that begins with the four-bar passage from exercise 1 and follows it with a four-bar consequent resolution 3. the same four-bar rhythmic passage from exercise 1 with pitches added, again creating impulse 4. the same eight-bar passage from exercise 2 with pitches added, so that the four-bar impulse can be played out consequently in a four-bar resolution.

FIFT H

SPECIES

Fifth species is termed florid; it combines characteristics of second, third, and fourth species. Fifth species ex-

RHTTHMIC

772

ercises introduce the element of rhythm as a source of impulse and resolution. Characteristics of Fifth Species Counterpoints 1. The co-line accompanies the cantus using characteristics of second, third, and fourth species co-lines. 2. Rhythmic characteristics of more than one species may occur within the same bar. Thus a single bar might include two successive quarter notes followed by a half note, or a half note followed by two successive quarter notes. The half note in either case must be consonant. • Listen to example 4.11, which illustrates these two rhythmic configurations. 3. The fifth species introduces the eighth-note. Eighth-

VARIETY

notes dramatically increase the tendency to create impulse. Therefore they are limited to no more than two in succession, and may occur only in the second or fourth quarters of the bar. Although the first eighth-note may be approached by leap, each eighth must be quit by step.2 • Listen to example 4.12, which illustrates eighthnotes in a fifth species context. 4. The downward stepwise resolution of the dissonant suspension may be ornamented. These ornamentations fall into four different categories, which are illustrated in succession in example 4.13. First (bar 2), Example 4.12

Example4.ll

2. More than two successive eighth-notes, eighth-notes quit by leap, or eighth-notes on strong parts of the bar almost always create more impulse than can be resolved over the course of our brief exercises.

RHYTHMIC

VARIETY

the dissonant suspended tone may leap down a fourth or a fifth to a consonance on the second quarter before sounding the resolution on the second half of the bar. Second (bar 3), the tone of resolution may be anticipated by a quarter note. Third (bar 4), after a descending leap of a fourth or fifth to a consonance the return to the tone of resolution may be ornamented by passing eighth-notes. Fourth and finally Example 4.13

113 (bar 5), the anticipated tone of resolution may itself be ornamented by an eighth-note neighbor figure.

5. Octaves or perfect fifths between the cantus and a coline in consecutive bars are not problematic if one occurs on the second eighth-note. • Listen to example 4.14. The linear progress is not interrupted by the octaves formed by the cantus G and the first co-line G in bar 1, and then by the cantus A and the final coline A of bar 2.3 6. Beginning. A fifth species co-line may begin in the same way as either a second, third, or fourth species co-line. 7. Ending. The last bar contains a single whole note. Counterpoint Write-Through

Example 4.14

The write-throughs illustrating the composition of twopart counterpoints in fifth species above and below begin on page 228.

3. Parallel fifths or octaves between the last tone of a bar (in the coline) and the first tone of the next bar are always problematic, however.

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RH TT H MI C VARI E T T

Exercises

• Using one of your cantus firmi, compose one fifth

• Using one of your cantus firmi, compose one fifth species exercise in which the co-line sounds above the cantus.

s

Pccics ^erase in which the co-line sounds below the cantus.

CHAPTER

FIVE

Three-Part Counterpoint

THREE CONJUNCT LINES In three-part counterpoint the cantus is accompanied by two co-lines, each of which is a primordial line.1 The counterpoint exercise is a whole, resulting from the conjunction of the cantus and two co-lines. The cantus may sound in any of the three positions: as the upper line (accompanied by two co-lines below), as the middle line (accompanied by one co-line above and one below), or as the lower line (accompanied by two co-lines above). Both upper and middle lines must form a successful two-part counterpoint with the lower line. To repeat, the counterpoint exercise is a whole, resulting from the conjunction of the three primordial lines. TRIADS The conjunction of three lines in our consciousness takes place in a more complex way than the conjunction of two lines does. The conjunction of two lines yields a 1. As with co-lines in two-part counterpoints, the component tones of each co-line in three-part counterpoint are related by step, and each tone participates discretely in the line. The entire exercise must be unitarily apprehensible; unitary apprehensibility of the individual lines sounding alone is irrelevant.

775

116

single harmonic interval at any given point. The quality of each of the interval's component tones is given by its relationship with its partner,2 in other words by the interval itself. • Listen to the four intervals represented in example 5.1. Each contains a D as one of its component tones. Note the difference in the quality of the D when it participates in different harmonic intervals. In two-part counterpoint the interval between a tone and the concurrently sounding tone of the other line is quality giving. At any given point, the conjunction of three lines results in a single chord that has three harmonic intervals: one between the upper and lower lines, one between the middle and lower lines, and one between the upper and Example 5.1

2. See p. 46.

T H RE E - PART

COUNTERPOINT

middle lines. The quality of the chord is defined by the relation to the lower line of the tones of both the upper and middle lines. The relation of the tones of the upper and middle lines to each other is one of clarification, or confirmation, but it is not quality giving. To restate, the two intervals of primary importance are (a) the one between the tones of the lower and upper lines, and (b) the one between the tones of the lower and middle lines. These intervals are quality giving; the interval between the tones of the two upper lines is of secondary importance-it is clarifying or confirming in nature. The simultaneous sounding of three tones allows for the formation of triads. A triad is formed by three different pitch classes that describe two consecutive thirds. Different triadic formations have varying levels of consonance and dissonance. All formations of three different pitch classes other than triads are dissonant. Triads may be consonant, or they may be dissonant. Stable consonant triads have a perfect fifth between the lowest tone and one of the upper tones. Unstable consonant triads have a diatonic third and sixth between the lowest tone and the two upper tones. Dissonant triads have one or more tones that stand for or represent other tones.

T H RE E - P A RT

COUNTERPOINT

There are only two stable consonant triads: the major and minor | triads. • Listen to the stable consonant triads represented in example 5.2. These triads derive their stable consonant quality from the stable consonant component interval-the perfect fifth. There are three unstable consonant triads: the major, minor, and diminished 3 triads.3 • Listen to the three unstable consonant triads represented in example 5.3. These triads derive their unstable consonant quality Example 5.2

3. The diminished 3 triad is actually a dissonant triad that functions as a consonant triad in three-part counterpoint exercises, and is treated as such. Although the tones of the middle and upper lines form an augmented fourth (or diminished fifth) that seems to require resolution, the component tones do not represent other tones. See p. 164 and chapter 7, nn. 13 and 14.

117

from the two unstable consonances: each has a sixth arid third above the lowest tone. There are seven dissonant triads available from the major and minor pitch fields: the major, minor, diminished, and augmented 4 triads, the diminished and augmented | triads, and the augmented 3 triad. • Listen to the seven dissonant triads represented in examples 5.4a-5.4g. Each is dissonant because each contains one or more tones with an obligation to move-tones that are agents of the consonant clients they represent. Because dissonant tones represent client tones, dissonant triads-

Example 5.3

Example 5.4

118

triads that contain dissonant agent tones-represent consonant triads that include the consonant client tones. During the sounding of a dissonant triad, the consonant client triad is functional. Any of the 4 triads (examples 5.4a-5.4d) may represent a 3 triad, with the dissonant fourth resolving down to a third. Or they may represent | triads, with both the fourth and the sixth resolving down to a third and a fifth. Additionally, in the diminished 4 triad (example 5.4c) the lower and middle tones D and Gtt may resolve out in contrary motion, with the upper tone B moving in either direction. And in the augmented 4 triad (example 5.4d) the D may resolve up to a consonant E\>. In the diminished | (example 5.4e) both D and A\> are obligated to resolve inward in contrary motion, with the F moving in either direction. In the augmented 3 (example 5.4f) the Ajt is obligated to resolve up. The augmented f (example 5.4g) is something of a special case. Neither the major third D:Ft nor the minor sixth D:Bl> is dissonant by itself. In conjunction with each other, however, the strong ascending obligation of the Ft conflicts with the descending tendency of the Bl>, to the degree that one or the other functions as a dissonance and must resolve. Either the

T H R E E - PART

COUNTERPOINT

Bl> is dissonant and must resolve to A, or both the Ftt and D are dissonant and must resolve to G and Et. Any chord with three different pitch classes that do not form a triad is dissonant. This is true even if both upper tones form consonances with the lowest tone. • Listen to the three-note configurations represented in example 5.5. The first is dissonant-even though both the A and the B form consonant intervals with the lowest tone D-because the three tones do not form a triad. The second is dissonant for the same reason, and also because the C forms a dissonant seventh with the lowest tone D. If a tone is doubled at the unison or octave, the result is an incomplete triad. Incomplete triads have the same basis for consonance or dissonance as complete triads. Any incomplete triad with a dissonance between the lowest tone and either the middle or the upper tone is a Example 5.5

THREE-PART

COUNTERPOINT

dissonance; any incomplete triad with a sixth above the lowest tone is an unstable consonance; and any incomplete triad with a third or fifth above die lowest tone is a stable consonance. The incomplete triad is more consonant if the tone that sounds like the root is doubled, less consonant if the other tone is doubled. Therefore if the chord has a third or fifth above the lowest tone, it is more consonant if the lowest tone is doubled. If it has a sixth above the lowest tone, it is more consonant if the sixth is doubled. • Listen to examples 5.6a-5.6c. Each example presents the same incomplete triads with different doubling. In example 5.6a the root of the Example 5.6

119

incomplete triad is E; the first version in which the E is doubled is more consonant than the second version in which the B is doubled. In example 5.6b the root of the incomplete triad is A; the first version in which the A is doubled is more consonant than the second version in which the CK is doubled. Finally, in example 5.6c the root of the incomplete triad is G; the first version in which the Bl> is doubled is less consonant than the second version in which the G is doubled. Tripling a tone (sounding the same pitch class simultaneously in all three lines) can be problematic. If a tone in the succeeding or preceding chords is doubled, parallel octaves will necessarily result; thus both preceding and succeeding chords must be complete triads. The

120

T H R E E - PA RT

COUNTERPOINT

shift in texture from three different pitches to one and again to three will tend to separate the tripled tone and prevent its members from participating discretely in their lines. • Listen to example 5.7a. The change in texture effected by the tripled D surrounded by complete triads tends to interrupt the linear progress. • Listen to example 5.7b, which represents the beginning of an exercise. A tripled tone at the beginning of an exercise is less likely to threaten the linear progress.

companying co-lines to form triads (complete or incomplete). For the conjunction of the lines to take place, the distance between the lines must not be too great and there must be no conflict of modal inflection.

CONJUNCTION

Mixed Modes

In a three-part counterpoint, each tone of the cantus combines with the corresponding tones of the two ac-

In three-part exercises in the minor mode, conjunction cannot take place if there are conflicting modal inflec-

Distance Again, distance between the lines affects conjunction. The greater the distance between two lines, the weaker their relationship. A distance of more than a thirteenth between two adjacent lines, or of more than three octaves between upper and lower lines, makes it difficult to perceive the three lines as joined.4

Example 5.7

4. A large interval between the middle and upper lines is more likely to be problematic than the same interval between the lower and middle lines.

THREE-PART

121

COUNTERPOINT

tions between any two lines. If an altered sixth or seventh is sounded, no unaltered sixth or seventh may be sounded in any line before the altered tone completes its motion between tonic and dominant.

of one of the co-lines or the cantus to become dissonant, it effectively eliminates the tone from participating discretely in its line. Thus one of the lines cannot be primordial.

PRIMORDIAL LINES

Parallel Stable Consonances

We have heard that it is possible for primordial lines to be interrupted by their conjunction. Primordial lines are interrupted when not all tones participate in a consonant triad. Primordial lines can also be interrupted by parallel stable consonances and simultaneous leaps.

The first type of harmonically generated interruption occurs when the same stable consonant interval (in either its simple or its compound form) occurs between the same two lines in two or more consecutive bars. No interruption occurs, however, when parallel stabilities are sounded in consecutive bars between two different pairs of lines. • Listen to example 5.8. In this example parallel fifths occur between different pairs of lines. The first chord contains the stable conso-

Consonant Triads Each tone of the cantus must participate in a consonant triad. For a cantus to join with two other primordial lines, each tone of the cantus must combine with the accompanying tones in the co-lines to form a consonant triad, complete or incomplete. An accompaniment resulting in a dissonant triad prevents conjunction, in effect by eliminating one or more of the lines. In a dissonant triad at least one tone is obligated to resolve; it represents its consonant resolution. Because the dissonant tone represents its tone of resolution, it is effectively no tone. If the conjunction of lines causes a tone

Example 5.8

722

nant fifth D:A between the lower and middle lines; it is followed by a fifth E:B between the lower and upper lines. No interruption is effected. Parallel Leaps Linear progress is interrupted when all three lines leap in the same direction simultaneously. • Listen to example 5.9a. All three lines leap up from the second chord to the third; this effectively interrupts the linear progress. If only one line moves by step the linear progress can continue uninterrupted, even if there are parallel leaps in the other two. Example 5.9

T H RE E - PART

COUNTERPOINT

• Listen to example 5.9b. Between the first and second chords there are parallel descending leaps in the middle and upper lines; the lower line repeats the G. Between the second and third chords there are parallel ascending leaps in the lower and upper lines; the middle line moves up by step. No interruption in the linear progress is effected in either case. Note that between the first and second chords the upper and middle lines leap to a perfect fifth. Because of the secondary relationship between the two upper lines, a parallel leap to a stable consonance between tones of the two upper lines does not threaten the linear progress.

THREE - PA RT

COUNTERPOINT

• Finally, listen to example 5.9c. The parallel leap to a perfect fifth between the lower and middle lines does effect an interruption. Exercises These exercises are for developing sensitivity to the conjunction of three primordial lines. • In conjunction with the three-note stepwise descending "cantus" represented in figure 5.1, compose and perform two "co-lines" for each of the exercises described below, singing one line and playing the others (the conjunction of the three lines must result in consonant triads, complete or incomplete): Figure 5.1

1. four different three-part exercises in which the "cantus" sounds in the lower line 2. four different three-part exercises in which the "cantus" sounds in the middle line

123

3. four different three-part exercises in which the "cantus" sounds in the upper line. • In conjunction with the six-note "cantus" (sounding in any line) represented in figure 5.2, compose and perform two "co-lines" for each of the exercises described below:

Figure 5.2

4. an exercise with the sole error of parallel octaves between the same two lines 5. an exercise with the sole error of parallel fifths between the same two lines 6. an exercise with the sole error of concurrent leaps in all three lines interrupting the linear progress 7. an exercise with the sole error of parallel leaps to a stable consonance 8. an exercise in which parallel leaps to a stable consonance do not interrupt the linear progress.

T H RE E - P ART

124 INDEPENDENCE

Three-part counterpoint results from the conjunction of three independent lines. As with two-part counterpoint, four successive like intervals between the same two lines result in the subjugation of one line under the other. Likewise, crossed voices and unisons (other than in the first and last bars) compromise the independence of lines in three-part counterpoint, as they do in two-part counterpoint. Complete Triads A triad is complete if all three pitch classes are sounded. Complete triads promote independence because each line sounds its own pitch class. Exercises These exercises are for developing sensitivity to lines losing their independence in a three-part texture. • Compose and then perform without music (singing one line and playing the others): 1. a three-part counterpoint with too many consecutive thirds 2. a three-part counterpoint with as many consecutive

3. 4.

5. 6.

COUNTERPOINT

thirds as possible, without losing the independence of the lines a three-part counterpoint with too many consecutive sixths a three-part counterpoint with as many consecutive sixths as possible, without losing the independence of the lines a three-part counterpoint with crossed voices a three-part counterpoint with a problematic unison.

UNITARY APPREHENSION

A counterpoint is apprehended as unitary when the resolution matches the impulse. As in two-part counterpoint, the direction of the upper-line motion generally determines the tendency to create or resolve impulse. There is a tendency to create impulse when the upper line ascends and to resolve impulse when the upper line descends. Exercises The following exercises are for developing sensitivity to impulse and resolution within three-part counterpoints.

THREE-PART

COUNTERPOINT

• Compose and perform without music: 1. two successive triads (complete or incomplete) resulting in impulse 2. three successive triads (complete or incomplete) resulting in impulse 3. four successive triads (complete or incomplete) resulting in impulse. • Adding to each of your three- and four-triad impulses (exercises 2 and 3, above), compose and then perform without music: 4. a resolution that is too small 5. a resolution that is too large 6. a resolution that is consequent.

The following exercises are for developing an awareness of the performer's contribution to impulse and resolution. • Both of your impulses with consequent resolutions (exercise 6, above) can also be performed in such a way that the resolution is not consequent to the impulse. Perform these four fragments so that the resolution is either too great or too small for the impulse.

125

FIRST

SPECIES

Three-part exercises accompany the cantus with two primordial co-lines. The cantus may sound as the lower line, the middle line, or the upper line. Three-part exercises in first species are note against note: each tone of the cantus is accompanied by one tone in each of the co-lines. There are no dissonances in first species exercises. Each tone of the cantus participates in a consonant triad (complete or incomplete) with the corresponding tones of the co-lines. Guidelines for All Three-Part Counterpoints 1. Beginning. The exercise must begin with a stable consonant triad (complete or incomplete). The lower line must begin on the tonic; the middle and upper lines may begin on the third or the fifth, and only the tonic may be doubled. This is because the exercise must first establish the most stable referent, and the stability of an incomplete triad is weakened if a tone other than the root is doubled. Thus the first bar must contain a tonic triad in root position, either complete or incomplete; neither third nor fifth may be doubled.

126

2. Ending. The exercise must end with a stable consonant triad (complete or incomplete). The lower line must end on the tonic; the middle and upper lines may end on the third or fifth. If the triad is incomplete the tonic must be doubled. As with the first chord, doubling either the third or the fifth of the final chord threatens its stability. (It is impossible to triple the tonic without creating parallel octaves or an unrecovered leap.) 3. Each tone of the cantus must participate in a single, consonant triad. The co-lines must join with the cantus so that each tone of the cantus participates in not more and not less than one triad, which is consonant. 4. Any co-line that accompanies the cantus in a first species texture of note against note may tie two successive notes on more than one occasion. The number of such ties is limited only by the ability of each tone to participate discretely in the linear progress. Counterpoint Write-Through The write-throughs illustrating the composition of three-part counterpoints in first species begin on page 239.

T H RE E - P ART

COUNTERPOINT

Exercises • Using your cantus firmi, compose and perform without the music two of the following: 1. a three-part exercise in first species in which the cantus sounds as the lower line 2. a diree-part exercise in first species in which the cantus sounds as the middle line 3. a three-part exercise in first species in which the cantus sounds as the upper line. SECOND

SPECIES

In three-part exercises in second, third, and fourth species, the cantus is accompanied by one active co-line, which unfolds the appropriate species, and one neutral co-line, which accompanies the cantus in a texture of note against note. Thus in a three-part exercise in second species, die cantus is accompanied by one active coline in second species and one neutral co-line (first species). Second, diird, and fourth species co-lines in threepart counterpoint are almost identical to those in twopart counterpoint. The sole difference is in their last bar, which may contain a fifth or a third. In three-part counterpoint (other than infirstspe-

T H R E E - P ART

127

COUNTERPOINT

cies), the active co-line may begin with a rest. If the cantus is in the upper line and the active co-line is in the lower line, the neutral co-line in the middle line must begin with the tonic. Otherwise the first sounded event would not have the tonic as the lowest tone, and therefore would not be the most stable referent. Review of Common Problems A leap to a dissonance occurs when the goal of a leap is dissonant with the prevailing consonant triad. Threepart exercises consist of successions of consonant triads, complete or incomplete, and not of harmonic intervals as in two-part exercises. When tones leap within a bar they must leap into a consonant triad-a triad that is consonant with the triad sounded in the first half of the bar. In other words, tones on the second half of the bar that are approached by leap must be consonant not only with the cantus, but with the consonant harmonic event as established on the first half note. • Listen to example 5.10. Assuming that the cantus is in the lower line, the cantus C participates in a complete C major triad in root position. The upper-line leap down to A results in a dissonant chord C:G:A; this is dissonant with the func-

tional consonant C major triad, even though the A would be consonant with the cantus if the two lines sounded alone. Counterpoint Write-Through The write-throughs illustrating the composition of three-part counterpoints in second species begin on page 247. Exercises • Using your cantus firmi, compose and perform without the music two of the following: 1. a three-part exercise in second species in which the cantus sounds as the lower line Example 5.10

128

T HRE E - PART

2. a three-part exercise in second species in which the cantus sounds as the middle line 3. a three-part exercise in second species in which the cantus sounds as the upper line.

THIRD

SPECIES

In three-part exercises in third species the cantus is accompanied by one active co-line in third species and one neutral co-line (first species).

COUNTERPOINT

Exercises • Using your cantus firmi, compose and perform without the music two of the following: 1. a three-part exercise in third species in which the cantus sounds as the lower line 2. a three-part exercise in third species in which the cantus sounds as the middle line 3. a three-part exercise in third species in which the cantus sounds as the upper line.

Counterpoint Write-Through

F O UR T H

SPECIES

The write-throughs illustrating the composition of three-part counterpoints in third species begin on page 258.

In three-part exercises in fourth species the cantus is accompanied by one active co-line in fourth species and one neutral co-line (first species).

Example 5.11

THREE -PART

COUNTERPOINT

Characteristics of Fourth Species Counterpoints 1. Three-part counterpoints in fourth species are essentially rhythmic variants of three-part counterpoints in first species. In two-part counterpoint dissonant suspensions represent and resolve to client tones that participate in consonant harmonic intervals. By contrast, dissonant suspensions in three-part counterpoint represent and resolve to client tones that participate in consonant triads, complete or incomplete. Although a dissonant suspension is sounding in the first half of the bar, the functional triad includes its client tone of resolution. • Listen to examples 5.11 a-5.lid. Each example represents the same two-chord segment of a three-part counterpoint. Example 5.1 la represents the two chords in a first species texture of note against note. The same succession of chords in a fourth species texture is represented in examples 5.1 Ib, 5.1 Ic, and 5.1 Id, in which the active co-line sounds respectively in the upper, middle, and lower lines. In example 5.lib the upper-line F is suspended over the bar line, where it becomes dissonant in the context of the functional triad G:B:E to which it resolves. In example 5.lie the middle-line C is sus-

129

pended over the bar line, where it becomes dissonant in the context of the functional triad G:B:E to which it resolves. In example 5.1 Id the lower-line A is suspended over the bar line, where it becomes dissonant in the context of the functional triad G:B:E to which it resolves. Although examples 5.1 la-5.1 lei each represent a succession of first-inversion triads, dissonant suspensions may represent and resolve downward by step to any consonant triads, complete or incomplete. Dissonant suspensions may not resolve to a dissonant harmonic event, because the functional harmonic event must be consonant. Because a dissonant suspension represents and resolves to its client, the client tone participates in the functional harmonic event of the bar and must be consonant. • Listen to example 5.12.

Example 5.12

130

THREE-PART

The upper-line E is suspended into the next bar, where it forms a dissonance with the F of the lower line. Although the D to which it resolves is consonant with the lower-line F, it participates in a dissonant harmonic event in conjunction with the F and C together. The upper-line E may not be suspended in this case. Dissonant suspensions may resolve up in two cases. First, when the co-line sounds in either the middle or the upper line a dissonant fifth above the tone of the lower line may resolve up to a consonant sixth. • Listen to examples 5.13a and 5.13b. In example 5.13a the upper-line G is suspended Example 5.13

COUNTERPOINT

into the next bar, where it becomes dissonant with the functioning triad C:E:A. The G could sound as part of the functioning triad, and the A as a dissonant passing tone. But because the A is quit by leap it must be consonant, and therefore must participate in the functional triad. The suspended G, which forms a dissonant fifth with the lower-line C, resolves up to the consonant sixth A. In example 5.13b the middle-line B is suspended into the next bar, where it becomes dissonant with the functioning triad E:C:G. The suspended B forms a dissonant fifth with the lower-line E, and resolves up to the consonant sixth C.

T H R E E - PA RT

COUNTERPOINT

Second, when the co-line sounds in the lower line a dissonant suspension may resolve up to the root of a root-position triad, complete or incomplete. • Listen to example 5.13c. The lower-line D is suspended into the next bar, where it becomes dissonant with the functioning triad E:B:G. The suspended D resolves up to the E, the root of the triad in root position. 3. A consonant suspension may be quit by leap, but only to a tone consonant with the functional harmonic event. • Listen to example 5.14a. The lower-line A is suspended into the next bar, Example 5.14

131

where it participates in a consonant incomplete triad with the F and A of the other two lines. If the lower line leaps to D the D must be consonant; then the functioning harmonic event is a D minor triad. When a D minor triad is the functioning harmonic event and the lowest tone is A, then die triad is a dissonant second-inversion triad. The A in this case may leap only to another A or to an F. • Listen to example 5.14b. The middle-line G is suspended into the next bar, where it participates in a consonant E minor triad. Although the C to which it leaps would be consonant with the lower-line E in a two-line context, this C is dissonant with the functioning E minor triad. Counterpoint Write-Through The write-throughs illustrating the composition of three-part counterpoints in fourth species begin on page 267. Exercises • Using your cantus firmi, compose and perform without die music two of die following:

T H RE E - PART

132

1. a three-part exercise in fourth species in which the cantus sounds as the lower line 2. a three-part exercise in fourth species in which the cantus sounds as the middle line 3. a three-part exercise in fourth species in which the cantus sounds as the upper line. FIFTH

SPECIES

A three-part exercise in fifth species consists of a cantus accompanied by one fifth species co-line and one neutral co-line (first species). Counterpoint Write-Through The write-throughs illustrating the composition of

COUNTERPOINT

three-part counterpoints in fifth species begin on page 276. Exercises • Using your cantus firmi, compose and perform without the music two of the following: 1. a three-part exercise in fifth species in which the cantus sounds as the lower line 2. a three-part exercise in fifth species in which the cantus sounds as the middle line 3. a three-part exercise in fifth species in which the cantus sounds as the upper line.

CHAPTER

Mixed Species z

SIX

£ is followed by its dissonant agent C, which combines with the consonant G in the upper line to form the consonant fifth C:G. Exercises These exercises are for developing sensitivity to the nature of consonant agents. Figure 6.1 is a model. The whole notes Clt-D (figure 6.la) represent a two-note "cantus"; the lines above (figure 6.1b) and below (figure 6.1c) represent active "co-lines." • For each of the four two-note "cantus firmi" below, compose and perform (without the music): Figure 6.1

MIXED

136

1. a three-note "co-line" unfolding a passing-tone or neighbor-tone motion, as in figure 6.1b 2. an additional "co-line," as in figure 6. Ic, that includes: (a) another passing-tone or neighbor-tone motion; or (b) a dissonant suspension resolving in the second half of the bar; or (c) a leap to a consonance. The tone in the second half of the bar must be consonant with the dissonant agent of the other coline. MULTILEVELED AGENCY

Agency creates hierarchy. Because they are representatives, or stand-ins, the agent tones do not function on the same level as their clients. In other words, they do not participate discretely in the same line. We have heard that functional harmonies consist of client tones. Their agents, even consonant agents, serve a subsidiary, ornamental role. The agents we have encountered so far create simple hierarchies: clients functioning on the

SPECIES

broadest level, agents functioning on a second, narrower level. We heard that a tone can be dissonant in one context and consonant in another.l When an agent tone becomes consonant it can itself be represented by an agent tone. The agency then becomes multileveled: on the broadest level the original client tone is represented by its agent, which functions on a second, narrower level; the agent itself becomes a client tone represented by its own agent, which functions on a third, still narrower level. • Listen to example 6.4a. The concurrent sounding of the lower line D-F, the middle line F-D, and the upper line Bl»-A results in two successive harmonies: D:F:B!> - F:D:A. • Now listen to example 6.4b. The lower line D-F is ornamented by a dissonant 1. Seep. 133.

MIXED

SPECIES

passing tone E, which is an agent of the D. • Finally, listen to example 6.4c. The upper-line B\> is ornamented by a neighbor cambiata. In the second half of the bar the dissonant C of the upper line joins with the dissonant E of the lower line to form a consonant sixth. In the context of the original functional harmony D:F:Bl>, both the E and the C are dissonant agents (of D and Bl>). In the context of each other these are both consonant. The C, consonant in the context of the E below it, is then represented by

137

its own dissonant agent, Bl>. 2 As a result there are three levels: (a) the broadest level of consonant client tones; (b) a second level, on which the original client tones D and B!> are represented by their agents E and C; and (c) a third level, on which the agent tone C is itself represented by its own dissonant agent, Bl>. In example 6.4c the consonant agents E and C sound together on the third quarter note. The dissonant agent Bl> sounds on the fourth quarter note, after its consonant client. It is also possible for a dissonant agent of

Example 6.4 2. That the Bl> would have been consonant with the harmony created by the original client tones is of no consequence and does not render the Bt consonant, because we hear it as functioning in its immediate context. This is true even though the original harmony is still

functional on the broadest level. A case might be made that we hear the second Bt as essentially the same tone as the first Bt, but because the immediate context has changed, the second Bl> functions as a dissonance.

138

an agent to sound together with the other consonant agent, before its consonant client has sounded. • Listen to example 6.5a. The concurrent sounding of the lower line D-B, the middle line B-G, and the upper line Ftt-B results in two successive harmonies: D:B:Ftt-B:G:B. • Now listen to example 6.5b. The middle line B-G is ornamented by a passing A. • Finally, listen to example 6.5c. The upper line Fit-B is ornamented by a passing cambiata. The middle-line A becomes consonant in the context of the upper-line A; the upper-line B on the third quarter note functions as a dissonant agent of the A that follows.

MIXED

SPECIES

Exercises These exercises are for developing sensitivity to multileveled agency. Figure 6.2 is a model. The whole notes D-Bl> (figure 6.2a) represent a two-note "cantus"; the "second species" line below (figure 6.2b) and the "third species" line below that (figure 6.2c) represent active "co-lines." • For each of the four two-note "cantus firmi" below, compose and perform (without the music): 1. a three-note "second species co-line" unfolding a passing-tone or neighbor-tone motion, as in figure 6.2b 2. an accompanying five-note "third species co-line" in which one of the two quarter notes in the second

Example 6.5

MIXED

139

SPECIES

half of the bar becomes consonant in the context of the second species dissonance, and is itself represented by the other quarter note in the second half of the bar, as in figure 6.2c.

tional harmonic event is sounded first; then one of the consonant tones is replaced by its dissonant agent. • Listen to example 6.6.

IMPLIED HARMONY

In the dissonances we have encountered so far, the agent tones sound either after the client tones or before the client tones. In either case the functional harmonic event is explicit: it sounds before or after the sounding of the agent. In passing and neighbor motions the func-

Example 6.6

Figure 6.2

140

MIXED

The functional triad C:E:A is sounded explicitly before the upper line sounds the dissonant agent B. In dissonant suspensions the suspended tone is a dissonant agent of the tone to which it resolves. Throughout the sounding of the suspension the client tone is functioning-the functional harmonic event includes the client tone. With the resolution of the dissonant suspension to its consonant client, the functional harmonic event is sounded explicitly. • Listen to example 6.7. The dissonant suspension B in the upper line represents the A to which it resolves. Thus the functional harmony throughout the first full bar is A:E:A. With the sounding of the resolution to A, the functional harmony A:E:A is explicitly sounded.

Example 6.7

SPECIES

It can happen that a functional harmony is never explicitly sounded. If in a three-part texture one component tone of a functional harmony is represented by a dissonant suspension, and another component tone of the functional harmony is quit before the suspension is resolved, then the functional harmony is never sounded explicitly-it is merely implied. • Listen to example 6.8. The functional harmony is the same as that of example 6.7. The sole difference is that the lower-line motion from A to E is now accomplished by ascending stepwise quarter notes: dissonant passing tone B, consonant Ctt, dissonant passing tone D. When the functional A of the upper line finally sounds in the second half of the bar, the functional A of the lower line is no longer sounding.

Example 6.8

MIXED

SPECIES

141

Figure 6.3 Thus the harmony A:E:A is never explicitly sounded, although it functions throughout the bar. A resolution can never be completed by an implied harmony. Implied harmony inhibits resolution, and thus generates impulse. Exercises These exercises are for developing sensitivity to implied harmony. Figure 6.3 is a model. The whole notes D-C (figure

6.3a) represent a two-note "cantus"; the "fourth species" line above (figure 6.3b) and the "third species" line below (figure 6.3c) represent active "co-lines." • For each of the four two-note "cantus firmi" below, compose and perform (without the music): 1. a three-note "fourth species co-line" unfolding a passing-tone or neighbor-tone motion, as in figure 6.3b 2. an accompanying five-note "third species co-line" such that the functional harmony is never explicitly sounded but merely implied, as in figure 6.3c.

742

COMPOUND MELODY

In our initial study of line we encountered the phenomenon of compound melody: two linear motions functioning concurrently within a single-line texture.3 Line Example 6.9

3. See p. 7.

MIXED

SPECIES

in its pure state, however, is singular: each component tone must participate discretely in a single linear motion. Thus in directing our study to the essential nature of line, we eliminated compound melody.

MIXED

143

SPECIES

That does not imply that compound melody is bad in any way. What it does mean is that a compound melody consists of two (or more) concurrent linear motions, each of which functions according to the essential nature of line. Having acquired a grasp of this essential nature, we can now turn to the common phenomenon of two linear motions functioning concurrently within the same melodic line. Examples 6.9a-6.9c illustrate two lines condensed into a single-line texture. • Listen to example 6.9a. A cantus in the lower line is accompanied by two colines, each of which is the skeletal framework for a fifth species co-line. Example 6,9 (continued)

4. These two fifth species co-lines include whole notes, borrowed from the first species. See p. 152.

• Now listen to example 6.9b. Each skeletal framework from example 6.9a is fleshed out into a fifth species co-line.4 • Finally, listen to example 6.9c. The two fifth species co-lines are combined into a single line, yielding a two-part counterpoint. Exercises These exercises are for developing sensitivity to two linear motions functioning concurrently within a singleline texture. The six-note line given below is a "cantus" for your exercises. • In a three-part texture with the "cantus" as the lower

MIXED

144

line, compose and perform (without the music): 1. two "co-lines" serving as fifth species skeletal frameworks 2. two fifth species "co-lines" based on the skeletal frameworks of exercise 1, above (you may use whole notes and dotted half notes) 3. a condensation into a single line of the two fifth species "co-lines" from exercise 2. • In a three-part texture with the "cantus" as the upper line, compose and perform (without the music): 4. two "co-lines" serving as fifth species skeletal frameworks 5. two fifth species "co-lines" based on the skeletal frameworks of exercise 4, above (you may use whole notes and dotted half notes) 6. a condensation into a single line of the two fifth species "co-lines" from exercise 5.

TYPE

1

In type 1 mixed species exercises the cantus is accompanied by two second species co-lines. As in all mixed

SPECIES

species exercises the cantus may sound as the lower, middle, or upper line. Type 1 mixed species exercises introduce the consonant agent. Guidelines for All Mixed Species Counterpoints Mixed species counterpoints are essentially similar to three-part counterpoints. 1. Both active co-lines must join with the cantus so that each tone of the cantus represents not more and not less than one functional consonant triad. 2. 'Beginning and ending. All considerations for beginning and ending mixed species exercises are the same as for simple three-part exercises. Characteristics of Type 1 Mixed Species Counterpoints 1. Each of the two co-lines has a relation of two against one with the cantus, and must accompany the cantus as the active co-line does in a simple three-part exercise in second species. Thus in each bar the first half note in each co-line must participate in a consonant triad (complete or incomplete). The second half note of the bar in each co-line may be consonant with the functional triad, or it may be a dissonant passing tone or neighbor tone. 2. The two co-lines also have a relation of note against

MIXED

SPECIES

note with each other, and must meet the requirements of consonance appropriate for a first species exercise. Thus the second half note must participate in a consonant harmonic event with the second half note in the other co-line (even if either or both are dissonant with the cantus). • Listen to example 6.lOa. The upper and middle lines are ornamented with dissonant passing tones, which together form a dissonant perfect fourth A:D in the second half of the bar. This dissonance requires a resolution that cannot occur; this configuration of parallel fourths is there -

Example 6.10

145

fore unacceptable (even though both dissonant passing tones function appropriately in the context of the lower-line cantus). A perfect fourth may sound between two co-lines in the upper and middle lines, but only if both components of the fourth participate in a consonant harmony. • Listen to example 6.10b. The functional harmony of the first bar is D:D:A. The upper line sounds a dissonant passing tone B, which represents A. The middle line leaps from D to an Fit, which is consonant with the cantus D. The two co-lines join in the second half of the bar to form a perfect fourth Flt:B. This fourth is perfectly acceptable because it is consonant: it participates with the D of the lower line (still sounding) in a consonant triad D:Flt:B. Although this triad is consonant, it does not replace D:D:A as the functional harmony of the bar. 3. In either co-line a tone may be suspended over the bar line. The suspended figure is borrowed from fourth species, and has the familiar considerations for consonance and dissonance. Although the number of occurrences is limited onlv by the need to maintain a

146

MIXED

SPECIES

steady half-note motion, it may occur in only one coline at a time. 4. Dissonances may be approached and quit only by step. A chord in the second half of the bar that is dissonant with the functional harmony of the bar is dissonant, even if in sounding alone it would be consonant. • Listen to example 6.11. Although the lower-line F participates in a consonant triad F:A:C, it is dissonant in the context of the functional harmony E:G:C, and may not be quit by leap. 5. Octaves or perfect fifths may sound between the two active co-lines if separated by a single interval.

• Listen to example 6.12. The upper and middle lines join to form a perfect fifth D:A, a sixth A:F, and a perfect fifth C:G. Because the two lines function in a first species context in relation to each other, no interruption is effected. This is true even though the fifth D:A and the fifth C:G both participate in their respective functional harmonies. 6. Four successive like intervals can sound between the two co-lines. This is because the nature of the harmonic context changes. • Listen to example 6.13a. Beginning with the second half note D:Ftt, the lower and upper lines sound four consecutive tenths

Example 6.11

Example 6.12

MIXED

SPECIES

without threatening the independence of the lines. This is due to the alternating dissonance and consonance of die successive thirds: the D:Ftt is dissonant with the harmonic context, the ensuing E:G participates in the consonant functional harmony, the ensuing Ftt:A is dissonant, and the ensuing G:B participates in the consonant functional harmony. • Now listen to example 6.13b. The same four thirds between the lower and middle lines are followed by yet another third, A:C. This fifth successive third causes the two lines to fuse together. Neither co-line may join with die cantus to form more than three consecutive like intervals, however, Example 6.13

14?

without compromising the independence of the lines. 7. Unisons function in mixed species exercises as they do in the appropriate species. Each co-line is in second species in relation to the cantus; thus unisons may occur between co-line and cantus on the second half note. The co-lines are in first species in relation to each other; thus unisons may not occur between the two co-lines except at the beginning and end of the exercise. Counterpoint Write-Through The write-throughs illustrating the composition of type 1 mixed species counterpoints begin on page 285.

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148

Exercises • Using your caiitus firmi, compose and perform without the music one of the following: 1. a type 1 mixed species exercise in which the cantus sounds as the lower line 2. a type 1 mixed species exercise in which the cantus sounds as the middle line 3. a type 1 mixed species exercise in which the cantus sounds as the upper line. TYPE

2

In type 2 mixed species exercises the cantus is accompanied by one second species co-line and one third species co-line. Type 2 mixed species exercises introduce multileveled agency. Characteristics of Type 2 Mixed Species Counterpoints 1. Each active co-line accompanies the cantus as it would in a simple three-part counterpoint. The second species co-line has a relation of two against one with the cantus, and accompanies the cantus as an active co-line would in a simple three-part exercise in

SPECIES

second species. The third species co-line has a relation of four against one with the cantus, and accompanies the cantus as an active co-line would in a third species exercise. 2. Another relation of two against one exists between the third species co-line and the second species coline. The relation between second species and third species co-lines is similar to the one between second species co-line and cantus. Each group of two quarter notes in the third species co-line joins with the corresponding half note in the second species co-line to form a consonant interval. (The second half note may join with either the third or the fourth quarter note to form a consonant interval.) 3. The third species co-line also makes use of cambiata figures, both neighbor and passing. The passing cambiata results from an elision of two passing motions; thus the third tone must participate in a consonant harmonic event. In type 2 mixed species exercises the third tone of the passing cambiata must be consonant either with the second species co-line or with the cantus. The neighbor cambiata has no such restrictions. (Remember that passing cambiatas cannot occur in the lower line.)

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4. Under certain circumstances, parallel stable consonances are not problematic. Stable consonances between the same two lines in successive bars do not interrupt the linear progress if they meet two conditions: (a) one of the stable consonances is both approached and quit by step and does not sound on the beginning of the bar; and (b) the sounding harmonic context for one of the stable consonances is dissonant. • Listen to example 6.14a, in which parallel octaves do not interrupt the linear progress. The middle and lower lines join to form an octave

Example 6.14

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G:G in the first bar, and an octave C:C in the third quarter of the second bar. The lower line C is approached and quit by step, and does not sound on the beginning of the bar. Also, its immediate harmonic context is dissonant: the consonant F in the upper line sounds concurrently with the octave C:C, resulting in a dissonant harmony C:C:F. If the sounding harmonic context is consonant, however, the parallel octaves do interrupt the linear progress. • Listen to example 6.14b. The middle and lower lines join to form an octave

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150

BkB!» in the beginning of the first bar, and an octave A: A in the third quarter of the second bar. The sounding harmonic context of both is consonant: the first sounds as part of a Bl> major harmony, the second as part of an A minor harmony. 5. Again, die two active co-lines can join with each other to form four successive like intervals, but not five. Neither co-line may join with die cantus to form more than three consecutive like intervals. 6. Unisons again function as appropriate for the particular species. One co-line is in second species in relation to the cantus; unisons may occur between the second species co-line and the cantus on the second half note. The other co-line is in third species in relation to the cantus; unisons may occur between the third species co-line and the cantus on the second, third, or fourth quarter note. The two co-lines relate to each other as second species does to cantus; unisons may occur between the two co-lines on the second or fourth quarter note only. Counterpoint Write-Through The write-throughs illustrating the composition of type 2 mixed species counterpoints begin on page 290.

SPECIES

Exercises • Using your cantus firmi, compose and perform without the music one of the following: 1. a type 2 mixed species exercise in which the cantus sounds as the lower line 2. a type 2 mixed species exercise in which the cantus sounds as the middle line 3. a type 2 mixed species exercise in which the cantus sounds as the upper line. TYPE

3

In type 3 mixed species exercises the cantus is accompanied by one third species co-line and one fourth species co-line. Type 3 mixed species exercises introduce implied harmony. Characteristics of Type 3 Mixed Species Counterpoints 1. Each active co-line accompanies the cantus as it would in a simple three-part counterpoint. The third species co-line has a relation of two against one with die cantus, and must accompany the cantus as an active co-line would in a simple three-part exercise in third species. The fourth species co-line has a syn-

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SPECIES

copated relation of one against one with the cantus, and must accompany the cantus as an active co-line does in a fourth species exercise. 2. The relation of the third species co-line to the fourth species co-line varies slightly, depending on the nature of the suspended tone. If the suspended tone on the beginning of the bar is dissonant, the first quarter note of the bar in the third species co-line must participate in a consonant harmonic event with the unsounded consonant resolution of the fourth species co-line. In other words, the tones of the third species co-line function as if the fourth species client tone were sounding instead of its dissonant agent. If the

Example 6.15

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suspended tone is consonant, the fourth species coline functions in second species, and the third species co-line joins it as it would in a type 2 mixed species exercise. • Listen to example 6.15a. The upper-line B is a dissonant suspension participating in the fourth species co-line; it represents the consonant A to which it resolves. In conjunction with the cantus C the functional harmonic event must be an A minor triad in first inversion. The middle-line consonant E participates in the functional A minor triad. • Listen to example 6.15b.

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The upper-line G is a consonant suspension. In conjunction with the cantus E in the lower line and the consonant C in the middle line, the functional harmonic event is a C major triad in first inversion. 3. The third tone of a passing cambiata in the third species co-line must be consonant. It may be consonant either in the context of the cantus tone, or in the context of the functional consonance of the fourth species co-line. Counterpoint Write-Through The write-throughs illustrating the composition of type 3 mixed species counterpoints begin on page 296. Exercises • Using your cantus firmi, compose and perform without the music one of the following: 1. a type 3 mixed species exercise in which the cantus sounds as the lower line 2. a type 3 mixed species exercise in which the cantus sounds as the middle line 3. a type 3 mixed species exercise in which the cantus sounds as the upper line.

MIXED TYPE

SPECIES

4

In type 4 mixed species exercises the cantus is accompanied by a single line that is condensed from two fifth species co-lines. Type 4 mixed species exercises introduce compound melody. The composition of type 4 exercises is a two-part process: first, the cantus is accompanied by two fifth species co-lines; second, the two active co-lines are condensed into a single-line texture. Characteristics of Type 4 Mixed Species Counterpoints 1. Each of the two co-lines relates to the cantus in almost the same way as a fifth species co-line would in a simple three-part exercise. Active co-lines in type 4 mixed species exercises also make use of whole notes (borrowed from the first species) and dotted half notes (resulting from a combination of species within the same bar). If the first half of a bar is a second species half note and the second half is two third species quarter notes, the half note may be tied to the first quarter note, resulting in a dotted half note. The degree of rhythmic activity in either co-line is limited by the necessity of resolving the impulse consequently. Two concurrently sounding fifth species

MIXED

SPECIES

co-lines result in a rhythmically dense texture, which tends to create more impulse than can be resolved. Whole notes and dotted half notes are useful to dilute the density of rhythmic activity. 2. Previously problematic configurations of tones will be commonplace in the condensed single line. For the single line to unfold two concurrent linear motions, it must contain unrecovered leaps, dissonant contours, and compound melody-configurations that have been problematic in earlier exercises. Individually, however, the two fifth species lines must be primordial lines, and may contain no such configurations. 3. When the two fifth species co-lines are condensed into a single line, the rhythm of the resultant twopart texture must approximate the rhythm of the original three-part texture. 4. The cantus may sound only in the lower line or the upper line, but not in the middle line. So that the active co-lines may be condensed into a single line, they must sound in adjacent lines. Thus either the cantus sounds in the lower line and the upper and

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middle lines are active co-lines, or the cantus sounds in the upper line and the middle and lower lines are active co-lines. 5. The concurrent sounding of two rhythmically free lines can have a dramatic effect on the dynamic forces. Two fifth species co-lines sounding concurrently can create substantially more impulse than a single such line can. Counterpoint Write-Through The write-throughs illustrating the composition of type 4 mixed species counterpoints begin on page 299. Exercises • Using your cantus firmi, compose and perform without the music one of the following: 1. a type 4 mixed species exercise in which the cantus sounds as the lower line 2. a type 4 mixed species exercise in which the cantus sounds as the upper line. • Combine into a single line the two active co-lines of your type 4 mixed species exercise.

CHAPTER

SEVEN

Consequences and Applications

754

CONSEQUENCES

OF AC E N CT

The phenomenon of agency requires two distinct objects with something in common. An actor can be represented by an agent. The two do not have the same physical being-they are different people-but they have an essential interest in common: the career of the actor. In music, agent tones represent client tones. An agent does not share the pitch of its client and may well have a different duration, volume, or timbre. Yet this agent and client also have something essential in common: they have the same harmonic and linear function. And because the structure of impulse and resolution is largely determined by the harmonic and linear functions, agents share the dynamic function of their clients. Imagine an original client tone participating in a progression of original client tones. The tone has a linear attribute, which results from its relation to the other members of the line in which it participates. It also has a harmonic attribute, which results from its relation to the other members of the harmonic event in which it participates. The progression of client tones suggests a structure of dynamic forces-impulse and resolutionbased primarily on the attributes of linear and harmonic

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function. When a client tone is represented by an agent, the agent tone represents not only the pitch of the client but also the linear function, the harmonic function, and the resultant dynamic function. In other words, if a tone in the client progression participates in impulse, its agent likewise participates in the same impulse. • Perform example 7. la so that the impulse is consequently resolved. The progression up to the soprano line A generates impulse, which is resolved consequently by the final three chords. • Now listen to example 7.1b, in which the climactic A is embellished by its agent Bl>. In the context of the progression of client tones, the Bl? has the same place in the dynamic structure as its cliExample 7.1

155

ent A: it climaxes the impulse. Note the dual function of the A. On the broadest level of the original client tones it participates in impulse, but in the context of the B!> it participates in resolution. Thus the agent Bl> carries the dynamic function of its client. The process of agents participating in the dynamic structure of their client tones is somewhat complicated, because agency itself may have an impact on dynamic function. Agency itself promotes impulse. When a given tone is sounding (the agent tone) and another is functioning (the client), there is an opposition between the tone sounding and the tone functioning. The tone sounding stands in opposition to the tone functioning. This opposition produces tension-it produces energy. In other words, it produces impulse. This is especially

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CONSEQUENCES

true of agent tones on strong beats (suspensions) and of agent tones approached by leap. If a client tone creates impulse within the progression of client tones, then that impulse is invested in its agent. The agent may itself have an agent, which also participates in the same impulse. And the agent's agent may in turn have an agent, which in turn participates also in the same impulse. With each additional level the conflict between tone sounding and tone functioning is intensified. J Not only is the impulse increased in degree, but it is extended temporally. By means of this multileveled agency a given impulse can be extended substantially.2 Although agency itself suggests impulse and can sub-

1. This is true if and only if the work is performed and experienced unitarily. A work is not experienced unitarily when the connections between tones are broken. If the connection between sounding tone and functioning tone is broken, then clearly there can be no opposition between them. An unmusical performance or a technically deficient performance will lead to a break in the connection of the tones in our experience. For example, a performance in which the impulse is not resolved consequently is experienced as unmusical precisely because it does not unfold the tones within a single indivisible unity-in other words, the tones are not connected in our experience.

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stantially extend a given impulse, it can also extend a resolution. To understand this, it may be helpful to return to the progression of original client tones, in which the resolution was consequent to the impulse. If the impulse is extended substantially by means of multileveled agency, then the original resolution cannot be consequent to the new, heightened impulse. When the force of resolution was first encountered it was heard that one primary determinant of the degree of resolution is temporal extension:3 the longer the resolution lasts, the more energy it releases. To allow the tones of resolution of the client progression to release the energy gathered in the expanded impulse, the resolution must itself be extended, again by means of multileveled agency. If agency itself promotes impulse (by creating conflict 2. Within that single extended impulse, smaller groups may be articulated by impulses and resolutions. We have heard this in the counterpoint exercises-an overall impulse itself encompassing smaller impulses and resolutions. And within the impulses (or resolutions) of those groups, yet smaller groups may be articulated by yet smaller impulses and resolutions. Overall, however, the entire passage will still unfold a single impulse. 3. Seep. 15.

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between the tone sounding and the tone functioning), then how can agency be used to extend resolution without building more impulse? First, a certain amount of impulse is necessary to keep any sound alive. Without energy there is silence. An extended resolution needs injections of energy simply to keep the sound going. Second, a resolution will have relatively few levels of agency in comparison with the impulse.4 To be experienced as indivisible, a primordial line must have an impulse and a consequent resolution. To be experienced as indivisible, any succession of tones must have an impulse and a consequent resolution. And thus, to be experienced as indivisible an entire work (or movement) must have an impulse and a consequent resolution-a single impulse, and a single, consequent resolution. Although both the extended impulse and the extended resolution will encompass smaller groups on narrower levels (formed by smaller impulses and resolutions), which may themselves contain smaller groups on 4. This explains the tendency in works that can be experienced unitarily for the tones of impulse of the client progression to be unfolded by elephantine portions of the work, while the tones of resolution are unfolded in relatively close succession.

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still narrower levels, the work that is experienced as indivisible will be articulated by a single extended impulse, which is released by a single extended resolution. The consequence of multileveled agency is enormous. The task of the composer of Western art music has been to write extended works that can still be experienced as indivisible.5 As we know, this is possible only if the impulse is played out by a consequent resolution. An extended whole work requires an extended impulse and an extended resolution; and it is multileveled agency that allows for the extension of the dynamic forces. It is thus no exaggeration to state that the phenomenon of multileveled agency lies at the foundation of Western art music.

P E R F O KM A N C E Music is a product of the human consciousness; it is not an object of the physical world. It is not "out there" 5. This is in contrast to most of the world's music (most thirdworld music, commercial or pop music, and jazz), which continues until all the formulas are played, or until the food or liquor runs out, or until the allotted time has passed.

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anywhere, but results from human experience of sounds. We do perceive physical objects-sounds-but until they are joined within the human consciousness they are not music. In this way music is similar to the optical illusion of figure l.l,6 in which four black shapes yield a perception of a white square. The square does not exist in physical reality, but is the connection that exists only in our consciousness of the black shapes. Similarly, music does not exist in physical reality; all that exists is sounds. Music is the connection of the sounds-a connection that cannot take place outside our consciousness. Thus music exists nowhere but in the consciousness of the listener. One requirement for music is therefore that the listener experience sounds. Another requirement is the piece of music, which suggests the succession of sounds. But music does not result from the listener's experience of the suggestions for sounds; it results from the experience of the sounds themselves. Between the listener on one side and the piece on the other is an essential link: the sounds-in other words, the performance. What is the performance? The performance is the un6. See page 6.

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folding of the sounds. But to what end? How should those sounds be unfolded? Under certain conditions, music can yield an extraordinary, transcendent, even spiritual experience, an experience of exquisite and indefinable beauty. The conditions for sublime musical beauty are simple: they occur when all the sounds join together in the listener's experience to form an indivisible whole. For this to occur the listener must be open to the sounds, the sounds must be presented so that they can be experienced as an indivisible unit, and the piece itself must allow such a presentation. The composer points us toward those conditions, those relations of sounds, not by specifying them precisely but by making general suggestions (for considerations of pitch, rhythm, volume, articulation, and tempo).7 Each part of the tripartite equation (listener, performance, piece) carries specific requirements for the occurrence of the sublime experience that is musical beauty.8 7. He cannot specify them precisely given our system of musical notation. More important, he would not want to, because the specific sounds necessarily change with each hall, with each performer, with each change in the weather, and so on. 8. The performer who is experiencing the work is also a listener.

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The first part of the equation, the availability of the listener's consciousness, may be considered but briefly. It is the responsibility of the listener to eliminate everything from consciousness, to come to the sounds open and free. The listener need not be "educated," or "competent," or "experienced," but only free to let the sounds seize, overcome, become him or her. Can the inexperienced listener and the accomplished musician listening to the same sublime performance have equally moving experiences? Certainly. There is only one difference: if die experience is not one of spiritual beauty, the accomplished musician will likely know the reason and the inexperienced listener will not.9 Another part of die equation, the piece itself, has been my primary focus so far. We have heard successions of tones that allow for their unfolding as a single, indivisible entity, and other successions that do not. Masterworks are pieces that allow for their performance 9. In fact, the accomplished musician is often likely to miss the sublime experience, for the musician's attention will often be focused on technical aspects of the performance. On the other hand, the inexperienced listener will more likely be open to focusing on the sounds alone.

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as such an indivisible entity; thus masterworks can yield the most sublime experience of musical beauty. The remaining part of this equation is the performance. In performing the counterpoint exercises, we have heard that the impulse must be released by a consequent resolution. We have also heard that the dynamic function results both from the attributes of the tone as composed and from the attributes of the tone as performed. Given a masterwork, it is the obligation of the performer to unfold the work as a whole, to create impulse and to release it consequently, so that it yields the sublime experience of musical beauty.10 How does the performer do this? Tempo (the quality of the motion) and volume are the two primary generators of impulse. On a small scale (on the level of individual phrases), volume is the primary generator of impulse. In general, greater volume results in greater impulse. On a broader scale (on the level of larger sections, or even entire works) it is die direction of the tempo-in other words the quality of the motion-that generates impulse. A forward-moving tempo or quality 10. See the discussion of performance on p. 21.

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of motion generally yields greater impulse; a restrained tempo or quality of motion generally effects resolution.11

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APPLICATIONS

Before examining passages from pieces, we must become familiar with some of the conditions of tonal

pieces that differentiate them from counterpoint exercises. 1. Bass motion. In counterpoint exercises all lines must be primordial. In tonal works the linear progress is carried by the soprano line.12 The bass line is not tied to the same linear considerations as the soprano. It sets the environment for the soprano line, but often the bass moves harmonically-by fifth (or by its inversion). Bass motion by fifth is not part of the study of counterpoint. Thus the bass line functions in a harmonic relation to the other lines, specifically the soprano, and not necessarily as a primordial line itself. 2. Silent client tones. In type 2 mixed species exercises

11. Tempo, or quality of motion, is a complex phenomenon, created only partJy by metronomic speed. The tempo is determined by the amount of time we have to process the information we receive: the more information, the faster the perceived tempo. Thus a faster metronomic speed will result in a faster tempo. But tempo is also affected by the size of the room, by the volume, by the articulations, in fact by every other attribute of the performance. For example, a passage performed in a small, resonant room sounds faster than if performed at the same metronomic speed in a large, dry hall. We have more information to process in the same amount of time, and therefore experience the passage as faster. A passage sounds faster if it is played

louder, or in a lower register, or if the degree of dynamic inflection is greater, or if the short notes are shorter, and so on. In each case more information to process results in a faster perceived quality of motion-in a faster tempo. 12. In the study of species counterpoint I have used the terms "lower line," "middle line," and "upper line." In tonal pieces I differentiate among the lines with the terms "bass," "tenor," "alto," and "soprano" lines, or voices. This is in recognition of the functional difference between lines in an exclusively linear context (as in species counterpoint exercises) and lines in a harmonic context (as in tonal pieces).

MUSICAL

PASSAGES

In this chapter we examine two passages from musical works, to consider whether an understanding of the issues raised so far can assist in the performance of these passages, and if so, how. PRELIMINARY CONSIDERATIONS

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we encountered two levels of the relation of client to agent. On one level the sounding fundamental tone in the cantus participates in a harmonic event, the component tones of which in the other lines are represented by dissonant agents. On the second level the two dissonant agents join to participate in a consonant harmonic event, one component tone of which is represented by yet another dissonant agent. In type 3 mixed species exercises we encountered implied harmonies, in which the functional harmony was never sounded explicitly by all three lines at the same time. But in all these cases, as long as a triad is functional at least one of its component tones is sounding. In extended tonal pieces, client-to-agent relations exist on more than two levels-there may be three or four, or even more. Thus it is most often the case that none of the tones of the harmony functioning on the broadest level are sounding. 3. Hierarchical levels determined by the conjunction of the soprano line with the harmonic environment. It is essential to note that levels of linear activity result from the consonance of tones in the context of the harmonic environment. For a single level of linear prog-

161

ress to exist, two conditions must be met: first, there must be stepwise motion in the soprano line;13 second, those tones of the soprano line that relate to each other by step must be consonant with the harmonic environment. • Listen to example 7.2a, which unfolds a single line with ten discrete components. • Now listen to example 7.2b, in which the line is unfolded within a harmonic context. Example 7.2c illustrates the effect of the harmonic environment on the upper lines. Consonances are notated as whole notes; dissonances are notated with diamond noteheads. The initial C is consonant, and the ensuing B and A's are dissonant passing tones. The second C moves to a B, which is consonant with the new harmony. That B is ornamented by the consonant Ftt. The final A and G are both consonant. The harmonic context separates the ten discrete components of example 7.2a into those tones that participate in a line on the broadest level, and those tones that are ornamental. A four-chord foundation with a 13. More precisely, there must be stepwise motion in the principal melodic line. This sounds only rarely in a line other than the soprano.

162 Example 7.2

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CONSEQUENCES

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soprano line of C-B-A-G is revealed, as illustrated in example 7.2d. This is because (a) these four soprano tones are consonant, and (b) they are separated by step. • Now listen to example 7.3a. The soprano line is identical to that of example 7.2, but because of the different harmonic context it is at the same time quite different. Example 7.3b Example 7.3

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illustrates the effect of the new harmonic environment on the harmonic function of the tones. The initial C is now dissonant; it is followed by a consonant B; the A and G are both dissonant. The second C is consonant; it moves to a consonant B. The Ftt and A of the second bar are dissonant; they are followed by a consonant G. The soprano line succession of consonant tones is B-A-B-G. Because the G does not

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CONSEQUENCES

relate to the B by step, the G does not participate in the same line-it has an ornamental function, as the dissonances do. Thus the client tone progression is B-C-B, as illustrated in example 7.3c. • Perform and listen carefully to examples 7.2b and 7.2d and then 7.3a and 7.3c, sensitive to how the client line is given by the harmonic context. 4. Dissonant chord members. Dissonant tones that particiExample 7.3 (continued)

Example 7.4

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APPLICATIONS

pate in dissonant chords share two characteristics of consonant tones: (a) they may be represented by agents; and (b) they can further the linear progress (although not on the broadest level).14 • Listen to the progression illustrated in example 7.4. The example unfolds a three-chord progression in G minor: i6-V7-VI. The soprano C (the seventh of the dominant seventh chord) is a dissonance, but it participates fully in the chord-it does not represent any other tone. Thus even though the C is dissonant, it can be represented by an agent or agents. Moreover, it can further the linear progress. In this case all three members of the soprano line D-C-Bl? participate equally in the progression.15 14. The seventh of a dominant seventh chord is a dissonant tone participating in a dissonant chord, as is any member of a diminished 3 chord. 15. A dissonant chord member is not as consonant as a consonant chord member (either a member of a consonant chord or a consonant member of a dissonant chord, such as the root, third, or fifth of a dominant seventh chord), even though it has some of the same attributes. Although both consonant and dissonant chord members may participate equally in the same linear progression on a local level, the dissonant chord member is unlikely to serve on the broadest level as an original client.

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PASSAGE A

This section examines a series of examples, beginning with a single chord that is then presented in successively Example 7.5

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more ornamented versions. The final version is a passage from a familiar piece of music. Consideration is given to the structure of impulse and resolution required for each ornamented version to be experienced unitarily, and also to the effect of each additional level of agency on the dynamic structure. This succession of increasingly ornamented versions of a chord is given in examples 7.5a- 7.5h. • Listen to the chord in example 7.5a. The chord is a stable consonance, with a fifth between bass and soprano. • Now listen to example 7.5b. The soprano B is followed by a leap to G. There is no linear progress; the single harmonic event-the E minor chord-is ornamented by a leap B-G in the soprano. (The inner voices are secondary to the outer voices. The tenor accompanies the soprano leap B-G with a complementary leap from G to B, while the alto accompanies the soprano descent in thirds.) To be experienced unitarily, this succession of chords must have an impulse generated by the initial chord, and resolving through the second chord. • Listen to example 7.5c. The progression is ornamented by stepwise dissonances in the outer voices. The bass and soprano sound

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a lower neighbor Dtt and a passing A. The soprano passing tone A is accompanied by a passing Fit in the alto line; the tenor anticipates the B. This results in a dominant seventh chord in first inversion, Dit :B:Ftt :A. Although its participant tones can be represented by agents, the chord is fundamentally dissonant; its members do not participate as client tones on the broadest level. Thus it serves as an agent of the first E minor chord. Again the first chord must generate the impulse, which is then played out by the two following chords. • Listen to example 7.5d. The pitches of example 7.5d are exactly the same as those of example 7.5c. In example 7.5d, however, the bass delays its motion to Dtt. The resultant suspension of E has a dramatic effect on the impulse. The impulse created by the sounding of the original soprano B is heightened by the sounding of the suspension. This heightened impulse must then be played out by the two following chords. • Perform example 7.5d so that the impulse is resolved consequently. You should have found that if you allow the first chord to carry the climax of the impulse by inflections of volume (as in the previous examples), the resolution

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will be too great. Instead the chord including the dissonant suspension must carry the climax of the impulse, which can then be resolved consequently by the final two chords. • Listen to example 7.5e. The impulse is heightened even further by the leap to C in the tenor voice. The C joins with the Dtt, Ftt, and A to participate in a diminished seventh chord. AlExample 7.5 (continued)

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though there is concurrent linear progress in both outer voices and the diminished seventh chord can be represented by agents (like the dominant seventh chord of example 7.5c), the chord is fundamentally dissonant. Therefore, on the broadest fundamental level the original E minor chord with B in the soprano is still functioning. The entire passage represents that fundamental E minor chord with B in the soprano. In performing this example as unitarily apprehensible, you should find that the impulse must be climaxed by the diminished seventh chord. Three things now come together in an interesting way. First, for the chords in example 7.5b to sound as an indivisible unit, the first chord generates the impulse, which is resolved through the sounding of the second chord. Second, for the chords in example 7.5e to sound as an indivisible unit, the impulse must climax with the second chord: the diminished seventh with the suspended E in the bass. Third, the suspension E is an agent of the Dtt that follows; the D# participates in a dominant seventh chord, which is an agent of the first E minor chord. Thus in the foundation, represented in example 7.5b, the impulse is effected by the first chord; and in the ornamented version of example 7.5e the im-

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pulse is effected by the agent of the agent of that chord. The dynamic function of the client is invested not only in the agent, but in the agent of the agent. This phenomenon was encountered earlier,16 when the dynamic structure of the primordial line was heard to reflect that of its stepwise foundation. Remember that in a primordial line sounding alone, the height of the impulse is carried either by the same tone that climaxes the impulse in its foundation, or within the indirect motion to that tone. In tonal music the height of the impulse will be carried either by the same chord that defines the impulse in the foundation, or by an agent of that chord. • Now listen to example 7.5f. The B and A of the soprano line are both ornamented by neighbor motions: the B by an upper-neighbor C, the A by a lower-neighbor G. The alto line accompanies the soprano in thirds. To be experienced unitarily, the impulse must again climax with the diminished seventh chord that has the suspended E in the bass. • Finally, listen to example 7.5g. You may recognize the passage as a four-part reduc16. Sec p. 24.

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tion of the opening bars of Bach's Brandenburg Concerto no. 4, second movement. A single ornamentation transforms example 7.5f into the passage represented in example 7.5g: the soprano upper-neighbor motion to C is repeated (the soprano activity is accompanied in thirds by the alto voice throughout), and is quit by leap. When the A that begins bar 2 is sounded, the second C

Example 7.5 (continued)

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hangs unresolved. Three eighth-notes at the end of bar 2 pick up the C and recover the leap C-A. The incomplete upper-neighbor motion effects an increase in the impulse. Because of the unresolved C, the soprano A of bar 2 carries with it a greater tension. The tension is exaggerated even further by the rhythmic stress produced when the high note C is resumed on a weak part of the bar. The ascending leap to C in the tenor voice has a similar intensifying effect. It is not possible to perform the passage represented in example 7.5g so that the impulse created is consequently resolved. Although the diminished seventh chord that begins bar 2 still climaxes the overall impulse, the increased energy within the resolution brings the passage to a state of imbalance: there is more impulse than can be resolved. Sacrilege? Is Bach's work faulty? Did he not hear the possibilities of unfolding this music as a unit? Certainly he did. • Listen to example 7.5h. This example represents the first four bars of the movement: the two bars represented in example 7.5g plus the two bars that follow. The latter two bars are an echo of the first two; the pitches are restated in a higher

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register, at a reduced volume, and without the tenor line. The overabundance of impulse allows for this twobar echo and in fact necessitates it. The echo functions as a resolution; it resolves the excess impulse gathered by the first, fuller two-bar statement. • Perform example 7.5h so that the impulse is consequently resolved. You should have found again that the overall impulse of the four-bar passage is climaxed by the diminished seventh chord in bar 2, and that the additional resolution effected by bars 3 and 4 is enough to resolve the impulse consequently. Both two-bar phrases climax with the diminished seventh chord that begins the second bar; the secondary impulse of bar 4 serves to extend the overall resolution. Example 7.5 (continued)

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PASSAGE B The second passage is a similarly ornamented version of the same chord. • Listen again to the successive ornamentations illustrated in examples 7.5a-7.5d. • Listen next to example 7.6a, which is a rhythmic variant of example 7.5e. • Next listen to example 7.6b. The soprano B is ornamented by an upper neighbor, as is the ensuing A. To be experienced as unitary the impulse again must be carried by the second chord. • Now listen to example 7.6c. The soprano B and A are each ornamented by a descending leap to the pitch of the alto voice. The soprano B leaps down to the alto G, and the soprano A leaps

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down to the alto Ft. To be consequently resolved, the impulse must still be carried by the second chord. • Finally, listen to example 7.6d. Example 7.6

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The passage might be more familiar in the key of G minor; it is a four-part reduction of the opening nine bars of Mozart's Symphony no. 40. Two simple ornamentations effect the transformation from example 7.6c. First, the half-note upper-neighbor motions are ornamented and intensified by repetition. And second, the pitches of the alto voice ornamenting the soprano are sounded an octave higher, with ascending leaps filled in by descending stepwise motion. The overall impulse must still be carried by the second chord. The ascending leap to Ftt in the seventh bar is recovered only on the most immediate level, for it is filled in by descending eighth-notes and quarter notes. It remains unrecovered in context of the harmonic activity. The hanging Fit participates in the dissonant dominant seventh chord, and thus must be resolved to a consonant E. The increased energy created by the hanging dissonance brings the

CONSEQUENCES Example 7.6 (continued)

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nine-bar passage to a state of imbalance (much as in the example from Bach above): there is too much impulse for the resolution. As illustrated in example 7.6e, the next six bars extend the resolution and allow the initial impulse to be played out consequently. FUNCTIONAL BASS

Has our understanding of the foundation of the passage by Bach helped us in performing it? Probably not that much. The structure of impulse and resolution necessary to experience this four-bar passage as an indivisible unit is fairly accessible. If it is not immediately clear, it should reveal itself with repeated listening. The passage by Mozart, however, is more complex. The structure of dynamic forces required to experience this passage as unitary is elusive; the ascending leaps, among other elements, are confusing. In this passage the structure of dynamic forces that enables unitary apprehension might be revealed through a conscious awareness of the underlying foundation. But how? We heard this passage in several stages of ornamentation. The structure of impulse and resolution required to hear each of the different stages as an indivisible unity differs. For instance, the progression repre-

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sented in example 7.5c serves as a foundation of the passage by Mozart. The progression represented in example 7.6a also serves as a foundation, even though it is somewhat more ornamented. For example 7.5c to be apprehended unitarily the impulse must climax with the first chord. For example 7.6a to be apprehended unitarily the impulse must climax with the second chord. If we were to assume that the dynamic structure of the musical surface paralleled that of the foundation represented in example 7.5c, we would expect the impulse in the Mozart itself (example 7.6e) to climax with the first beat of bar 3. This is not the case, however. If the impulse climaxed at that point it would be impossible to create enough impulse to carry through the ensuing resolution. We can turn for assistance to the fundamental progression at the level of the functional bass.17 In other words, the functional bass participates in a harmonic

17. This is most often the actual sounding bass. If the bass voice is ornamented, the functional bass is that which participates in the fastest-moving progression in which all voices participate. If the passage sounds over a pedal, the functional bass for this purpose is the sounding bass, disregarding the pedaJ.

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progression with the other voices. We find that the dynamic structure required for unitary apprehension of the musical surface mirrors that required for the unitary apprehension of its fundamental progression at the level of the functional bass. This is true of the two passages above. In the passage by Bach represented in example 7.5h, the functional bass participates with the other voices in a progression first given in example 7.5e. For this foundation to be experienced as unitary, the impulse must climax with the second chord. For the passage by Bach to be experienced as unitary, the impulse must climax with this same chord, which sounds on the first beat of the second bar. In the passage by Mozart represented in example 7.6e, the functional bass participates with the other voices in a progression illustrated (in transposition) in example 7.6a. For this foundation to be experienced as unitary the impulse must also climax with the second chord. For the Mozart to be experienced as unitary the impulse must climax with this same chord, which sounds on the first beat of thefifthbar. The performer seeking the dynamic structure that allows a passage to be unitarily apprehended may turn for guidance to the dynamic structure of the fundamental

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progression at the level of the functional bass. But it is essential to note that this cannot take the place of any open listening to the musical sounds; at best it can only aid in the preparation of performances. The consequent resolution of impulse can be achieved only when the performer is sensitive to the impulse actually created in a given performance. A conscious knowledge of the fundamental progression at the level of the functional bass, or any conscious knowledge, cannot be part of the listener's or the performer's consciousness if the experience of beauty is to result. Exercises These exercises are for developing sensitivity to the phenomenon of dynamic structure mirrored by that of the functional bass progression. • Find a passage from the repertoire in which the structure of dynamic forces required for unitary apprehension is elusive. By stripping away levels of ornamentation, find the fundamental progression that has the same bass line as the functional bass of the musical passage. Determine the dynamic structure required for unitary apprehension of the fundamental progression. Finally, perform the musical passage with a similar dynamic

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structure, so that the impulse created is consequently resolved. WHOLE WORKS

We have heard that brief passages can provide an experience of sublime beauty, given an impulse that is resolved consequently. We have also considered the phenomenon of dynamic forces extended by the phenomenon of agency, so that an entire work may be unitarily apprehensible. The process of performing entire works (or movements) as indivisible wholes involves expanding the object of consciousness from a limited segment to

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the entire continuum of sounds. Whether the object is a small segment or the entire work, it must have an impulse that is consequently resolved to be apprehensible as an indivisible whole. The performer who is sensitive to the nature of the dynamic forces will be able to perform the entire work so that the impulse created is resolved consequently. And it is the performance in which an entire work (or movement) is apprehended as an indivisible whole in an uninterrupted, single moment that yields the highest, most sublime experience of musical beauty.

APPENDIX tfis

Cantus Write-Through

Write-1 hrOUghS

from inidal conception to finished line. Although

This section illustrates the composition of three cantus there may be as many good ways to go about composing a cantus as there are good musicians, I suggest a four-part process: 1. sketch a stepwise foundation 2. compose the impulse 3. compose a consequent resolution 4. listen to the entire cantus and make corrections as needed.1 I begin with a cantus in G major, keeping in mind that it must meet the three guidelines for all cantus firmi: (a) it must begin and end on the tonic; (b) it must end with a stepwise descent to the tonic; and (c) it should consist of not fewer than eight tones and not more than fourteen. I start the process with a stepwise palindromic foun1. As you become more comfortable with composing single lines, you will probably find it unnecessary to begin by consciously sketching the foundation.

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dation.2 As notated in example A. la, I let the foundation be an ascent to the third scale degree B, with a return to the tonic G. Within this foundation, impulse is created by the ascent to B and resolved by the return to G. Example A. la

Example A. Ib

Example A.lc

2. The stepwise foundation of any primordial line that sounds alone and begins and ends with the tonic will be palindromic. See n. 9.

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As illustrated in example A.lb, I next compose the actual impulse of the line. In the foundation, the height of the impulse comes with the motion to B. For the cantus proper, I ornament that step A-B by a leap up to E, followed by a stepwise descent to B. The height of the impulse of the cantus will be the motion to E, which is serving to accomplish die fundamental B. Next I must see that the resolution can play out the energy gathered by the impulse. If the tones of example A.lb were followed by the fundamental tones A and G alone, there could not be sufficient resolution to dissipate all the energy gathered by the impulse. As illustrated in example A.lc, the resolution will be sufficient if I ornament the step B-A with a descending leap to Eft and a stepwise return through G. • Now listen to this cantus, as notated in example A.lc. Although this cantus can resolve its impulse conse-

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quently, it is problematic. The descending motion from the highest tone E down to the lowest tone Fit creates the dissonant contour of a minor seventh. This problem can be eliminated simply by adding an A after the C, as notated in example A. Id. The E now participates in a descending motion only until A, eliminating the dissonant contour. And the resolution can still be consequent to the impulse.3 • Listen again to the cantus as given in example A. Id,

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and remain sensitive to the requirements of primordiality: all tones must be connected by stepwise motion, each tone must participate discretely, and the entirety must be apprehensible as a unit. This final version of the cantus is given in example A.le. If you empty your mind of all considerations and focus exclusively on the sounds, you may gain from it an experience of sublime musical beauty.

Example A.Id

Example A.le

3. It may seem that this additional A creates a conflict between the dynamic structure of the line and that of its foundation. It appears that the E, D, and C sound between two statements of A, and that the height of the impulse (E) may be accomplishing some kind of embellishment of the A alone rather than the motion to B. If this were the case, then the impulse would climax with the B, not the E. This is not

the case, however, because the two A's are functioning on different levels. The first fundamental stepwise motion is accomplished with the first A. The second A is embellishing an indirect stepwise motion CB, which on the next level participates in the descent from E. On the highest level the E participates in the indirect stepwise motion A-B.

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1 7Q 1/0

For the second cantus I choose F major. As illustrated in example A.2a, I begin by outlining a stepwise foundation ascending to the fourth scale degree, Bl>, and returning to the tonic, F. The ascent to Bl> creates impulse; the return to F allows the impulse to resolve. The height of the impulse of the foundation is given by the motion to B!>. The height of the impulse of the line itself must be controlled by that fundamental impulse. As notated in example A.2b, I begin to build the

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impulse of the line with a leap from F to Bl>, then return to the fundamental G. After this subsidiary impulse I insert a climactic C between the fundamental A and Bl>. Next I compose a consequent resolution. The descent from Bl> to F is not quite sufficient to resolve the impulse that has been created. Adding a statement of F after the Bl», as illustrated in example A.2c, extends the resolution enough so that it can be consequent to the impulse.

Example A.2a

Example A.2b

Example A.2c

Example A.2d

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• Listen to the entire cantus represented in example A.2d. This follows the three guidelines for all cantus firmi: it begins with the tonic, ends with a stepwise descent to the tonic, and consists of nine tones. All the tones participate discretely in the line and are connected directly or indirectly by step. And the cantus can be performed so that its impulse is matched by a consequent resolution. This line too can provide an experience of musical beauty, given both that the performance allows it and that the attention of the listener is focused exclusively on the totality of the sounds. Finally I will compose a cantus in the minor mode. I choose the key of D minor. As illustrated in example

A. 3 a, I again begin by outlining the stepwise foundation-this time ascending to the dominant, A, before returning to the tonic, D. The ascent to A creates impulse, which can be resolved by the return to D. Next I compose the actual impulse of the line. The height of the impulse of the foundation comes with the motion G-A. As illustrated in example A.3b, I ornament the motion G-A with an intermediary Bt, and ornament the earlier motion E-F with an intermediary G as well. As the line stands, therefore, the leap to G creates impulse and the leap to Bt> creates even greater impulse. This mirrors the dynamic structure of the foundation. The resolution begins after the sounding of the Bk Again it cannot quite be effected by the stepwise descent

Example A 3a

Example A.3b

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to D. As illustrated in example A.3c, the resolution can be extended sufficiently by the addition of a statement of the tonic, D, after the A. • Listen carefully to this cantus, notated in example A.3c. This cantus is problematic. Not all the tones participate discretely, because the two leaps of a minor third create a pattern: an ascending minor third followed by a descending second. This disruptive pattern can be elimiExample A.3c _fl

Example A.3d

Example A.3e

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nated by inserting an A before the first G, as notated in example A. 3d. With this emendation, the D minor cantus becomes a primordial line that follows the three guidelines for cantus firmi. This cantus, given in its final version in example A.3e, can be performed so that it is apprehended as an indivisible whole. Listening to such a performance openly will yield an experience of musical beauty.

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Two-Part First Species This section illustrates the composition of twofirstspecies counterpoints using the cantus firmi from chapter 1. The first counterpoint will accompany the G major cantus with a co-line above, and the second will accompany the D minor cantus with a co-line below. As with the cantus firmi, there may be many good ways to go about composing a counterpoint. I recommend the following steps: 1. Notate the cantus. Then sing it, precisely in tune and with rhythmic accuracy, without looking at the music. 2. In a separate staff, notate all the pitches that would form consonances with each tone of the cantus. 3. Compose a counterpoint that you find pleasing. You may choose to do this in smaller segments of two, three, or four tones. In composing each segment keep in mind that the entire co-line must be primordial, and that it must join with the cantus to become an indivisible whole. In two-part counterpoints the final bar of the coline is given and the penultimate bar either is given or has limited possibilities. Therefore it is advisable to compose the co-line from back to front.

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4. As you compose the counterpoint, perform it without the music, listening with a particular sensitivity to those conditions that might prevent primordiality or conjunction. Listen again, with a particular sensitivity to the forces of impulse and resolution. 5. Change the co-line to eliminate any problems. Resist the temptation to hold on to segments of the co-line that are particularly pleasing (keep in mind that changing any one tone of the co-line changes the attributes of every other). 6. Perform the completed exercise without the music. Focus only on the sounds, allowing yourself to experience the counterpoint as an extraordinarily beautiful indivisible whole. If the counterpoint cannot provide this experience, go back and check for problems you might have missed. (You will get stuck; it will seem at times that there is no solution. Rest assured that for most good cantus firmi there are many acceptable co-lines for any given species. When you get stuck go back over the entire process, systematically considering all options. Eventually you will come to a solution.) Examples A.4a-A.4h illustrate the composition of a first species co-line above the G major cantus. I begin by

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notating the cantus in the lowest line (example A.4a). • Sing the cantus without the music.4 In the uppermost line I notate the possible consonances-those tones that form consonant intervals with each tone of the cantus.5 Note that the first tone of any co-line above the cantus may be either the tonic or the dominant, and that the final two tones must be the leading tone and the tonic (ending an octave above the cantus). As illustrated in example A.4a, I begin composing the co-line with the given final tones, Fit and G. After I con-

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sider the possible consonances, I try to accompany the cantus A-B-Flt-G of bars 6-9 with the ascending stepwise motion Fti-G-A-B. • Listen to example A.4a.6 The leap B-Ftt in bars 9-10 is left unrecovered, so I try to eliminate this problem. Because the Fit is given, the B of bar 9 must be changed. From among the possible consonances, neither a high D nor a low D would solve this problem. As illustrated in example A.4b, I try replacing the B with a G. • Listen to example A.4b.

Example A.4a

4. If the cantus lies outside your vocal range, I suggest that you transpose both it and the co-line by an octave. 5. The possible consonances are notated without regard to octave

placement. They may sound in the co-line in any octave. 6. When you listen to any co-line or part of a co-line, it is advisable to perform it along with the complete cantus.

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Although it has no leap to go unrecovered, this version creates another problem. The stepwise activity centered on G results in a linear embellishment of G. This problem of linear embellishment is exacerbated by the three final tones of the cantus, G-A-G, sounding in conjunction with G-Fti-G in the co-line. Example A.4c illustrates another solution: replacing the co-line B of bar 9 with an E.

• Listen to example A.4c from bar 6 to the end. The leap A-E of bars 8-9 eliminates the linear embellishment and is recovered by the final F# and G. Next I try accompanying the cantus A-E-D-C of bars 2-5 with the descending motion A-G-Fit-E. • Listen to example A.4c. There are two problems that prevent the co-line from being primordial. First, the palindromic stepwise mo-

ExampleAAb

Example A.4c

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tion from A to E and back, along with the following leap A-E, all serve to promote the perception of a single fourth E:A. Second, the sounding of E-Ftt-G in bars 5-7 and again in bars 9-11 results in a repetition. Each of these configurations precludes the discrete participation of the tones. As illustrated in example A.4d, I try replacing the G-Flt-E of bars 3-5 with B-B-A. • Listen to example A.4d. Although both the problematic E:A and the repetition from example A.4c are eliminated, new problems arise. First, the Ftt-G-A of bars 6-8 followed directly by the E-Ftt-G of bars 9-11 creates a pattern of three ascending steps, which disrupts the progress of the line and prevents primordiality. Second, the independence of

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the lines is compromised by the four successive sixths in bars 4-7 (D:B-C:A-A:Ftt-B:G). As illustrated in example A.4e, I try to eliminate these problems by replacing the Ftt-G of bars 6-7 with C-B. • Listen to example A.4e from bar 3 to the end. The disruptive pattern no longer exists, and the succession of consecutive sixths is broken. Next I try D-C for the first two bars. • Listen to the entire counterpoint given in example A.4e. Here there are no problems of primordiality or conjunction. There is a problem, however, with the structure of impulse and resolution. Regardless of which dynamic inflections are chosen,7 it will not be possible to perform

Example A.4d

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this counterpoint with an impulse that is resolved consequently. As illustrated in example A.4E, I replace the opening descent D-C-B-B with G-Ftt-G-A.

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• Listen to the new version of the complete counterpoint, given in example A.4f. The ascent to the co-line C in bar 6 climaxes an im-

ExampleAAe

Example A.4f 7. The term "dynamic inflections" refers to volume, and is not to be confused with the term "dynamic forces," which refers to impulse and resolution.

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pulse that can be consequently resolved. And the altered opening creates no problems of primordiality or conjunction. The final version of the co-line is presented in example A.4g. The impulse of any counterpoint is generally given by the activity of the upper line,8 and in exercises Example A Ag

cf

Example A.4h

8. One exception occurs when the upper line is essentially static. This is true of example A.6 (see p. 200).

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in which the impulse can be resolved consequently, the dynamic structure of the upper line mirrors that of its stepwise foundation. The stepwise foundation (G-AB-A-G) is given in example A.4h. When this counterpoint exercise is performed so that the impulse is resolved consequently, the impulse climaxes with the

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187

upper-line C of bar 6. This climactic C participates in the indirect motion to the B, which climaxes the stepwise foundation of the upper line.9 This brings up an interesting point. To allow the highest experience of musical beauty, the counterpoint must be experienced unitarily-it must have an impulse that is consequently resolved. To be consequently resolved, the impulse of this counterpoint must climax with the C of bar 6; if the impulse climaxes with any of the other tones it cannot be resolved consequently. The impulse is effected in part through inflections of volume, which grows to the C and recedes from it. Thus the loudest tone is the C. What is interesting is that the C, the loudest tone, does not participate in the stepwise foundation.

So what? It is a common assumption that the performer's function is to communicate the structure of the work to the audience. This assumption justifies the study of music theory by the performer: the more that is known about the structure of the work, the better the performer can do the job of communicating that structure. The performer is expected to communicate the structure by "bringing out" its most important tones or events, and it is an underlying assumption that these tones or events are to be brought out by playing them louder. It seems reasonable to intuit that the highest level of structural events should be brought out, in other words played louder or with some distinguishing characteristic. This is not the case at all, however.10 To perform

9. If a line begins and ends on the same pitch, we tend to understand its stepwise foundation as an ascent away from the initial tone followed by a return, or, less commonly, as a descent away from the initial tone followed by a return. If the first and last tones of a line are different pitches, we tend to understand the foundation as a single descent or (rarely) ascent. Where the initial stepwise motion occurs in a direction opposite from the climactic tone, we tend to understand the initial motion as secondary and embellishing, and thus do not include it as part of the foundation of the line. In this co-line the first stepwise

motion away from the initial tone is descending, but the climax of the impulse comes with the motion to a tone higher than the initial tone. Thus we understand the Fit of bar 2 as ornamental and secondary, and exclude it from the stepwise foundation. 10. If there is a tendency at all, it is that the ornamental or secondary tones that often effect the impulse may tend to be louder (see example 7.2). But as can be discovered by performing and studying the counterpoints in this appendix, there are no hard and fast rules-every case must be considered on its own merits.

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the counterpoint of example A.4 so that it yields the highest musical experience, the impulse must be resolved consequently. The loudest tone then is the C, which is not a member of the stepwise foundation. When the performer's goal is beauty, the structural level on which a tone functions is irrelevant to its performance. Examples A. 5 a-A.5i illustrate the composition of a first species co-line below the D minor cantus. I begin by notating the cantus in the uppermost line (example A.5a).

Example A.5a

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• Sing the cantus without the music. In the lowest line I notate the possible consonances. Note that the first tone of any co-line below the cantus must be the tonic, and that the final two tones must be the leading tone and the tonic (ending either an octave below the cantus or on a unison). As illustrated in example A.5a, I begin composing the co-line with the given final tones, Ctt and D, ending with a unison. After I consider the possible consonances, I try to accompany the cantus G-F of bars 1011 withE-D. • Listen to example A.5a.

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Although these final four tones present no problems, they could not participate in a co-line that successfully joined the cantus. The E of bar 10, which is higher than the cantus D of the previous bar, necessitates crossed voices (if the co-line tone of bar 9 is higher than the cantus D), or a unison (if the co-line tone of bar 9 is a D), or problematic parallel leaps (if the co-line tone of bar 9 is lower than the cantus D).

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I could try to eliminate this problem by transposing down an octave these final four tones. Instead I try changing the E to a B^l, as illustrated in example A.Sb.11 • Listen to example A.Sb from bar 10 to the end. The B^l of bar 10 is the sixth degree in the ascending position, and must participate in the ascent from the dominant through the leading tone to the tonic. It participates in such an ascent with the Ctt-D of bars 12-

Example A.Sb

11. There is no particular reason for choosing this option. To compose counterpoints is to make choices. Those options not selected remain possibilities to be tried, should the one selected prove problematic. Keep in mind that even if a given option proves problematic and

is therefore eliminated from consideration, the same option may become perfectly acceptable as the other tones of the co-line are changed.

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13, but it must also be preceded by a dominant A. That A can sound with the cantus A of bar 8; the cantus D of bar 9 can then be accompanied by another B^. • Listen to example A.5b. There seem to be no problems with this example. As illustrated in example A.5c, I try next to accompany the cantus F-G-Bt of bars 5-7 with D-C^l-G. • Listen to example A.5c. The leap C-G is recovered by the A and B^l of bars 8 and 9. There is however a problem of mixed modes. The C& of bar 6 is an altered seventh degree, in the descending position. It must participate in a descending stepwise motion from D to A through the descending sixth degree, Bl>. The B that follows the C is however a ExampleA.Sc

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fill, in the ascending position. The conflicting alterations of sixth and seventh degrees divide the line and prevent primordiality. As illustrated in example A.5d, I try a co-line of A-Bt-G for these three bars. • Listen to example A.5d. The new solution seems to present no problems. Note that there is no conflict of modes. The unaltered sixth degree Bl> of bar 6 moves to the fifth degree A of bar 8 before the altered B^ of bar 9 is sounded. For the first four bars, illustrated in example A.5e, I begin with the given D followed by G-C^-Bk • Listen to the completed counterpoint given in example A.5e. The first problem is one of conjunction: the succes-

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sion of five consecutive sixths in bars 2-6 prevents the two lines from remaining independent. To eliminate this problem I try replacing the G-C'i of bars 2-3 with C^F, as illustrated in example A.5f. Example A.5d

Example A.Se

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• Listen to the version of the complete counterpoint given in example A.5f. The leap C^l- F is recovered by the Bl>, A, and G of bars 4-7. Another problem should become evident: the

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linear embellishment of bars 4-6. The conjunction of the co-line Bl>-A-B!> with the cantus G-F-G causes both three-note figures to become linear embellishments. Replacing the Bt-G of bars 6-7 with G-D, as Example A.5f

Example A.5g

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illustrated in example A.5g, eliminates this problem. • Now listen to the final version of the counterpoint, represented in example A.5h. If the co-line were sounding alone, its stepwise foun-

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dation would be D-Ctt-Bt-A-G-A-B'l-Cll-D, as given in example A.5i. Remember that the dynamic structure of the counterpoint is generally determined by the upper line. (When the counterpoint given in example A.5h is performed so that the impulse is resolved consequently, the impulse climaxes with the upper-line Bl? of bar 7.) Therefore in counterpoints with co-lines below the cantus, the dynamic structure that the co-line Example A.5h

Example A.5i

12. In three-part exercises this is true of the stepwise foundation of a co-line sounding in either the middle or the lower line.

193

might have were it sounding alone is irrelevant to the dynamic structure of the counterpoint itself; the relation of its dynamic structure to the dynamic structure of its stepwise foundation is similarly irrelevant. The stepwise foundation of a co-line below the cantus is of interest only to the extent that it concerns the primordiality of the co-line.12 Thus in the write-through exercises that follow, I will indicate the stepwise foundations only of

194

those co-lines sounding in the upper line, which do concern the dynamic structure of the entire counterpoint. Two-Part Second Species This section illustrates the composition of two second species counterpoints, one with a co-line above the D minor cantus and one with a co-line below the G major cantus. Examples A.6a-A.6j illustrate the composition of a second species co-line above the D minor cantus. I begin by notating the cantus in the lowest line (example A.6a).

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• Sing the cantus without the music. Then I notate the possible consonances in the uppermost line. The first tone of each bar of the co-line will always be one of these consonances; the second will be either another consonance (if it is approached by leap) or a dissonant passing or neighbor tone (if it is approached by step).13 Note that the penultimate bar of the co-line may contain a whole note, in which case the functional consonance is the sixdi formed between the cantus E and the ascending seventh degree CK. Or it may contain two half notes, in which case the functional consonance is the perfect fifth formed between the

Example A.6a

13. The sole exception to this is the occasional dissonant suspension, which is borrowed from the fourth species. In this case the first

tone of the bar, tied to the preceding tone, is dissonant, and the second tone is consonant even though it is approached by step.

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cantus E and the ascending sixth degree B^, which is followed by a passing CD. I begin by trying the whole note CK for the penultimate bar, preceded by the half notes A-B^l-E-E-D, as illustrated in example A.6a. • Listen to example A.6a. There are no problems with this passage. The second E is a dissonant suspension that resolves to the consonant D on the second half of the bar. The B^l participates with the A before it and the Clt-D of the final two bars in an ascending motion from the fifth scale degree to the tonic. The first tone of bar 9 presents a problem, however. If it is a Bl> the functional consonance of bar 9 will be D:Bl>, which conflicts with the GiB^l of bar 10. Example A.6b

195

• Listen to the example A.6a from bar 9 to the end, inserting an unwritten Bl> in the co-line in the first half of bar 9. The conflicting positions of the sixth degree in two successive bars cause a disruptive mixture of modes. • Now listen from bar 9 to the end, inserting an unwritten D at the beginning of bar 9. This results in a compound melody: D-A-B^-E. An F on die top line would also yield a compound melody: F-A-B^l-E. An F on the bottom space would create an unrecovered leap to A. Finally a B^, followed by the A, would not participate in an ascent from dominant to tonic. I put aside this solution and try again, as illustrated in example A.6b.

196

I begin this time in bar 9 with a descent from a consonant D to a passing C^, to a consonant Bt and G in bar 10, and finally to a consonant A in bar 11. The leap to the consonant D is recovered by the B^l-CH of bar 12. The C'l of bar 9 participates in a stepwise descent from tonic to dominant; likewise the B^l of bar 12 participates in a stepwise ascent from dominant to tonic. • Listen to example A.6b. As illustrated in example A.6c, I next try accompanying the cantus Bt>- A of bars 7-8 with a leap from a consonant G down to a consonant D, recovered by the consonant F and passing E. • Listen to example A.6c. Example A.6c

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The descending motion from F (bar 8) to G (bar 10) creates a dissonant contour. To remedy this problem I try reversing the D and G in bar 7, as illustrated in example A.6d. As a result the descending motion begins with a G, creating the contour of an octave. For bars 56 I try a leap A-D followed by B^ and then a passing Cl. • Listen to example A.6d. The successive consonances G:B^-BkD of bars 6-7 constitute a disruptive modal mixture. As illustrated in example A.6e, I try to solve this problem by changing the offending B^. I replace bars 5-6 with a leap from a consonant D to a consonant F suspended over the bar

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line, where it becomes a dissonant seventh in conjunction with the cantus G and then resolves to the consonant E. For bars 2-4 I try a leap from E to a consonant Example A.6d

Example A.6e

197

G, followed by a stepwise descent to the lower neighbor tone Cl. • Listen to example A.6e.

198

Several problems become evident. First, the passing E moving to the consonant D of bars 3-4 forms successive perfect fifths with the cantus A and G. Also, the descending motion from G of bar 2 to the Ctt of bar 4 creates a dissonant contour of a diminished fifth; likewise the ascending motion from dial Ctt to the F of bar 5 creates the dissonant contour of a diminished fourth. As illustrated in example A.6f, I try to eliminate this problem by replacing the D of bar 4 with a B^l, preceded by a consonant A. I then try opening the counterpoint widi a stepwise ascent from A to a consonant E in bar 3. • Listen to this version of the completed counterpoint given in example A.6f. Example A.6f

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The opening is problematic. The stepwise ascent of bars 1-3 from A to E followed directly by the stepwise ascent of bars 3-5 from A to D creates a disruptive repetition. As illustrated in example A.6g, I try replacing bars 1-2 with a leap from a consonant D to a consonant A, followed by a stepwise descent to the E of bar 3. • Listen to the completed counterpoint given in example A.6g. This version is acceptable, but it could be better. The dramatic leap to the highest tone of the co-line, sounding in the very first bar, creates impulse. But the impulse of the entire co-line must climax with the G of bar 7 if it is to be consequently resolved. The climactic G is lower

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S

than the A of bar 1. The structure of dynamic forces could be more clearly unfolded if the high A were eliminated. As illustrated in example A.6h, I try replacing bars 1-2 with a half rest followed by the half notes DC#-D. Example A.60

Example A.6h

199

• Listen to die version of the complete counterpoint presented in example A.6h. This counterpoint is given in its final version in example A.6i, and is successful. The impulse builds to the climactic G of bar 7 and can be consequently resolved. An

200

examination of the foundation proves interesting, though. The stepwise foundation of this co-line or any other consists of consonant tones only. Dissonant tones sound while other tones are functioning; thus the very existence of dissonance creates a hierarchy.14 A dissonant tone on a narrower hierarchical level than that of the consonant tone it represents could not participate on the broadest hierarchical level-the stepwise foundation. The consonant tones of the co-line of example A. 6 are

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indicated by asterisks in example A.6i. The stepwise foundation, D-E-D-E-D, is derived from these tones, as illustrated in example A.6j.15 This foundation constitutes an extended embellishment of the tonic tone, D. But within the extension of that single tone are three embellishments that are successively more intense: DE-D (bars 1-5), F-E-D (bars 5-7), and G-F-E-D (bars 7-9). The height of these embellishments, the G of bar 7, coincides with the cantus Bl>. If the cantus were to sound alone, the Bl> would constitute its climax.

Example A. 6i

14. See p. 154. 15. The stepwise foundation obviously moves by step. The E of bar 3 participates in the stepwise foundation, and not the Ctt of bar 2, however. The Ctt would move by step to the B^ of bar 4, where it

would then be left dangling. It could not move to the Ctt of bar 4, because it is dissonant. Nor could it move to the Bt> of bar 10 or the A of bar 11, because either would create a mixture of modes.

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On the most global level, this successful counterpoint thus consists of an essentially static upper line extending a single tonic tone, supported by a lower line with a structure of dynamic forces that generates the structure of dynamic forces of the entire counterpoint. Examples A. 7a-A. 7j illustrate the composition of a second species co-line below the G major cantus. I be-

Example A.6j

Example A.7a

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gin by notating the cantus in the uppermost line (example A. 7a). • Sing the cantus without the music. Then I notate the possible consonances in the lowest line. As with co-lines above, the first tone of each bar will always be one of these consonances; the second will either be another consonance or a dissonant passing or neighbor tone. In second species co-lines below, the

202

penultimate bar must contain a whole note that sounds the leading tone. I begin with the given final two tones, Ftt-G, as illustrated in example A.7a. Turning next to bar 7,1 try a consonant G followed by a leap down to a consonant D, which is tied into bar 8. A leap to a consonant Ftt is followed by a consonant E and G in bar 9. • Listen to example A.7a. This passage is full of problems. First, the leap G-D of bar 7 is problematic. Both the G and the D are consonant. Thus either they both participate in the same harmonic event, which is a dissonant triad in second inversion,16 or they each participate in a different harExampleAJb

16. Seep. 87.

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monic event. Because each tone of the cantus must participate in a single consonant harmonic event, both G and D could not participate in a dissonant secondinversion triad, nor could they each represent a different interval. A second problem is the compound melody created by the D-Ftt-E-G of bars 8-9. A third problem is caused by the parallel octaves in the same two bars. As illustrated in example A. 7b, I try replacing the E-G of bar 9 with an F$ tied to the preceding bar, followed by a resolution to E. • Listen to example A.7b. These four bars present no problems. As illustrated in

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example A.7c, I next consider bars 6-7. I try a leap down from A to C, followed by a neighbor motion DE-D. The leap A-C is recovered ultimately by the final G. • Listen to example A. 7c. Example A. 7c

Example A.7d

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This passage seems fine. Example A.7d gives a solution for bars 3-5. I try an ascent from E to a consonant B in bar 4, followed by a lower-neighbor motion embellishing A. • Listen to example A.7d.

204

The three successive neighbor motions A-G-A (bars 5-6), D-E-D (bars 7-8), and Ftt-E-F)t (bars 9-10) promote stasis. Otherwise the line seems to present no problems. I then try G-F#-D for the first two bars. • Listen to the completed counterpoint given in example A. 7e. This co-line is a primordial line. It presents a problem of conjunction, however, which is strained by the fifteenth between the lines in bar 3. It also presents a problem of unitary apprehension. The impulse of this counterpoint, which climaxes in bar 3, is not sufficient to play out the extended resolution. That resolution is

Example A.7e

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promoted by the return to a stable consonance built on the tonic in bar 4. I return to the final two tones, which are given, and try again. Instead of approaching the final two tones from below, I try a descent from B in bar 8, as illustrated in example A.7f. • Listen to example A.7f from bar 8 to the end. This segment of the counterpoint seems to present no problems. The previous material will have to be changed, because the leap from the dissonant E of bar 7 is problematic. I try replacing the C-D-E of bars 7-8 with Ftt-G-A, as illustrated in example A.7g.

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• Listen to the complete counterpoint presented in example A.7g. Although the leap from the dissonant E has been Example A.7f

Example A.7jj

205

eliminated there is still a problem of unitary apprehension, for the impulse is not sufficient to play out the resolution. As illustrated in example A.7h, I erase the

206 opening five and a half bars. For bars 4-6 I try a descent from a high G to a C, then move to a D in bar 6. • Listen to example A.7h. Exa.mpleA.7h

Example A.7i

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The leap D-Fjt in bar 6 is left unrecovered. I try replacing the Fit- G-A of bars 6-7 with E-D-C, as illustrated in example A.7i. I then complete the counter-

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point, beginning with a high G in bar 1 and leaping down to a C in bar 2. The line ascends to a consonant G in bar 3 tied to the consonant G in the following bar.17 • Listen to the completed counterpoint represented in example A. 7j. The ascent of the co-line to the high G in the second half of bar 3, and the syncopation of bars 3-4, combine to generate significantly more impulse than in earlier versions. This impulse can be consequently resolved. Two-Part Third Species This section illustrates the composition of two third species counterpoints, one with a co-line above the F Example A.7'j

17. If the G in bar 4 were dissonant, the Fl would join with the C of bar 5 to create a problematic augmented fourth.

207

major cantus and one with a co-line below the G major cantus. Examples A.8a-A.8h illustrate the composition of a third species co-line above the F major cantus. I begin by notating the cantus in the lowest line (example A.8a). • Sing the cantus without the music. In the uppermost line I notate the possible consonances. As illustrated in example A.8a, I begin composing the co-line with the final two bars. I try accompanying the cantus G of bar 10 with a neighbor cambiata embellishing E, before the final F of bar 11. With the cantus Bt of bar 7 I try G embellished by a leap to a

208

consonant Bt>; with the A of bar 8 I try an embellished F; and with the F of bar 9 I try a neighbor cambiata embellishing D. • Listen to example A.8a. Two problems can be heard. First, the repetition GF-G-F in bars 7-8 interrupts the line. Second, the coline in bars 7-10 (from the final quarter note G in bar 7 to the E of bar 10) combines with the cantus Bl> - A-FG to form four consecutive sixths. As illustrated in example A.8b, I try replacing bar 8 with an embellished C, bar 9 with a stepwise descent from a consonant A, and bar 10 with an embellished D. • Listen to example A.8b. Example A.8a

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The descending motion from the D in bar 8 to the C in bar 10 describes a major ninth and creates a dissonant contour. As illustrated in example A.8c, I remedy this problem. In bar 9 I leap down from the consonant F to D; in bar 101 try a neighbor cambiata embellishing E. Next I accompany the C of bar 6 with an ornamented G, moving in bar 7 to a consonant Bt> embellished by a leap down to a consonant G. • Listen to example A.8c. This example presents no problems, so I turn to the beginning of the exercise. I begin the counterpoint in bar 1 with a leap from the consonant F down to an A,

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as illustrated in example A.8d. Bar 2 unfolds a neighbor cambiata embellishing D, followed in bar 3 by a consonant E leaping to a consonant A. A stepwise descent to an embellished E in bar 4 is followed in bar 5 by an F, which leaps to a consonant C. Example A.8b

Example A.8c

209

• Listen to the completed counterpoint represented in example A.8d. Although there are no problems here with either primordiality or conjunction, there is a problem with unitary apprehensibility. The gradual ascent to the C and

210

then the D of bar 8 creates an impulse that cannot be played out by the final three bars. This problem can be corrected if the impulse climaxes earlier in the counterpoint, thus extending the resolution. Example A.8e illustrates an effort to climax the impulse earlier. I try a leap to a consonant F in bar 5, and also try replacing the opening F with a C. This clarifies the impulse, allowing for a gradual ascent from bar 1 to bar 5. Finally, if bar 4 were to sound as in example A.8d, there would be a dissonant contour between the E in bar 4 and the F in bar 5. Instead, I make a minor alteration to the neighbor cambiata of bar 4, exchanging the Exa.mpleA.8d

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position of the embellishing F and D. This allows the ascending motion to begin with D, resulting in a consonant contour of a minor tenth (D up to F). • Listen to the counterpoint represented in example A.8e. The counterpoint is successful in terms of primordiality and conjunction, and its impulse can be played out consequently. I am a bit uncomfortable with the extreme range, from the low A of bar 1 to the high F of bar 5. As illustrated in example A.8f, I try replacing bar 1 with a C on the second quarter, followed by a leap up to F. To avoid the pattern created in bars 1-3 and then in bars 3-4 (F-E-D-E-C-D-E and then G-F-E-

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F-D-E-F), I exchange the position of the ornamental tones in bar 2. • Listen to the completed counterpoint represented in example A.8g. Example A.8e

Example A.8f

211

This counterpoint is successful. Its impulse climaxes with the first upper-line E of bar 6, and can be resolved consequently. The consonant tones are indicated by asterisks; they

212

yield a stepwise foundation of C-D-E-F-G-F, as illustrated in example A.8h.18 The E of bar 6 that climaxes the counterpoint participates in the motion to the

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G of bar 7, which climaxes the upper-line stepwise foundation. Examples A.9a-A.9h illustrate the composition of a

Example A.8$

Example A.8'h

18. The stepwise foundation could include either the A or the F of bar 9 after the G of bar 7. I consider the F as participating in the foundation and the A as not participating, for two reasons. First, if the foundation includes the A it will consist of a somewhat amorphous ascent from the fifth degree to the third. Second, the impulse of

the counterpoint climaxes with the E of bar 6, which participates in the motion to the G. If the G is followed by F, then it is the climactic tone of the foundation; if the G is followed by A, then it is not the climactic tone of the foundation.

APPENDIX:

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UGH S

third species co-line below the G major cantus. I begin by notating the cantus in the uppermost line (example A.9a). • Sing the cantus without the music. In the lowest line I notate the possible consonances. I begin composing the co-line with the final four bars, as illustrated in example A.9a. After a consonant B in bar 9 I try a stepwise ascent of an octave from G to the final G. I try an embellished D to accompany the cantus Ftt of bar 8. • Listen to example A.9a. This segment of the co-line seems to be without

Example A.9a

213

problems. I proceed as illustrated in example A.9b, accompanying the A of bar 6 with a neighbor cambiata ornamenting Ftt. This is followed by a consonant G in bar 7, which ascends by step to bar 8. • Listen to example A.9b. The ascending motion from the Ftt of bar 6 to the ornamental E of bar 8 creates a dissonant contour. This can be eliminated by exchanging the position of the ornamental E and G in bar 6, as illustrated in example A.9c. Next I try accompanying the cantus D of bar 4 with a consonant B, which leaps down to a consonant G. This is followed by an ascent to a C in bar 5 and

274

then by a consonant E, which leaps down an octave. • Listen to example A.9c. Although there are no problems of conjunction, there are problems with the motion from the E of bar 6 to the Example A.9b

ExampleA.9c

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final G, which is almost completely stepwise. As illustrated in example A.9d, I try embellishing the B of bar 9 with a neighbor cambiata. This breaks up the stepwise octave ascent G-G in the final three bars. Then I com-

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plete the exercise by composing the first three bars. I begin with a G on the second quarter of bar 1, followed by a descent to G in bar 3, and then an ascent to the B of bar 4. • Listen to example A.9d. Again the co-line is static and overly conjunct. Its meandering nature prevents any real linear progress and results in insufficient impulse. As illustrated in example A.9e, I begin anew. This time I try beginning with the opening of the counterpoint and concentrate on creating an impulse large enough to be played out consequently. Remember that the impulse is given primarily by the upper-line activity, in this case the ascent to E in

Example A.9d

215

bar 3. I now compose the co-line, attempting to intensify this impulse. I open with an embellished G on the second quarter. An ascending leap to a D in bar 2 creates impulse; the descending leap from A to C in bar 3 creates more impulse. I follow die C of bar 3 with a neighbor cambiata ornamenting D in bar 4, and then an ascent from a consonant E in bar 5. As the final tone of bar 5, the A forms a consonant tenth with the cantus C; I leap up to an embellished C in bar 6, and then again down an octave to the low C. I accompany the cantus B of bar 7 with a D, ascending to the embellished FK in bar 8. I end with the final three bars from example A.9d.

216

• Listen to example A.9e. Although the impulse is clearly defined in bar 3 and can then be consequently resolved, there are problems of conjunction. First, the low C in bar 3 forms a seventeenth with the cantus E. This interval is large enough to break the conjunction of the lines. Second, the leap from the seemingly consonant A of bar 5 is problematic. The cantus C and the first co-line tone E are necessarily consonant. If the A is consonant, either it participates with the other consonances in a dissonant secondinversion triad (E:A:C), or there are two harmonies in Example A.9 e

19. Sec p. 87.

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the bar.19 Because each tone of the cantus must participate in a single consonant harmonic event, the A must be a dissonance and therefore cannot be quit by leap. As illustrated in example A.9f, I try again, this time beginning with a high G on the second quarter of bar 1. I descend by step to an embellished D in bar 2, which ends with an octave leap down to D. The climactic cantus E of bar 3 is accompanied in the co-line by an ascending leap to G, which intensifies the impulse and participates in the recovery of the octave leap D-D of bar 2. After a motion to an embellishing E in bar 3, the

APPENDIX:

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UGHS

G is regained in bar 4 and is again embellished by a leap to the consonant B. The cantus C of bar 5 is accompanied by a descending octave C-C, followed by an ascent to an ornamented Ftt in bar 6. For the final five bars I try the final five bars from example A.9c. • Listen to example A.9f. This counterpoint is close to being successful, but it has two small problems.20 First, the low C in bar 5 forms a fifteenth with the cantus C, straining the conjunction of the lines. Approaching the fifteenth by large descending leap exacerbates the problem. Second, the Example A.9f

20. You will find that although problems may seem small, solving them can require extensive changes in your counterpoint. In this case it fortunately does not.

217

consonant C of bar 5 is followed by a stepwise ascent to a consonant Ftt in bar 6. The resultant augmented fourth is problematic. As illustrated in example A.9g, both problems can be eliminated by replacing the octave leap C-C of bar 5 with a stepwise ascent to E, followed by a leap down to a low E. The low E forms an acceptable thirteenth with the cantus C, and the Ftt of bar 6 does not participate in the same ascending motion with theC. • Listen to this final, successful version of the counterpoint, as represented in example A.9h.

218

Two-Part Fourth Species This section illustrates the composition of two fourth species counterpoints, one with a co-line above the G major cantus and one with a co-line below the F major cantus. Example A.9^

Example A.9h

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Examples A.lOa-A.lOi illustrate the composition of a fourth species co-line above the G major cantus. I begin by notating the cantus in the lowest line (example A.lOa). • Sing the cantus without the music.

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Then I notate the possible consonances in the uppermost line. I begin composing the counterpoint by trying a B with the cantus B in bar 7, as illustrated in example A.lOa. The B of bar 7 becomes a dissonant suspension resolving to the consonant A in bar 8. This A is followed by a resolution to a consonant G in bar 9, then by a resolution to a consonant Fit in bar 10, before moving to the final tonic. • Listen to example A. lOa. This segment of the counterpoint presents no problems. As illustrated in example A.lOb, I try accompany-

Example A.lOa

219

ing the cantus C of bar 5 with a G. The G becomes a dissonance in bar 6 with the sounding of the cantus A, and then resolves to a consonant Ftt. The Ftt functions as a consonant suspension in bar 7, forming a consonant perfect fifth with the cantus B. Because the Fit is consonant, it can be quit by leap. • Listen to example A.I Ob. The next step is illustrated in example A. lOc. I try accompanying the cantus E-D-C of bars 3-5 with a succession of thirds: G in bar 3, to Ftt in bar 4, to a consonant E on the first half of bar 5.

220

• Listen to example A.IOc. I complete the co-line as illustrated in example A.lOd. 1 begin with a D on the second half note, followed in bar 2 by a consonant C, then by a leap to a consonant A. Example A.I Ob

Example A.IOc

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• Listen to example A. lOd. The leap C-A of bar 2 is unrecovered; thus the coline cannot be primordial. I attempt to eliminate this problem, as illustrated in example A.lOe. Instead of

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leaping up from the E in bar 5,1 continue the descent to a consonant C in bar 6. The consonant C is followed by a consonant D on the first half of bar 7. To avoid another unrecovered leap in bar 7,1 replace the co-line Example A.10A

Example A.10e.

221

with a leap to a consonant G in that bar, followed by the consonant Fit in bar 8, E in bar 9, and Ftl in bar 10. • Listen to example A.lOe. Akhough the leap C-A of bar 2 is now recovered,

222

other problems arise. The C of bar 6 is the lowest tone of a descending motion that includes Ftt; the resultant augmented fourth is problematic. Also problematic is the succession of thirds in bars 3-7: E:G-D:Ftt-C:EA:C-B:D. As illustrated in example A.I Of, I attempt to eliminate these problems. I try replacing the descent to C in bar 6 with a stepwise ascent to G. I then follow the consonant G of bar 7 with a leap down to a consonant D, which becomes a consonant suspension in bar 8 and is followed by a consonant Ftt. • Listen to example A.I Of. Although the original leap C-A is recovered and the

Example A.I Of

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problematic thirds and augmented fourth from example A.lOe are eliminated, the co-line is still unsatisfactory. The Fl-E-Ftt-G of bars 4-7 is repeated with the same rhythmic values in bars 8-11. This repetition prevents the tones from participating discretely, and thus prevents primordiality. As illustrated in example A.lOg, I try eliminating this repetition by replacing the half notes Ftt-G of bars 6-7 with a suspended A, followed by a suspended G. Removing the low D prevents the leap C-A from being recovered, however. I try to correct the problem by beginning the co-line on the first half of the bar with a D, then leaping up to B on the second half.

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• Listen to example A.IOg. This completed counterpoint, presented again in example A.lOh, is successful. The stepwise foundation of the co-line (D-E-Ftt- G) is rare in that it ascends exclusively; it is given in example A.lOi. Example A.I Og

Example A.I Oh

223

• Perform the stepwise foundation of the upper line along with the corresponding tones of the cantus (thus G:D-C:E-A:Ftt-G:G), so that the impulse is consequently resolved.

224

You should have found that the height of the impulse must come with the second interval, the third C:E.21 Again, the upper-line fit that climaxes the impulse of the counterpoint participates in the motion to the E of bar 5 that in turn climaxes the impulse of the stepwise foundation. Examples A.lla-A.llh illustrate the composition of a fourth species co-line below the F major cantus. I begin by notating the cantus in the uppermost line (example A.lla). • Sing the cantus without the music. Example A.IOi

21. There is an interesting parallel here with harmonic progressions, where subdominant function chords often serve as the height of the impulse.

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Then I notate the possible consonances in the lowest line. I begin composing the co-line as illustrated in example A.lla. I first try a consonant D with the cantus A of bar 8. The D becomes a consonant suspension in bar 9; I follow it with a leap to a consonant F. The F, which becomes a dissonant suspension in bar 10, resolves to a consonant E before moving to the final tonic in bar 11. • Listen to example A. 1 la. These final four bars seem fine. As illustrated in example A.lib, I try accompanying the cantus C-Bl>-A of bars 6-8 with a succession of sixths: E, resolving to D, moving to a consonant C on the first half of bar 8.

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• Listen to example A.lib. Bar 8 of this example is problematic. The C sounding on the first part of the bar is not suspended; thus it functions in second species and must be consonant. The Example A.I la

Example A.I Ib

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D on the second half of the bar must also be consonant, for it is suspended into the next bar. The two consonant intervals, the sixth C:A and the fifth D:A, conflict with each other. Therefore, either (a) one of the co-line tones

226

must be dissonant, which it cannot be, or (b) the cantus A participates in two harmonic events, which it cannot do. Example A. He illustrates an attempt to eliminate this problem. I suspend the D of bar 7 into bar 8, where it resolves to a consonant C. The D of bar 9 sounds on the first half of the bar. Next I try accompanying the cantus A-G-A of bars 3-5 with C-Bl>-A. The C becomes a dissonant fifth in bar 4, resolving to the consonant Bl>; the Bl> becomes a dissonant seventh in bar 5, resolving to the consonant A. • Listen to example A. 1 Ic. Bar 6 presents two problems. First, the leap A-E is

Example A.I Ic

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unrecovered. Second, the cantus C participates in either two harmonic events or a single dissonant one. In example A. 1 Id I eliminate this problem by accompanying the cantus G of bar 4 with a G, followed by a resolution to F in bar 5. The F becomes dissonant in bar 6, resolving to the consonant E. • Listen to example A. 1 Id. Both the problems from the previous version of bar 6 have been eliminated. Next I complete the exercise, as illustrated in example A.lie. I open with an F on the second half of the bar, followed by a resolution to E, which moves to a consonant F on the first half of bar 3. The F leaps to a consonant A.

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• Listen to this complete counterpoint represented in example A. He. The first three bars have two problems. First, the E of bar 2 is not a consonance but a dissonant diminished Example A.I Id

Example A.I le

227

twelfth. Second, the leap to A in bar 3 followed by the resolution to G in bar 4 results in the parallel octaves A:A-G:G. I try a different opening, illustrated in example A.I If.

228

This time I begin with an F on the first half of the bar, followed by a leap to A. I accompany the Bt-A-G of bars 2-4 with a series of 9-10 suspensions. The descending motion ends with a D on the first half of bar 5. • Listen to the completed counterpoint as given in example A. 1 If. Although this counterpoint is not bad, there is a disturbing repetition of the D-F-E, which is sounded first in bars 5-6 and then again with the same rhythmic values in bars 9-10. I try to remedy this problem by moving the leap of an ascending third to the preceding bar, as illustrated in example A.I Ig. • Listen to example A.I Ig. Example A.I If

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The completed counterpoint is given in example A.llh. The succession of ascending leaps followed by stepwise descents tends to be a common characteristic of fourth species co-lines. It does not create a problematic pattern here, probably because the first descent encompasses four tones (A-G-F-E in bars 1-4), the second encompasses five tones (G-F-E-D-C in bars 4-8), and the third encompasses only two (F-E in bars 910). Two-Part Fifth Species This section illustrates the composition of two fifthspecies counterpoints, one with a co-line above the F

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major cantus and one with a co-line below the D minor cantus. Examples A.12a-A.12i illustrate the composition of a fifth species co-line above the F major cantus. I begin by notating the cantus in the lowest line (example A.12a). • Sing the cantus without the music. Example A.I Iff

Example A.I Ih

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Then I notate the possible consonances in the uppermost line. I begin composing the co-line by sketching an outline, as illustrated in example A.12a. This outline may have one or more tones in a bar, and need not be primordial. • Listen to example A.I2a.

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This outline suggests an impulse that climaxes with the D in the second bar. As illustrated in example A.12b, I attempt both to increase the impulse and to Example A. I2a

Example A.I2b

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have it climax later in the co-line. I try descending from an opening F, and then leaping up to the C in bar 3. • Listen to example A. 12b.

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I then begin to fill in the outline, as shown in example A. 12c. I ornament the D in bar 2 with a neighbor cambiata. Next I transform the C-Bt-A of bars 3-5 into a series of suspensions. The consonant A of bar 6 is followed by a stepwise descent to a low D, recovering the octave leap C-C. The G of bar 7 sounds on the second half of the bar and is suspended into bar 8, where it resolves to the F. The F becomes a consonant suspension in bar 9 and is ornamented there by the successive neighbor tones E and D. The E of bar 10 is ornamented with a neighbor cambiata. • Listen to example A. 12c. One problem with this counterpoint lies in the initial

Example A.12c

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impulse. The gathering of energy resulting from the quarter notes of bar 2 is weakened by the longer half note C on the first half of bar 3. As illustrated in example A.12d, I attempt to permit the impulse to grow unimpeded until the upper octave C. I try sounding the opening F on the second half of the bar, suspending it into bar 2, and sounding the D as a half note on the second half of the bar. A second problem lies in the overall resolution, which is too great for the impulse gathered. To remedy this, I try to create additional impulse in bar 6.1 resolve the dissonant B\> of bar 5 to the consonant A on the second quarter, and ornament the A with a descent to a conso-

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nant F. The A of bar 6, as the height of a small ascending motion, creates a degree of impulse. • Listen to example A.I2d. Example A.I 2d

Example A.l2e

22. Although the volume would decrease somewhat over the first two and a half bars in a musical performance, the rhythmic activity would further the impulse.

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The impulse of the counterpoint is more clearly defined,22 and the resolution can come closer to playing out the impulse consequently. But the energy still seems

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to die out with the suspended half notes in bars 8-9. As shown in example A.12e, I try to provide additional energy with an eighth-note neighbor ornamentation in bar 8. • Listen to example A. 12e. A minor problem with this counterpoint lies in the overly conjunct nature of the final four bars. I try to corExampleA.Uf

Example A.I2g

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rect this by replacing the E of bar 9 with a leap to a consonant C, as shown in example A.12f. • Listen to this version of the completed counterpoint, which is given in example A.12g. There is still a problem with the dynamic structure of this otherwise successful counterpoint. The resolution is still too great for the impulse. An examination of the

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stepwise foundation of the co-line proves helpful. In fact, the stepwise foundation is not that suggested in example A.12b. To find the stepwise foundation, I first consider the consonant tones only (indicated by asterisks in example A. 12g). Beginning with the consonant F and moving by step, the consonant tones yield a stepwise foundation of F-E-D-E-F, as illustrated in example A.12h. If the C of bar 3 were to climax the impulse, it would not participate in the motion to D, which climaxes the stepwise foundation; instead it would participate in the motion to E. Example A.12h

Example A.12i

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The impulse then must climax either in the motion to the D from the tone before, with the D itself, or somewhere widiin its unfolding. The E of the stepwise foundation directly precedes die D widiin the co-line proper; thus the impulse must climax either with die D itself, or perhaps with die G of bar 7. • Listen to example A.12g, performed so that the G of bar 7 climaxes the impulse. The resolution is now insufficient to the impulse. • Now listen to example A.12g performed so that the D of bar 7 climaxes the impulse.

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This is better, but still not all the energy that has been gathered is released. As illustrated in example A. 12i, I remove the ornamentation from the final four bars, returning them to the version of example A.12d. • Listen to example A.I2i. This version of the completed counterpoint is successful. The impulse does climax with the D of bar 7, which also climaxes the impulse of the stepwise foundation (example A.12h). The final four bars allow the impulse to be resolved. Examples A.13a-A.13g illustrate the composition of a fifth species co-line below the D minor cantus. I begin

Example A.I3a.

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by notating the cantus in the uppermost line (example A.13a). • Sing the cantus without the music. Then I notate the possible consonances in the lowest line. Example A.lSa gives an outline sketch of the coline. The Bl> of bar 7 that climaxes the cantus is accompanied by the lowest tone of the co-line sketch, the G. • Listen to example A.I3a. I attempt to create more impulse through the first three bars. As shown in example A.lSb, I begin the coline on the second half of the bar. I flesh out the Cit of bar 2 with a lower neighbor B^. On the second half of bar 3 I leap to an F, which I suspend into bar 4, where

23d it resolves to an E. I ornament the D of bar 5 and the G of bar 7 with neighbor cambiatas, the A of bar 8 with a leap down to F, and the Bl> of bar 9 with a leap up to a consonant D. • Listen to example A. 13b.

Example A. 13b

Example A. 13c

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This co-line has two problematic contours. The first is in bars 2-3, where the ascending motion from B^l to F describes a diminished fifth. Also, the altered B^l does not participate in an ascending motion from the fifth scale degree. The second problematic contour is in bars

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5-7, where the descending motion from E to F describes a major seventh. As illustrated in example A.13c, I attempt to solve the problem in bar 2 by leaping down to an A. I then change the position of the ornamental C and E in the neighbor cambiata of bar 5, so that the descending motion begins with the final D of the bar. • Listen to example A.I3c. A minor problem is created by the perfect fifths between the co-line A and cantus E of bar 2, and co-line D and cantus A of bar 3. A bigger problem lies with the dynamic structure, which is fairly nebulous. As shown in example A. 13d, I attempt to solve the first problem by descending by half notes from the opening D to an A in bar 3. Then I attempt to create more impulse into

Example A. 13d

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the climactic bar 7 by adding eighth-notes in bar 6. • Listen to example A. 13d. The opening seems to present no problems. A problem of overall dynamic structure remains, however, for the additional impulse created in bar 6 is still insufficient to carry through to the end of the counterpoint. Example A.13e illustrates an attempt to remedy this problem with an octave leap up to a G on the second half of bar 7. The G is suspended into bar 8, where it resolves to F and then descends to the Bt of bar 9. I attempt to increase the impulse within the final three bars by ornamenting the A of bar 11 with a lower neighbor G. This delays the ascent to the final D, and allows for quarternote motion in bar 12.

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Example A.I3e

Example A.I 3f

Example A.lSg

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• Listen to example A.I3e. The final three bars are still unsatisfactory, because the energy is still played out before the end of the exercise. Moreover, the G that begins the ascending motion of bars 11-13 participates with the Ctt in creating a problematic augmented fourth. Both problems can be eliminated by ornamenting the A of bar 11 with a leap to F, as shown in example A.13f. The succession of four quarter notes injects additional impulse, sufficient to carry the energy through to the end of the exercise. I also try a minor improvement in bar 4. The resolution of the dissonant suspension to the half note E in bar 4 is followed somewhat suddenly by quarter-note motion in bar 5. Anticipating the resolution E by a Example A. 14a

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quarter note better prepares the growth of impulse, which climaxes in bar 7. • Listen to the final version of this counterpoint, given in example A.13g. Three-Part First Species This section illustrates the composition of two threepart counterpoints in first species, one with the G major cantus sounding in the middle line and one with the F major cantus sounding in the upper line. Examples A.14a-A.14g illustrate the composition of a three-part exercise in first species with the G major cantus sounding in the middle line. I begin by notating the cantus in the middle line (example A.14a).

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• Sing the cantus without the music. Next I sketch in the more restrictive of the two colines. The co-line in the lower line is more restrictive, for it must begin and end with the tonic. As illustrated in example A. 14a, I sketch in a co-line in the lower line. I think of it as a sketch because it may remain or it may be changed, depending on the possibilities it allows for the upper line.23

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• Listen to example A. 14a. As illustrated in example A.14b, I next sketch in the consonant possibilities for the upper line. Keep in mind that if changes are made in the lower line, the consonant possibilities for the upper line may change as well. I then begin composing the upper line. I try leaping from G to D in bars 7-8, then descending through C to A and then to B.

Example A.14b

23. As you compose more three-part counterpoints, you will find yourself keeping considerations of the other co-line in mind as you

sketch the first co-line. Eventually you may well find yourself composing both lines simultaneously.

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• Listen to example A. 14b. The D-G-C-A of bars 7-10 creates a compound melody. As illustrated in example A. 14c, I replace bars 8-9 with repeated B's. I also try a C in bar 6. • Listen to example A. 14c. Although this segment of the counterpoint is not problematic, bar 5 is. There cannot be repeated C's in bars 5-6 because of the concurrent repeated C's in the lower line. An A in bar 5 would result in an unrecovered leap A-C; a G would result in a harmonic embellishment (a leap to and from the same note); and an E would result in a triad E-C-G.

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As illustrated in example A. 14d, I begin again, using the G in bar 7, which I follow with a stepwise descent to a final D. I try beginning with a G, moving to A in bar 2, and leaping to a C that climaxes the impulse in bar 3. This C begins a descent to a repeated A. The repeated A's defeat what would otherwise be a dissonant contour, from the C of bar 3 down to the final D. • Listen to the completed counterpoint given in example A. 14d. Two problems arise. First, the upper line G and A form octaves with the first two tones of the cantus. Second, the resolution is too great for the amount of im-

Exa.mpleA.14c

242 Example A.14d

Example A.14e

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pulse generated. Example A. 14e gives a simple solution for each of these problems. To solve the first problem, I replace the opening G with B. This eliminates the parallel octaves. To eliminate the second problem, I replace the repeated A of bar 6 with a leap down to E. The succeeding leap to G energizes the resolution sufficiently to allow it to carry through to the end of the exercise.

Example A.14f

Example A.14$

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• Listen to example A. 14e. This counterpoint is shown in its final version in example A. 14f; the stepwise foundation of the upper line appears in example A.14g. The climax of the counterpoint comes with the upper-line C of bar 3; the climax of the upper-line stepwise foundation comes with the B ornamented by the C.

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Examples A.15a-A.15d illustrate the composition of a three-part exercise in first species with the F major cantus in the upper line. I begin by notating the cantus in the upper line (example A.15a). • Sing the cantus without the music. Again, I next compose the more restrictive of the two co-lines. Because the co-line sounding in the middle line generally has a smaller range than those in the two outer lines, it seems more restrictive. As illustrated in example A.15a, I sketch in a co-line in the middle line. Note that the middle and upper lines of a three-part counterpoint are different from the two lines of a twopart exercise. First, the intervals created by the simultaneous sounding of the two lines are not themselves the final harmonic event-they participate in that event. Thus an interval that has been problematic in two-part counterpoints may occur between tones of the two upper lines, if its component tones participate in a consonant harmonic event. For example, diminished fifths (as in bar 7) may occur between the middle- and upper-line tones if diey participate in a first-inversion diminished triad. Likewise, perfect fourths (as in bars 1, 3, 4, 8, and 11) may occur between middle- and upper-line tones if diey participate in a major or minor triad in first inversion or root position. Also, both lines may leap in the

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same direction simultaneously (as in bars 5-6), provided that the other line does not also leap in that direction. • Listen to example A. 15a. As illustrated in example A.15b, I next sketch in the consonant possibilities for the lower line. Again, should the middle line change, the consonant possibilities for die lower line may change as well. I then begin composing the lower line, as shown in example A.15b. I try a line that ascends to the tonic by step from a C in bar 8. I open die co-line with an ascent to G, a leap to C, and a stepwise return to G. • Listen to example A.I5b. The first two bars have parallel fifths between the lower and middle lines. Also, I am not entirely satisfied widi the repeated A in bar 6. The cantus C in bar 6 climaxes the impulse, but diis climax is reduced somewhat by the repeated tone in the lower line. As illustrated in example A. 15c, I replace die opening two bars of the middle line with F-E. Because of the repeated E's in bars 2-3,1 try to eliminate the repeated E's in bars 7-8. I try replacing the middle-line E of bar 7 with a D. To increase the impulse slighdy, I replace the lower-line A of bar 6 with a leap to F. Note that two of these middle-line alterations change

APPENDIX: Example A. 15a

Example A.I 5b

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245

246 Example A.I5c

Example A.I5d

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the consonant possibilities. In bar 2 the E:Bl> can be consonant only with a G in the lower line. And in bar 7 the D:Bl> can now be consonant with a Bl> in addition to the G. In bar 1, however, the lower line must still sound F. • Listen to this version of the counterpoint, as given in example A.15c. This successful counterpoint is shown in example A. 15d. The upper-line C of bar 6 climaxes the impulse, which can be consequently resolved.

Example A.I6a

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Three-Part Second Species This section illustrates the composition of two threepart counterpoints in second species, one with the G major cantus sounding in the lower line and one with the F major cantus sounding in the middle line. Examples A.16a-A.16g illustrate the composition of a three-part exercise in second species, with the G major cantus sounding in the lower line and the active co-line sounding in the upper line. I begin by notating the cantus in the lower line (example A.16a).

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• Sing the cantus without the music. Next I sketch in the more restrictive of the two colines. Again the middle line seems somewhat more restrictive, because of its more limited range. As illustrated in example A.16a, I sketch in a co-line in the middle line. • Listen to example A.I6a. Next I sketch in the consonant possibilities for the upper line, as shown in example A. 16b. I then compose the active upper line. I begin in bar 7 with an orna-

Example A. 16b

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mented Ft, moving to E in bar 9, to C in bar 10, and up to the final D. • Listen to example A.I6b. The upper-line D and E of bars 8-9 form parallel octaves with the middle-line D and E. As illustrated in example A.16c, I replace bars 7-8 with a leap up from D to a consonant G, suspended over the bar and resolving to a consonant Fit. Next I try an A in bar 4, ornamented by an upper neighbor B and followed by a stepwise descent to the D of bar 7.

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To have the consonant A in bar 4,1 must change the passive co-line in the middle line. I try replacing the B in bar 4 with an A. To avoid three successive A's, I then move to a C in bar 5, repeated in bar 6. This necessitates a change in bars 7-8, which I replace with B-A. The resultant leap down to E is recovered by the final Fl-G. This alteration necessitates altering the opening three bars as well, so I leave them blank for now. And whenever I change the passive co-line, I must reexamine the

Example A.I 6c

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consonant possibilities for the active co-line. In this case the consonant possibilities for the active co-line change in bars 4 and 5, but not in the other bars. • Listen to example A. 16c. There seem to be no problems with this segment of the counterpoint. Example A.16d illustrates the completion of the exercise. I begin the new passive co-line with D-C-G. As a result, the consonant possibilities for the active co-line change in bars 1 and 3. I try opening the active co-line with a suspended B on the second half of

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the bar. The resolution to A in bar 2 is followed by a return to B in bar 3. • Listen to the completed counterpoint given in example A. 16d. One problem is created by the parallel octaves G:GA:A in bars 3-4 between the active and the passive colines. A second problem results from an almost complete absence of impulse. As illustrated in example A.16e, I first replace the middle-line tones of bars 3-4 with repeated B's. I try leaping down from the opening D to an A in bar 2. These changes result in new consoExa-mpleA.16d

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nant possibilities for bars 3-4. I try replacing the opening four bars of the active co-line with an ascent to B in bar 4, a leap down to D, and a return to A in bar 5. • Listen to example A.I6e. You should have found that the impulse in this counterpoint must come with the C:C:A of bar 5 to be resolved consequently. The dynamic structure of this counterpoint might cause some confusion. A visual inspection of the upper line may suggest that it should climax with the highest tone B, sounding on the first half of bar 4. Confirma-

APPENDIX: Example A.I6e

Example A.I6f

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251

252

tion of the requisite dynamic structure lies only in our experience of the counterpoint as an indivisible unit. An examination of the stepwise foundation can at best suggest how to find that experience, or in this case offer some support for the logic of a dynamic structure. The asterisks in example A. 16f indicate the consonant tones. As illustrated in example A.16g, the stepwise foundation is G-A-G-Flt-E-D. The A then climaxes the stepwise foundation; thus it is logical that it is also the climax of the co-line itself. Examples A.17a-A.17J illustrate the composition of a three-part exercise in second species, with the F major cantus sounding in the middle line and the active co-line sounding in the lower line. I begin by notating the cantus in the middle line (example A. 17a). • Sing the cantus without the music. Next I sketch in the more restrictive of the two coExample A.l6g

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lines. This time the lower line seems more restrictive because of the given first and last tones, and because of the difficulties presented by a second species co-line below the cantus. As illustrated in example A.17a, I sketch in the active co-line in the lower line. • Listen to example A.I7a. The ascending motion that begins with the consonant B!> of bar 2 and includes the consonant E of bar 4 unfolds a problematic augmented fourth. To avoid this I try opening with an F one octave lower, as illustrated in example A.lTb. Next I sketch in the consonant possibilities for the upper line. I then consider the passive upper line. I begin the upper line with repeated D's in bars 89 and repeated C's in bars 10-11. • Listen to example A.I7b. The repeated D's are especially problematic because they sound in conjunction with two D's in the lower line. Instead, as illustrated in example A.17c, I try end-

APPENDIX: Example A.17a

Example A. 17b

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253

254 Example A.I 7c

Example A .17d

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ing the exercise on the third, A. I replace the D of bar 8 with an F, which moves to A, then G. For bars 6-7 I try F and G. • Listen to example A.I7c. The six tones of the upper line suggest two linear embellishments, one of F and one of A. The second embellishment is reinforced by the cantus F-G-F sounding concurrently. I try to correct this, as shown in example A.17d, by replacing bars 6-7 with A and Bt, and try another A in bar 5. • Listen to example A.I7d. Exa.mfleA.17e

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The A and Bl> of bars 6-7 form parallel octaves with the A and Bt of the active co-line. The solution to this problem is difficult. If I replace the upper-line B!> of bar 7 with F, then I can precede it in bar 6 only with another F, which would result in three successive F's, or with an A, which would result in a leap to and from the same note, or with a D, which would create an unrecovered leap and a triad D-F-A. As illustrated in example A. 17e, I can try a G in the upper line in bar 7.1 then try to complete the opening of the exercise with F-EC-G.

256 • Listen to example A.I7e. The G and A of bars 4-5 form parallel octaves with the cantus G and A. And the leap C-G of bars 3-4 is unrecovered. Replacing the G of bar 4 with an E would result in a triad C-E-A. Also, this upper line could not provide enough impulse. As illustrated in example A. 17f, I erase these first four bars. I return the upper line of bar 7 to Bl>, and try changing the lower line to avoid the parallel octaves in bars 6-7.1 try a low D in bar 7 in the lower line, moving to an ornamented F in

ExampleA.17f

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bars 8-9, E in bar 10, and finally F in bar 11. Next I try approaching the D of bar 7 with a stepwise descent from a Bl> in bar 4. I try opening with an F, moving to an ornamented D in bars 2-3, and descending by step to a low Bl> in bar 4. • Listen to example A. 17f. The active co-line has two problems. The descending motion beginning with E in bar 2 and ending with Bl> in bar 4 forms a dissonant contour, and the octave leap Bl>-Bl> is unrecovered. Example A.17g illustrates an at-

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tempt to correct these problems. I try replacing the stepwise descent to D in bar 7 with a descending leap A-C in bar 6.1 approach the A of bar 6 with a stepwise descent from E in bar 4. Having changed the lower line, I must reexamine the upper line to ensure that its tones are still consonant. • Listen to example A.I7g. This segment seems to present no problems. I then complete the active co-line, as illustrated in example A. 17h. I follow the initial tonic of bar 1 with an orna-

Example A.I

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mented D in bars 2-3. Considering the new consonant possibilities for the upper line, I begin with repeated F's and leap up in bar 3 to a D, which I follow with a C in bar 4. • Listen to example A.I7h. The impulse climaxes with the upper-line D of bar 3, and is resolved from there to the end. The resolution is given additional energy by die secondary impulse that climaxes with the upper-line Bl> of bar 7. Although the low C in bar 6 forms an interval of a fifteenth with the

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cantus C in the middle line, it does not destroy the conjunction of the lines. This is perhaps because (a) the first interval of bar 6 is the tenth A:C, (b) the A is approached by step, and (c) the C has an ornamental character. It is followed in bar 7 by the unproblematic thirteenth D:Bk The final version of this successful counterpoint is given in example A.17i. Example A.17J gives the stepwise foundation of the upper line along with the corresponding tones of the other lines. This three-chord fundamental progression reaches a climax with the second Example A. 17h

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chord. Thus the G climaxes the stepwise progression of the upper line; the climactic D of the counterpoint participates in the motion to that G that climaxes the stepwise foundation. Three-Part Third Species This section illustrates the composition of two threepart counterpoints in third species, one with the G major cantus sounding in the middle line and one with the D minor cantus sounding in the upper line. Examples A. 18a-A. 18f illustrate the composition of a

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three-part exercise in third species with the G major cantus sounding in the middle line and the active co-line sounding in the upper line. I begin by notating the cantus in the middle line (example A.18a).

Example A. 17i

Example A.17j

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• Sing the cantus without the music. As illustrated in example A.18a, I next sketch in the passive co-line in the lower line. I then notate the consonant possibilities. Because of the great number of

260

tones involved, composing third species co-lines can be fairly easy. I sketch in an active third species co-line in the upper line. • Listen to example A. 18a. Although there are no problems of either primordiality or conjunction, the line meanders with no particular structure of dynamic forces. I attempt to redo the active co-line with a more clearly defined dynamic structure, as illustrated in example A.18b. I decide to replace bars 5-7 with an impulse-giving octave leap in bar 5, moving to an ornamented Fit in bar 6 and then to G in Example A.I8a

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bar 7.1 try descending from an Ftt in bar 8 to an ornamented B followed by a leap to E in bar 9. The E participates in a stepwise resolution that carries from the climactic A of bar 5 through the G of bar 7, the Ftt of bar 8, and the E of bar 9 before reaching the final D. In example A.18a the lower-line tone of bar 6 was C. This would form a dissonant compound tritone with the new upper-line Ftt. Thus I replace the lower-line C with a consonant A. To avoid parallel octaves between the lower and middle lines in bars 6-71 replace the lower-line B with a G.

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• Listen to example A.I8b. I complete the exercise as illustrated in example A.18c. To avoid two successive repeated tones in the lower line (G-G in bars 3-4 and A-A in bars 5-6) I replace bars 2-4 with D-C-B. I reconsider the consonant possibilities for the upper line, which change for bars 3 and 4. I then compose an active co-line that begins with D and moves through passing cambiatas in bars 2 and 3 and a neighbor cambiata in bar 4 to the A of bar 5. • Listen to example A.I8c. Example A. 18b

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Again there are no problems of line or conjunction, but the dynamic structure is weak. As illustrated in example A.18d, I attempt to define the impulse more clearly. I begin with an ornamented B, followed by a gradual ascent to the climactic A, which I sound on the first quarter of bar 6. • Listen to the completed counterpoint given in example A.I 8d. The final version of this particularly beautiful exercise is presented in example A. 18e. It climaxes with the first A of bar 5 in the upper line. The asterisks indicate the

262 Example A.I8c

Example A.18d

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consonant tones, which yield the stepwise foundation illustrated in example A.18f. The climax of the foundation comes with the E of bar 9. The A that climaxes the co-line participates in the motion to the climactic E of the foundation. Examples A.19a-A.19e illustrate the composition of a three-part exercise in third species, with the D minor ExampleA.lSe

Exa.mpleA.18f

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cantus sounding in the upper line and the active co-line sounding in the lower line. I begin by notating the cantus in the upper line (example A. 19a). • Sing the cantus without the music. Next I sketch in the more restrictive of the co-lines. The lower line is the most restrictive, because it cannot leap a fourth or fifth within a bar, and because its many notes must meet the requirements for chromatic alter-

264

ation of the minor mode. As illustrated in example A.19a, I begin to compose the lower line with an ornamented G in bar 7, gradually moving down to the final Example A.19 a

Example A. 19b

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D. In the uppermost staff I then notate the consonant possibilities for the passive co-line sounding in the middle line. Next I sketch in an accompanying middle line,

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beginning in bar 7 with G and followed by A-G-GD-E-F. • Listen to example A.I9a. The active co-line seems overly conjunct. As shown in example A.19b, I try to remedy this by replacing bar 7 with an impulse-giving leap and then by having neighbor cambiatas ornament F and E in bars 5 and 6. The altered lower line in bar 7 changes the consonant possibilities for the middle line. I replace the middle-line G widi an F. I then try D and E in bars 5-6. • Listen to example A. 19b. Keeping in mind the necessity to create impulse, I continue to compose the exercise, as illustrated in examExample A.19c

265

ple A.19c. I take out bar 5 of the active co-line, instead creating an ascent from a low F in bar 3 to the D of bar 7. This alteration of the active co-line in bar 5 changes the consonant possibilities for the middle line. I try an E in the middle line in bar 4, and replace bars 5-6 with A and G. • Listen to example A.I9c. As illustrated in example A.19d, I complete the active co-line with a descent from the tonic D to an ornamented F in bar 3. I then complete the passive co-line, considering the consonant possibilities. • Listen to the completed counterpoint given in example A. 19d.

266

Although the low F in the active co-line in bar 3 is more than three octaves from the cantus A, it does not prevent conjunction. This is probably because the tones between adjacent lines do not form any intervals greater Example A.19d

Example A.I9e

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than a thirteenth, and because the initial tone of bar 3 is the upper F, which is approached by step. The lower F is ornamental in nature; it moves by step to the G of bar 4. Neither the upper F of bar 3 nor the G of bar 4

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forms an interval greater than three octaves with the upper-line tones. The final version of this successful counterpoint is given in example A. 19e. The exercise climaxes in bar 7 with the cantus E\> in the upper line; the climax of the impulse is supported by the ascending leap to Bl> in the active co-line. Three-Part Fourth Species This section illustrates the composition of two threepart counterpoints in fourth species, one with the D minor cantus sounding in the lower line and one with the F major cantus sounding in the upper line. Example A.20a

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Examples A.20a-A.20h illustrate the composition of a three-part exercise in fourth species, with the D minor cantus sounding in the lower line and the active co-line sounding in the upper line. I begin by notating the cantus in the lower line (example A.20a). • Sing the cantus without the music. Next I sketch in the more restrictive of the two colines. Because of the requirement that dissonant suspensions resolve by descending, the active co-line is more restrictive. As illustrated in example A.20a, I sketch in an active co-line in the upper line. • Listen to example A.20a. This co-line has one problem. The B'l-E-D-Ctt of

268

bars 9-12 joins with the cantus D-G-F-E to form four consecutive thirteenths. As illustrated in example A.20b, I try a leap up to A in bar 5 descending to D, followed by a leap up to F in bar 9 descending to the final Cl-D. To avoid a repetition of the untied half notes E-D, I try moving to F in the first half of bar 5. • Listen to example A.20b. Unfortunately the F in bar 5 and the G in bar 6 form parallel octaves with the cantus F-G. Also, the octave leap G-G of bar 2 is unrecovered. As illustrated in example A.20c, I eliminate the octave leap, beginning instead with untied half notes F and E. To avoid the parallel octaves I replace the F of bar 5 with a D, as in Example A.20b

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example A.20a. Note that bars 8-9 seem to contain parallel fifths, but in fact do not. The F of bar 8 must be consonant because it is suspended, but the E of bar 9 is a dissonant passing tone following the consonant F. The disruptive effect of the untied half notes E-D sounding in both bars 4-5 and 8-9 can be lessened if they participate in two completely different harmonic events. I keep this in mind as I compose the passive coline. I sketch in an A in bar 5 and a Bl> in bar 9. • Listen to example A.20c. As shown in example A.20d, I flesh out the passive co-line around these two tones. The upper-line E and D of bars 4-5 participate in a first-inversion E diminished

APPENDIX: Example A.20 c

Example A.20A

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triad and a first-inversion D minor triad. By contrast, the upper-line E and D of bars 8-9 participate in a root-position A minor triad and a first-inversion Bl> major triad. • Listen to example A.20d. The opening of the passive co-line is problematic, for it is little more than a linear embellishment of A. Unfortunately, the repetition of E-D in the upper line is still bothersome as well. As shown in example A.20e, I try to solve the latter problem by replacing bar 9 with F-

Example A.20e

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D. The D is suspended into bar 10, where it becomes a dissonant fifth resolving up to the consonant sixth. I then try to eliminate the linear embellishment in the middle line by beginning with a D, and then descending by step to the A of bar 5. • Listen to example A.20e. This counterpoint is acceptable, but the fifteenth between the first tones of the lower and middle lines threatens their conjunction. To solve this problem I try bringing both co-lines down an octave, as illustrated in

APPENDIX: Example A.20f

Example A.20$

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example A.20f. This works in all but bars 7 and 10, where the Bl> and G would form unisons with the tones of the cantus. I try replacing the Bl> of bar 7 with a repetition of D, and the G of bar 10 with a repetition of Bk To avoid four successive thirds between the passive coline and the cantus in the last four bars, I replace the final F of the passive co-line with A. • Listen to example A.20f. A final version of this successful counterpoint is given in example A.20g. The stepwise foundation of F-E-DCtt-D in the upper line is given with the corresponding tones of the lower and middle lines in example A.20h. The foundation climaxes with the first of the upper-line D's; the counterpoint itself is climaxed by the A of bar 5, which participates in the unfolding of the fundamental D. Example A.20h

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Examples A.21a-A.21e illustrate the composition of a three-part exercise in fourth species, with the F major cantus sounding in the upper line and the active co-line sounding in the middle line. I begin by notating the cantus in the upper line (example A.21a). • Sing the cantus without the music. Next I sketch in the more restrictive of the two colines. Because of its requirement to descend, the active co-line is again somewhat more restrictive. As illustrated in example A.21a, I sketch in a co-line in the middle line. • Listen to example A.21a. The resolutions to E in bar 2 and to D in bar 3 are possible only if the tones are consonant-in other words, if they participate in consonant harmonic events. Thus the E of bar 2 must participate in a first-inversion

APPENDIX: Example A.21 a,

Example A.21b

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diminished chord, and the D of bar 3 must participate in a D minor chord, either in root position or in first inversion. As shown in example A.21b, I try an opening for the passive co-line, to be sure that these tones will be acceptable. • Listen to example A.llb. The passive co-line can include G and F in bars 2-3 without any problem. After I consider the consonant possibilities I complete die passive co-line, as illustrated in example A.21c.

Example A.21c

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• Listen to the completed counterpoint given in example A.21c. Bars 6-7 unfold parallel fifths between the lower and middle lines. To avoid this problem I try a new passive co-line that does not include A in bar 6. As illustrated in example A.21d, I try replacing the F of bar 3 widi the other possibility, the D, and ascending through E and F in bars 4-6. The new passive co-line changes the harmonic environment for the active co-line. The B\> of bar 4 in example A.21c is no longer consonant. I replace the

APPENDIX: Example A.21 d

Example A.21e

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275

276 opening of the active co-line after I consider the new consonant possibilities. • Listen to the completed counterpoint given in example A.2 Id. The final version of this counterpoint is given in example A.21e. The cantus C of bar 6 climaxes the impulse, which can be consequently resolved. Three-Part Fifth Species This section illustrates the composition of two threepart counterpoints in fifth species, one with the F major cantus sounding in the lower line and one with the G major cantus sounding in the upper line. Examples A.22a-A.22g illustrate the composition of Example A.22a

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a three-part exercise in fifth species, with the F major cantus sounding in the lower line and the active co-line sounding in die middle line. I begin by notating the cantus in the lower line (example A.22a). • Sing the cantus without die music. As illustrated in example A.22a, I begin to sketch in the middle line a skeletal outline for the activefifthspecies co-line. Beginning in bar 7,1 try a G descending to a final A. • Listen to example A.22a. As illustrated in example A.22b, I flesh out die final five bars of the active co-line with neighbor cambiatas in bars 7, 9, and 10. After notating the possible consonances I sketch in the final five bars of the passive co-

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line in the upper line, leaping from D in bar 7 down to A in bar 8, and returning by step to the final C. Finally, I sketch the skeletal framework for the active co-line for bars 4-6. • Listen to example A.22b. Next I flesh out the active co-line in bars 4-6, as shown in example A.22c. I ornament the D of bar 5 with a lower neighbor C, followed by a leap down to Bl>. I fill in the leap C-F of bar 5, and ornament the E of bar 6. Considering the possible consonances for the passive co-line, I sketch in B!>- A-G for bars 4-6. I then sketch the framework for the opening three bars of the active co-line. Example A.22b

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* Listen to example A.22c. Example A.22d illustrates the completion of the counterpoint. I flesh out the opening three bars of the active co-line, leaping to E in bar 3. The E is suspended into bar 4, where the quarter notes D-C are compressed into eighth-notes. I complete the passive co-line after I consider the consonant possibilities; the co-line opens with A-G-C. • Listen to this completed counterpoint, as given in example A.22d. Each of the co-lines has a small problem. The active co-line has a dissonant contour from the F of bar 2 up to the E of bar 3. And the passive co-line has a disturb-

278 Example A.22c

Example A.22d

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ing repetition of A-G, which sounds in bars 1-2 and again in bars 5-6. As illustrated in example A.22e, I solve the first problem by exchanging the ornamental A and F in bar 2. The second problem is somewhat more complicated. Although it is solved easily enough by replacing the G of bar 6 with a high E, this solution creates another problem with the structure of dynamic forces. In example A.22d the leap up to D in bar 7 climaxed the impulse. The climax was supported by the gradual ascent to a cli-

Example A.22e

279

mactic G in the active co-line. As a result of leaping to an E in bar 6, the impulse climaxes one bar earlier. So diat the active co-line contributes to the structure of dynamic forces of the counterpoint, I try replacing widi a descent the active co-line ascent to G in bar 6. I also add an eighth-note passing tone in bar 8. This gives a bit of increased energy, necessary to carry the resolution through an extra bar. • Listen to this version of die completed counterpoint, as given in example A.22e.

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Example A.22F presents the final version of this successful counterpoint. The stepwise foundation of the upper line, given in example A.22g, is A- Bl> - C. In the context of the corresponding tones of the lower and

Example A.22f

Example A.22g

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middle lines, the climax of the foundation comes within the unfolding of the upper-line A. In the counterpoint itself, the climactic E participates in the unfolding of the A.

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Examples A.23a-A.23g illustrate the composition of a three-part exercise in fifth species, with the G major cantus sounding in the upper line and the active co-line sounding in the lower line. I begin by notating the cantus in the upper line (example A.23a). • Sing the cantus without the music. As illustrated in example A.23a, I begin to sketch a skeletal framework for the active co-line. Starting with a G in bar 7,1 try an ascent to the final G. • Listen to example A.23a. I flesh out the final five bars with neighbor cambiatas in bars 9 and 10, and begin in bar 7 with an octave leap

Example A.23a

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down to the low G, as illustrated in example A.23b. I then notate the consonant possibilities, and sketch in ED-C-A-B for the final five bars of the passive co-line. For the active co-line framework I sketch in a low A to join the impulse-giving cantus E of bar 3, and follow it with an ascent to the G of bar 7. • Listen to example A.23b. Next I flesh out the framework for the active co-line in bars 3-6, as shown in example A.23c. I ornament the A and Bl> of bars 3-4, and carry the ascent to an A in bar 6 by means of stepwise motion. I also ornament the G of bar 7, and compress the final two quarter notes

282

into eighth-notes. After I consider the consonant possibilities, I try C-B-C-E for bars 3-6 of the passive coline. I complete the framework for the active co-line, opening with G and then Fit. • Listen to example A.23c. There is a dissonant contour in the active co-line, from the C of bar 4 up to the B of bar 6.1 try replacing the neighbor tone B of bar 6 with a leap up to C, as illustrated in example A.23d. I also revise bar 7 of the active co-line. To create more impulse I suspend the A of

Exa.mfleA.23b

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bar 6 into bar 7, compressing the first two quarter notes into eighth-notes. To avoid the repetitions of A-B in bars 7-8,1 then eliminate the last eighth-note A. I flesh out the framework for the opening two bars with a dissonant suspension in bar 2 followed by two groups of eighth-notes leading to the climactic bar 3. Finally, I finish out the opening of the passive co-line with B-D in bars 1-2. • Listen to example A.23d. The active co-line contains another dissonant con-

APPENDIX: Example A.23c

Example A.23d

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tour, from the C of bar 6 to the Ft of bar 7. As shown in example A.23e, I correct this by leaping down to a consonant E.24 Also, I would like to clarify the structure of dynamic forces. The impulse of this exercise climaxes with the cantus E of bar 3. This is supported in the active co-line by the descent to the low A. Within the

Example A. 23e

24. This is not a problematic leap of a fourth in the lower line. The dissonant suspension A represents a G. Functionally, the leap is from a G down to an E.

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overall resolution a secondary impulse climaxes with the cantus B of bar 7. To support this dynamic structure I try to eliminate the half note E in bar 5, which halts the progress toward the secondary impulse. I substitute a quarter note and two eighth-notes in a stepwise ascent. I must then change bar 4 as well, to avoid a dissonant

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contour (from the C of bar 4 in example A.23d to the B of bar 5 in example A.23e). This necessitates a change in bar 3 as well. • Listen to example A.23e. I try one additional detail affecting the dynamic structure. The succession of rising quarter notes following the A of bar 3 suggests more impulse. For the impulse of the counterpoint to be resolved consequently, this secondary impulse should not start again until bar 5. As illustrated in example A.23F, I alter bar 4 so that it ends

Example A.23f

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with a half note, to halt the gathering of energy. • Listen to example A.23f. The final.version of this counterpoint is presented in example A.23g. The cantus E of bar 3 climaxes the impulse of this exercise. Mixed Species, Type 1 Examples A.24a-A.24j illustrate the composition of a type 1 mixed species counterpoint, with the G major cantus sounding in the lower line and the two second

286

species co-lines sounding in the middle and upper lines. I begin by notating the cantus (example A.24a). • Sing the cantus without the music. Next I begin to compose the two co-lines, as illustrated in example A.24a. I start in bar 7 with a suspended A in the upper line, which is followed by a descent to the final D. I accompany this in the middle line beginning widi an octave leap C-C, then a descent to the final G. • Listen to example A.24a. Bars 8-9 of the middle line unfold a compound melody. Also, the conjunction of the two co-lines presents a problem in bar 9, where the suspended Ftt in the upper Example A.23$

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line forms a dissonant seventh with the G of the middle line. Because each tone of the two co-lines must join to form a consonant interval, as in two-part counterpoints in first species, this is problematic. As illustrated in example A.24b, I replace the middle-line G with a suspended D, which is consonant with the upper-line Ftt. Next I try composing the opening of the upper line. I begin with a suspended D; I move down to B in bar 3, up to a consonant Ftt in bar 4, and then to the consonant E in bar 5. I accompany this in the middle line with an initial B and move gradually down to the D of bar 6. • Listen to example A.24b.

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As in bar 9 of example A.24a, the dissonant suspension A of bar 3 forms a dissonant second with the upper-line B. Moreover the concurrent suspensions in both co-lines in bars 8-9 interrupt the linear progress Example A .24a

Example A.24b

287

and threaten the independence of the lines. I scrap example A.24b and begin again with a new middle line, given in example A.24c. • Listen to example A.24c.

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The succession of four consecutive tenths (bars 3-6) fuses this line to the cantus. As illustrated in example A.24d, I replace the C-B of bars 2-3 with A-E, and change bars 6-9 as well. Example A.24c

Example A.24d

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• Listen to example A.24d. The octave leap A-A of bar 2 is unrecovered. Otherwise, this may prove successful. As illustrated in example A.24e, I replace the octave leap with a stepwise figure

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and then begin to compose the upper line. I begin in bar 5 with what may become a climactic Bl>, depending on what precedes it. After a descent to D in bar 8, the line leaps to A before moving to the final G. • Listen to example A.24e. I then try to complete the exercise, as shown in example A.24f. I borrow the opening three bars of the middle line and transpose them up an octave into the upper line, completing the ascent to the Bl> of bar 5 with stepwise motion. I then complete a new opening for the middle line. • Listen to the completed counterpoint given in example A.24f. The fifteenth between the middle and upper lines on the second half of bar 3 threatens their conjunction. Example A.24e

289

That this interval is achieved by leap in both lines exacerbates the problem. As illustrated in example A.24g, I try another new opening for the middle line. • Listen to example A.24g. This version of the counterpoint is fine, except that the fifteenth between the lower and middle lines to open the exercise strains their conjunction. I try transposing the cantus up an octave, but this produces crossed voices and an unacceptable unison with the middle line in bars 4 and 5. As illustrated in example A.24h, this solution can work if I revise the middle line in bars 4-6, leaping up to B and descending by step. • Listen to example A.24h. Example A.24i presents the final version of the counterpoint. It is climaxed by the upper-line A of bar 6. Ex-

290

ample A.24J illustrates the stepwise foundation of the upper line, given in the context of the corresponding tones of the lower and middle lines. The upper-line foundation of D-E-Ftt-G climaxes with the Fit. The A of bar 6 that climaxes the counterpoint participates in the unfolding of the fundamental Fit. Example A.24f

Example A .24g

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Mixed Species, Type 2 Examples A.25a-A.25h illustrate the composition of a type 2 mixed species counterpoint, with the F major cantus in the middle line, the second species co-line in the upper line, and the third species co-line in the lower line. I begin by notating the cantus (example A.25a).

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• Sing the cantus without the music. Next I begin to compose the two co-lines. As illustrated in example A.25a, I begin with an octave leap Bt-Bb in bar 4 in the upper line. A descent to D in bar 7 is followed by a leap to G, and then a descent to the final C. I try accompanying this with a third species coExamfle A.24h

Example A.24i

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line in the lower line, ascending from D in bar 4 to the Bl» in bar 9, and then moving down to the final tonic. • Listen to example A.25a. The lower line unfolds a dissonant contour from the Bl» of bar 9 down to the E of bar 10. As illustrated in example A.25b, I correct this by exchanging the orna-

292

mental F and D of bar 10. Next I complete the opening of the counterpoint. In the upper line I try descending from an opening F to the Bl> of bar 4. In the lower line I try opening with a leap down from F to A, followed by an ascent to the G of bar 4. Example A.24'j

Example A .25a

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• Listen to example A.25b. There is a dissonant contour in the upper line from the E of bar 2 to the Bt of bar 4. There are also parallel octaves between the two active co-lines in bar 2, where the upper-line D joins with the lower-line D to form an

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octave, and the upper-line E joins with the lower-line E on the fourth quarter note to form an octave. Moreover, the lower line unfolds a triad G-B!>-D in bars 4-5. I attempt to correct these problems, as illustrated in example A.25c. I replace bars 1-4 of the upper line Example A.25b

Example A.25 in bar 7, followed by an ascent by step through C in bar 8 to the final F. • Listen to example A.26a. Example A .26a

Example A.26b

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As illustrated in example A.26b, I flesh out the lowerline framework of the final five bars with neighbor cambiatas. Returning to the middle line, I try a C in bar 4 followed by a leap up to the unison G, then by a descent

298

to the half note D. Then I sketch a framework of E-AA for the lower line of bars 4-6. • Listen to example A.26b. I try to revise the final five bars of the lower line, where the successive neighbor motions create an overly conjunct texture. As shown in example A.26c, I try replacing the neighbor cambiata of bar 7 with an ornamental motion to D. Next I flesh out the lower line for bars 4-6 with a stepwise descent from E to A. I ornament the A of bar 5 with a stepwise motion up a third to C, and I ornament the A of bar 6 with a leap down a third to F. Returning to the middle line, I open with half notes C-BI>, followed by a leap to D. Finally I sketch a framework for the first three bars of the lower Example A.26c

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line. The first tone must be F, and because of the consonant possibilities given by the middle and upper lines the second bar must be D or Bl>. To avoid parallel octaves with bar 1, the second bar must be D. I sketch in an F for bar 3. • Listen to example A.26c. The lower line is still problematic. First, bars 7-9 unfold three successive statements of B\>- C-D. As shown in example A.26d, I correct this by exchanging the positions of the ornamental D and Bl> of the neighbor cambiata in bar 8. The final C of bar 8 is a consonance, which I quit by leap to F over the bar. Second, the leap to F in bar 6 of example A.26c makes a consonance of the suspended F in the middle

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line. Thus the harmonic event of bar 6 is an F major triad in first inversion, as is that of the preceding bar. This stasis threatens the linear progress of the lower line and deflates the overall impulse of the counterpoint, which climaxes with the C of bar 6. Ornamenting the A of bar 6 with a neighbor cambiata, as in example A.26d, allows the middle-line F to be a dissonant sixth resolving to the consonant fifth of an A minor triad. Next I flesh out the lower-line framework of the first three bars. Finally I replace the opening C of the middle line with an A. This results in an ascent to the C of bar 3, which is less static than die previous version. • Listen to example A.26d.

Example A.26A

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This final version of the counterpoint is presented in example A.26e. Its impulse is climaxed by the cantus C of bar 6 in the upper line. Mixed Species, Type 4 Examples A.27a-A.27f illustrate the composition of a type 4 mixed species counterpoint, with the D minor cantus sounding in the upper line and the twofifthsp cies co-lines sounding in the lower and middle lines. I begin by notating the cantus (example A.27a). • Sing the cantus without the music. Next I begin to compose the two co-lines, as illustrated in example A.27a. I begin with the middle line in

300

bar 9, moving from B^ up to D in bar 11 and then descending to a final A. In the lower line I begin with an ornamented G, followed by a descent to an ornamented Ctt in bar 12 and then by the final D. Example A.26e

Example A.27a

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• Listen to example A.27a. This segment has two problems. First, bars 11-12 unfold parallel octaves D:D and A:A. These octaves are separated by intervening tones, which would have dif-

APPENDIX:

WRITE-THROUGHS

fused the problem in earlier mixed species exercises. In this case, however, each bar constitutes a fifth species co-line against a first species co-line. In bar 11 the middle line sounds a fifth species texture against the first species whole note of the lower line; in bar 12 the lower line sounds a fifth species texture (here it is four quarter notes borrowed from the third species) against the first species whole note of the middle line. Thus the octaves function much as they would in a simple two- or threepart counterpoint in fifth species. A second problem lies in the amount of activity in bar 9. Remember that the events of both lines must combine into a single-line texture, and therefore the amount of concurrent activity in both lines must be limited.

Example A.27b

301

With the middle line leaping up to D, it will be difficult to accommodate the eighth-notes of the lower line in a single-line texture. As illustrated in example A.27b, I correct the first problem by eliminating the lower-line A in bar 12. The second problem is also solved easily, by eliminating the ornamentation of G in bar 9. Next I compose a simple middle line for bars 5-8. I accompany it in the lower line with a quarter-note descent from F to an ornamented fit in bar 6 and G in bar?. • Listen to example A.27b. This segment presents four problems. Again there is too much activity in bars 5-6. Also, the lower-line leap G-G of bar 7 is unrecovered. Third, the lower line un-

302

folds a dissonant contour from the F of bar 10 to the Cfl of bar 12. Finally, the B'l of bar 12 does not participate in a stepwise ascent from dominant to tonic. As shown in example A.27c, I eliminate the middle-line quarter notes in bar 5 and reduce the activity of the lower line in bar 6. In bar 7,1 leap up to an E in the lower line instead of to G; this leap is recovered in the following bar. The octave leap A-A at the end of bar 8 is recovered eventually with the sounding of the eighth-note B^l in bar 12. The dissonant contour of bars 10-12 has been eliminated, and the fi'l of bar 12 participates in a stepwise ascent from the A of bar 8 to the final tonic. Finally, I eliminate the lower-line dissonant contour of bars 10-12 by leaping down to D in bar 9. • Listen to example A.27c. Example A.27c

APPENDIX:

WRITE-THROUGHS

Now it seems that there is too little activity in bars 57, especially if the cantus Bk of bar 7 climaxes the impulse. As shown in example A.27d, I add two quarter notes to the middle line of bar 6. Then I complete the middle line, which opens with an A, moves to the lowered sixth degree fit in bar 2, and returns to A in bar 3. The leap to D in bar 3 is recovered by the ornamental descent to fill, which returns through Ctt to the D of bar 5. The B^l and Ctt participate in the ascending motion from the A of bar 3. Finally I complete the lower line, beginning with a leap up from D to G and then descending by step back to the D of bar 5. • Listen to the completed counterpoint, as given in example A.27d. This counterpoint, presented in example A.27e, is

APPENDIX:

WRITE-THROUGHS

successful as a three-part exercise. The final step entails combining the two active co-lines into a single-line texture, as illustrated in example A.27f. The first bar is fairly straightforward. In bar 2,1 use the middle-line Bl> as an ornamentation of the lower-line resolution. I add a passing eighth-note A for increased impulse and greater Example A.27d

Example A.27e

303

fluidity. In the third bar I delay the middle line D until the fourth quarter; bar 4 sounds without adjustment. I sound the middle-line D on the first quarter of bar 5, and compress the lower-line D and E into eighth-notes. I do the same with the quarter notes G and F of the middle line in bar 6. In bar 7,1 do not sound the

304

middle-line G, making use instead of the lower-line G that sounds one octave lower. Again I compress the lower-line quarter notes E and D into eighth-notes. Bar 8 sounds the lower line as is; the middle-line A is expressed by the final quarter note. In bar 9 the middleline B^ sounds in the beginning of the bar and the D is sounded by the cantus, whereas the lower-line half note

APPENDIX:

WRITE-THROUGHS

G is sounded as a quarter note on the second quarter of the bar. With the exception of the lower-line D in bar 11, which is delayed until the second quarter, the remaining three bars sound without adjustment. • Listen to the final two-part version of this type 4 counterpoint, as given in example A.27f.25 The exercise climaxes with the cantus B\> of bar 7.

Example A.27f

25. Note that the active co-line in example A.27F is replete with unrecovered leaps and dissonant contours. These were problematic in

earlier exercises for precisely the same reason they are necessary here: because they create two functional lines in one.

INDEX

Page numbers in italics refer to examples from the write-thoughs.

Brandenburg Concerto no. 4 (J. S. Bach),

Agency defined, 67, 154 effect on dynamic structure, 155–56, 167 silent client tones and, 160-61 dissonant chords and, 164, 166 hierarchy and, 166, 167 see also Client tone; Consonant agent; Dissonance; Dissonant agent; Hierarchy; Implied harmony; Multileveled agency Agency, multileveled. See Multileveled agency Agent, consonant. See Consonant agent Agent, dissonant. See Dissonant agent Attributes of tones, xviii, 17-21. See also Dynamic function; Harmonic function; Scale degree Augmented fourth. See Tritone

Cambiata. See Neighbor cambiata; Passing cambiata Cantus firmus denned, 31-32 guidelines for, 32 co-line joined with, 43 in two-part counterpoint, 62-63, 84, 91, 98, 112 in three-part counterpoint, 115, 125-28, 132 in mixed species, 144, 148, 150-53 composing, 175-80 see also Primordial line Client tone denned, 67, 154 of passing tones, 68 of neighbor tones, 70 of dissonant suspensions, 81-82, 129 in functional harmonies, 117-18, 129, 134, 136 in multileveled agency, 138 silent, 160-61 in hierarchies, 164, 166 see also Agency; Consonant agent; Dissonant agent; Multileveled agency; Triad

Bach, J. S., 168-69, 172, 173 Balance, 46-48 Bass line soprano differing from, 160 functional, 172-73 Beauty in music, xiii, xv-xvi, xvii, xx, 20, 23, 158, 159, 173, 174, 181

168-69, 172, 173

305

306 Co-line defined, 43 cantus joined with, 43, 48, 62 in two-part counterpoint, 62-63 first species, 63-64, 125 second species, 84-90, 126-27 third species, 91-97, 128 fourth species, 98-104, 129-31 fifth species, 112-13, 132 in three-part counterpoint, 115 active or neutral, 126 in mixed species, 133, 143-53 see also Primordial line Composition. See Work, musical Compound melody defined, 7-8, 10 discrete participation prevented by, 10, 34, 195, 202, 241, 286, 294 exercises for, 10-11, 40-41, 143-44 in mixed species type 4, 133, 142-43, 152-53 see also Primordial line Conjunct line, overly defined, 37 exercises for, 40-41 in third species, 214-15, 265, 294, 297 in fifth species, 233 in mixed species, 302 see also Dynamic structure; Impulse; Resolution, force of

INDEX Conjunction (joining) of two lines, 48 positive effects of, 49-50 primordiality prevented by, 50 independence and, 53 of three lines, 115-16, 120-21, 244 dissonance created by, 118 exercises for, 123 see also Distance between lines; Mixed modes Consciousness of sound, 2 objects constituted by, 11-12 of a work, 15-17 of music, 157-58, 159, 173, 174 see also Perception; Unitary apprehension Consequent resolution in a musical work, 15-17, 157, 174 in a line, 15-16 exercises for, 17, 24, 62, 111, 124-25, 173-74 performer's contribution to, 20-21, 23, 173 in conjunct lines, 61 in a musical passage, 166, 168-69, 172 in a cantus, 776-77, 178, 179-80 in two-part counterpoint, 184-86, 187, 198-99, 204-07, 209-10, 231-33, 23436, 239 in three-part counterpoint, 241-43, 279

in mixed species, 295 see also Dynamic forces; Dynamic function; Dynamic structure; Impulse; Resolution, force of; Unitary apprehension Consonance. See Agency; Client tone; Consonant agent; Dissonance; Triad Consonance, stable. See Stable consonance Consonance, unstable. See Unstable consonance Consonant agent defined, 133-35 exercises for, 135-36 in multileveled agency, 136-38 Consonant suspension. See Suspension, consonant Contour, dissonant. See Dissonant contour Crossed voices confusion of lines caused by, 54 defined, 55 exercises for, 58, 124 in two-part counterpoint, 64, 189 in three-part counterpoint, 124 in mixed species, 289 see also Independence of lines Diminished fifth. See Tritone Discrete participation defined, 5-10 exercises for, 10-11

307

INDEX in a line, 15», 26n2, 142, 750, 182, 18384 in three-part counterpoint, 120, 121, 126 of tones in a hierarchy, 136 see also Compound melody; Dissonant contour; Dissonant leap; Harmonic embellishment; Pattern; Primordial line; Repetition; Steps, too many; Triad; Tritone Disjunct line, overly defines, 37 exercises for, 40-41 see also Dynamic structure; Impulse; Resolution, force of Dissonance harmonic intervals, 44-45 preventing primordiality, 50 defined, 67 in second species, 84 in third species, 91-94 in fourth species, 98-100 triads, 116, 117-18, 121, 202, 216 dissonant chord members, 164 see also Agency; Dissonant agent; Harmonic interval; Neighbor cambiata; Neighbor tone; Passing cambiata; Passing tone; Successive dissonances; Suspension, dissonant Dissonant agent defined, 9-10 dissonant suspension as, 81, 100-101

in dissonant triads, 116, 117-18, 121 in multileveled agency, 137, 138 functional harmony and, 139 hierarchy and, 200 see also Agency; Client tone; Consonant agent; Hierarchy; Multileveled agency Dissonant contour denned, 9-10, 34 exercises for, 11, 40-41 unproblematic with repeated tone, 63-64 in mixed species, 153, 304n preventing discrete participation, 777, 196, 198, 208, 210, 213, 238, 256, 277, 28285, 291, 292-93, 301-02 see also Discrete participation Dissonant leap defined, 9 exercises for, 11, 40-41 see also Discrete participation Dissonant suspension. See Suspension, dissonant Distance between lines preventing conjunction, 48, 204, 216, 217, 270, 289 in two-part counterpoint, 64 in three-part counterpoint, 120 see also Conjunction Dynamic inflections (dynamics). See Volume Dynamic forces (impulse, resolution) defined, 13-17

exercises for, 17, 61-62, 111, 124-25 dynamic function and, 19 factors affecting, 105n shared by agent and client, 154-57, 174 see also Dynamic function; Dynamic structure; Impulse; Resolution, force of Dynamic function defined, 19-23 shared by agent and client, 154, 167 see also Dynamic forces; Dynamic structure; Impulse; Resolution, force of Dynamic structure created by performer, 20-21 of foundation, 24, 26-28, 186, 252 in two-part counterpoint, 61, 184-85, 193, 201, 238 of a work, 157 hierarchical levels and, 165-73 exercises for, 173-74 in three-part counterpoint, 250-52, 256, 260, 261, 279, 284 in mixed species, 299, 301 see also Dynamic forces; Dynamic function; Impulse; Resolution Eighth-notes in fifth species, 112, 113 promoting impulse, 111, 233, 279, 282, 301, 303 in musical passages, 168, 169

INDEX

308 Fifth, perfect, 45, 84, 130, 226, 270. See also Harmonic progress; Suspension, dissonant Fifths, parallel. See Parallel stable consonances Foundation, stepwise dynamic structure of, 24, 26-28, 167, ' 172-73 of a cantus, 775-76 177n, 178, 179 dynamic structure of, 776, 178, 179, 186, 193, 200, 212, 223, 234, 243, 252, 263, 272, 280, 290, 296 ascending, 223 see also Hierarchy; Stepwise motion, direct and indirect Fourth, augmented. See Tritone Fourth, perfect as dissonance, 9, 44, 68 in mixed species, 145 Functional bass, 172-73 Harmonic embellishment defined, 35 interrupting linear progress, 247, 255 see also Discrete participation Harmonic function attribute of, 4, 116 defined, 46 differing for similar tones, 50 shared by agent and client, 154-55 see also Attributes of tones Harmonic interval

interrupting linear progress, 6-7, 10, 18384 categorized, 44-46 exercises for, 61-62 in first species, 63 from two lines, 116 in three-part counterpoint, 116 see also Dissonance; Harmonic function; Stable consonance; Unstable consonance Harmonic progress, 87, 89, 90-91, 202, 216, 225-26 Harmony, implied. See Implied harmony Hierarchy (levels of linear activity) in a line, 4, 25 dynamic function and, 21 implied by successive neighbors, 72n6 agency and, 136, 200 silent client tones and, 160-61 given by conjunction, 161-64 dissonant chord members and, 164 performance and, 187-88 see also Agency; Client tone; Foundation, stepwise; Stepwise motion Implied harmony defined, 139-41, 161 exercises for, 141 Impulse defined, 13-17 dynamic function and, 19-23

in a foundation, 26-28 from ascending motion, 13, 16, 22-23, 26, 37, 59, 60, 90, 107, 124, 168, 777, 179, 198, 207, 215, 216, 230, 232, 265, 279 in a line, 27 in a cantus, 32, 37, 775, 776, 177», 178 in two-part counterpoint, 58-59, 61, 186 in a performance, 81, 173 meter and, 89-90 rhythm and, 105-12, 152-53, 168, 237 from implied harmony, 141 shared by agent and client, 155 agency and, 155-57 tempo and, 159-60 volume and, 159, 187 in hierarchical levels, 165-72 see also Disjunct lines, overly; Dynamic forces; Dynamic function; Dynamic structure; Resolution, force of Independence of lines in two-part counterpoint, 43, 53-58, 89, 124, 184, 190-91 in three-part counterpoint, 124 in mixed species, 287 see also Crossed voices; Parallel stable consonances; Unison Interval, harmonic. See Harmonic interval Joining. See Conjunction

309

INDEX Leap tones quit by, 2, 4 in harmonic embellishment, 35 in second species, 84 over the bar line, 84, 87, 94-95 problems in co-lines below caused by, 8687, 202 in resolution of suspensions, 113 to a dissonance, 127 to a consonant agent, 135 in mixed species, 146 dynamic forces and, 166 from a dissonance, 204, 216 see also Dissonant leap; Neighbor cambiata; Parallel leaps; Passing cambiata; Suspension, consonant; Unrecovered leap Leaps, parallel. See Parallel leaps Levels of linear activity. See Hierarchy Like intervals, too many defined, 54 exercises for, 58, 124 in two-part counterpoint, 65-66, 184, 190-91, 208, 222 in second species, 88-89 in three-part counterpoint, 124 in mixed species, 147, 148 in fourth species, 267- 68, 272 in mixed species, 288 Linear function attribute of, 4

denned, 18 differing for similar tones, 50 shared by agent and client, 154-55 see also Attributes of tones Linear embellishment defined, 35-36 unproblematic because of conjunction, 50 preventing conjunction, 182, 191-92, 255, 270 Meter, affecting dynamic forces, 90, 94, 97, 103, 105 Minor mode major distinguished from, 28-31 exercises for, 40-41 cantus in, 179-80 counterpoints in, 188-94, 195-201, 23639, 263-67, 267-72, 299-304 Mixed modes (modal inflection) defined, 36 exercises for, 40-41 in two-part counterpoint, 48-49, 790,

exercises for, 138-39 consequences of, 157 dynamic structure and, 167 Neighbor cambiata defined, 74-75 exercises, 76-77 in third species, 91, 92-93, 207-70, 213, 265, 297, 299 in multileveled agency, 137, 148 in fifth species, 238, 276 Neighbor tone defined, 68 exercises for, 70-71, 136, 138, 141 in second species, 83, 83n, 84, 248 in fourth species, 100 in suspension resolutions, 113 as agent, 134, 139 in mixed species, 144 in a musical passage, 167, 169 in fifth species, 277 see also Successive dissonances

195, 196

in three-part counterpoint, 120-21 see also Conjunction Mode, minor. See Minor mode Modes, mixed. See Mixed modes Mozart, W. A., Symphony no. 40, 170-73 Multileveled agency defined, 136-38

Octave unfolding, in third species, 95 Octaves, parallel. See Parallel stable consonances One harmony per bar. See Harmonic progress Ornamentation, layers of, 164-72. See also Hierarchy

INDEX

310 Overly conjunct line. See Conjunct line, overly Overly disjunct line. See Disjunct line, overly Parallel leaps confusion of lines caused by, 54 defined, 56-58 exercises for, 58 in two-part counterpoint, 65, 189 in three-part counterpoint, 122-23 See also Independence of lines; Leap: over the bar line Parallel stable consonances (perfect fifths and octaves) cantus and, 32n28 preventing primordiality, 51-52 in first species, 64 in second species, 88, 198, 202, 248, 250 in third species, 95-97, 260 in fourth species, 101-02, 227, 265, 274 in fifth species, 113, 238 in three-part counterpoint, 119, 121-22,

125, 247, 244, 255, 256 exercises for, 123 in mixed species, 146, 149-50, 294, 297, 300-301 see also Independence of lines Passing cambiata defined, 75-76 exercises for, 77-78 in third species, 91-92, 93-94

prohibited in co-lines below, 94, 148 in multileveled agency, 138, 148, 152 Passing tone defined, 67 exercises for, 67-68, 136, 138, 141 in second species, 83, 84 in suspension resolutions, 113 as agent, 133-34, 135, 139, 144-45 in fourth species, 100, 265 see also Successive dissonances Pattern defined, 37 exercises for, 40-41 preventing discrete participation, 180, 184 see also Discrete participation Perception, 2, 11n simplifying nature of, 6-7, 11-12, 34, 57, 81-82 see also Consciousness Perfect fourth. See Fourth, perfect Performance beauty in, xx, 23n, 157-60, 188 directions for performing exercises, xxixxii effect of on dynamic structure, 20-21, 23, 173, 186, 234, 295-96 exercises for, 23-24, 125, 173 of musical passages, 160, 167, 168, 169,

172-73

of a work, 174 function of, 186

see also Dynamic structure; Unitary apprehension Pitch field defined, 28 of minor mode, 29-31 Primordial line defined, xviii, 1, 5, 10, 16 foundation of, 28 in minor mode, 31 cantus as, 31, 32, 33, 50 dynamic forces in, 24, 26, 32, 46 co-line as, 43, 50, 62 prevented by parallel stable consonances, 52 junction with other primordial line of, 58 in three-part counterpoint, 115, 121-23 in mixed species, 153 in counterpoint, 160 bass line and, 160 see also Discrete participation; Foundation, stcpwise; Stepwise motion; Unitary apprehension; Unrecovered leap Recovered leap. See Unrecovered leap Repetition defined, 37-38 preventing discrete participation, 754, 795, 208, 222, 228, 268-70, 278-79, 297-98 see also Discrete participation Resolution, consequent. See Consequent resolution

311

INDEX Resolution, force of defined, 13-14 dynamic function and, 19-22 in a foundation, 26 from descending motion, 13, 22, 59, 60,

107, 124, 215 in a line, 28 in a cantus, 32, 37, 775, 178 in a performance, 81 aftected by meter, 89-90 affected by rhythm, 105-12 inhibited bv implied harmony, 141 extended bv agency, 156-57 in hierarchical levels, 165-72 see also Conjunct line, overly, Consequent resolution, Dynamic forces, Dynamic function, Dynamic structure, Impulse, Unitary apprehension Resolution of dissonance See Suspension, dissonant Scale degree attribute of, 18-19, 23 relative tension of, 22 in minor mode, 27-29 Sixth, dissonant, 45, 81-82, 84, 130, 299 See also Harmonic progress. Suspension, dissonant Sixths, too many See Like intervals, too many Stable consonance defined, 44

beginning a counterpoint, 62, 125 in triads, 116-17, 118 in a hierarchy, 165 Steps, too many defined, 38 unproblematic in second species, 85-86 see also Discrete participation Stepwise motion, direct and indirect primordiality and, 1-5, 24-25 exercises for, 5 see also Foundation, stepwise; Hierarchy Successive dissonances defined, 71 neighbor tones, 71, 72, 231 passing tones, 71-72 exercises for, 73-74 Suspension, consonant in second species, 85 in fourth species, 98, 279, 222, 224 in three-part counterpoint, 131 in consonant agency, 145 in multileveled agency, 151-52 promoting impulse, 207 in fifth species, 237 Suspension, dissonant defined, 78-82 exercises for, 82-83 in second species, 85, 796-97, 248 in fourth species, 98-99, 101-02, 129-

31, 224, 226, 225

as agent, 81, 100-101, 134-35, 156

downward resolution of, 81-82, 86, 87, 99, 112, 129-30 upward resolution of, 81-82, 130-31 in fifth species, 113, 231, 235, 238 in implied harmony, 140 in multileveled agency, 151 promoting impulse, 166, 282 in a hierarchy, 166, 167 see also Dissonance Tempo, generating impulse, 159-60. See also Performance Thirds, too many. See Like intervals, too many Triad. See also Discrete participation; Dissonance; Stable consonance; Unstable consonance interrupting linear progress, 8, 247, 255, 295 exercises for, 40-41 in three-part counterpoint, 116-20, 121 incomplete (complete), 118, 119, 121, 125, 126, 129, 144 consonant, 121, 125, 126, 127, 128, 14445 in implied harmony, 139-41 functional, 161 Tritone (augmented fourth; diminished fifth) preventing discrete participation, 9, 33, 39, 217, 222, 239, 252 as double dissonance, 45

INDEX

372

Tritone (cont.) unproblematic in second species, 86 see also Discrete participation Two harmonies in a bar. See Harmonic progress Unison confusion of lines caused by, 54, 55-56 exercises for, 58, 124 in two-part counterpoint, 66, 189 in second species, 89 in third species, 97 in fourth species, 102-03, 272 in three-part counterpoint, 118, 124 in mixed species, 147, 150, 289 see also Independence of lines Unitary apprehension (wholeness) defined, 11-17 performance and, 20-21, 23n

of stepwise foundation, 26-28 of a counterpoint, 58-61, 186 of a three-part counterpoint, 124, 252 and multileveled agency, 156n1 of a work, 157, 174 of hierarchical levels, 164-73 ofacantus, 177, 180 see also Consequent resolution; Dynamic structure; Impulse; Performance Unrecovered leap (recovered leap) defined, 4-5, 39 exercises for, 40-41 tripled tone and, 126 in mixed species, 153, 304n in a musical passage, 170 preventing primordiality, 182, 195, 206, 220, 226, 241, 255, 256, 268, 288, 30102 see also Primordial line

Unstable consonance defined, 44 in triads, 117, 118 Volume. See also Performance; Unitary apprehension affecting dynamic structure, 20-21, 24, 46, 105n, 166 exercises for, 23-24 Wholeness. See Unitary apprehension Work, musical, xvi, 2, 23 contribution to dynamic forces, 20, 21, 105 unitary apprehension of, 157 suggesting sounds, 158 allowing beauty, 159 performance of, 174