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Explaining Tonality
Schenkerian Theory and Beyond
Matthew Brown
A catalogue record for this title is available from the British Library. This publication is printed on acid-free paper.
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no part of this work may be photocopied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recorded, or reproduced in any form or by any means,
without the prior permission of the copyright owner. First published 2005
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ISBN: 1–58046–160–3
Library of Congress Cataloging-in-Publication Data
Brown, Matthew,
1957-Explaining tonality : Schenkerian theory and beyond / Matthew Brown. p. cm.
Includes bibliographical references and index. ISBN 1-58046-160-3 (hardcover : alk. paper) 1. Schenkerian analysis. 2. Tonality. I. Title.
MT6.B87E9 2005 781.2⬘58–dc22
Figures vii
Preface xiii
Introduction. Theoretical and Meta-Theoretical Issues 1
Basic Goals and Assumptions 2
Building and Testing Theories 12
Six Criteria for Evaluating Theories 18
1. Schenker and the Quest for Accuracy 25
Fux and Strict Counterpoint 27
“The Heinrich Maneuver” 41
“The Complementarity Principle” 56
2. Semper idem sed non eodem modo 66
Conceptual Origins 67
Prototypes 72
Transformations 76
Levels 83
Fallout 91
3. What Price Consistency? 99
Sequences Reconsidered 103
Sequences and Counterpoint 117
Analytical Implications 126
4. Schenker and “The Myth of Scales” 140
Modes and Scales in Traditional Theory 142
Schenkerian Theory and Scales 146
Schenkerian Theory and Modal Inflections 151
Schenkerian Theory and Exotic Inflections 158
Schenkerian Theory and the Emergence of
Functional Tonality 162
5. “Pleasure is the Law” 171
The Limits of Schenkerian Theory 172
Debussy, “C’est l’extase langoureuse” 186
Debussy, “La mort des amants” 192
6. Renaturalizing Schenkerian Theory 209
Naturalizing Schenkerian Theory 211
Schenkerian Theory as a Model of Expert
Functional Monotonal Composition 222
Conclusion 234
Notes 239
Bibliography 267
I.1. Explaining tonality 3 I.2. A procedure for composing typical tonal melodies 5
I.3. Five forms of passing tone 6
I.4. Counterfactual conditionals 7
I.5. ‘The Covering-Law Model’ 8
I.6. Explaining suspensions 9
I.7. A procedure for generating 7–6 suspensions 11
I.8. ‘The Hypothetico-Deductive Method’ 13
I.9. The logic of falsification 16
I.10. Six criteria for evaluating theories 19
1.1. From strict counterpoint to functional tonality 26
1.2. Fuxian cantus firmi 30
1.3. First Species counterpoint 32
1.4. Prototypical counterpoints in Fifth Species 33
1.5. Dissonances in florid counterpoint 34
1.6. Triads in three- and four-voice textures 36
1.7. First Species in three voices 37
1.8. Cadence patterns in two, three, and four voices 39 1.9. Parallel and direct perfect octaves and fifths in
three and four voices 40
1.10. Differences in the behavior of triads and Stufen 42
1.11. The major-minor system 44
1.12. Laws of melodic motion and closure 45
1.13. Polyphonic melodies 47
1.14. Parallels by doubling and figuration 48
Beethoven, Piano Sonata, Op. 2, no. 3, 1st movement, mm. 47–51
1.15. Parallels by combinations of harmonic and
non-harmonic tones 49
1.16. Laws of relative motion and closure 50
1.17. Laws of vertical alignment 51
1.18. Consonant non-harmonic tones and dissonant
1.19. Neighbor tones and suspensions as passing motions 52
1.20. Implied tones and the nota cambiata 54
1.21. Successive seventh chords 55
1.22. Displacement and accented dissonances 55
1.23. Beethoven, Piano Sonata, Op. 81a, 1st movement,
mm. 230–42 56
1.24. Laws of harmonic classification 59
1.25. Chord function vs. chord derivation 60
1.26. Laws of harmonic progression 61
1.27. Laws of chromatic generation 62
1.28. Rectification of Phrygian II 63
2.1. Schenker’s concept of prototypes, transformation,
and levels 69
2.2. The non-recursive nature of Fuxian species
counterpoint 71
2.3. Schenkerian Ursätze in C Major 73
2.4. Horizontalizing transformations 78
2.5. Filling in transformations 80
2.6. Harmonizing transformations 81
2.7. Reordering transformations (non-recursive) 82
2.8. Composing out 85
2.9. Schenker’s deep-middleground paradigms 86
2.10. Divided Urlinien 88
2.11. Derivation of divided Urlinien 90
2.12. The explanatory scope of Schenkerian theory 92 2.13. Schenker’s sketch of “The Representation of Chaos”
from Haydn’s Creation 94
2.14. Alternative sketch of “The Representation of Chaos” 97
3.1. Sequences 102
3.2. A typical ascending-fifth sequence 104
3.3. Deriving ascending-fifth sequences 105
3.4. Restacking ascending-fifth sequences 107
3.5. Deriving ascending-third sequences 108
3.6. Deriving descending-fifth sequences 109
3.7. Deriving descending 5–6 sequences 111
3.9. Deriving ascending 5–6 sequences 113
3.10. Simple mixture 114
3.11. Double mixture 115
3.12. Fux’s prototypical cantus firmi 118
3.13. Typical two-voice counterpoint in First Species 118 3.14. Typical two-voice counterpoints in Fourth Species 119
3.15. Fux’s three-voice prototypes 120
3.16. Typical three-voice counterpoints in Fourth Species 120 3.17. Parallel motion in mixed species with three and
four voices 122
3.18. Parallel linear progressions within a single Stufe 123 3.19. Parallel linear progressions between different Stufen 125 3.20. Schenker’s analysis of Bach’s Little Prelude in
C Major, BWV 924 127
3.21. Alternative analysis of Bach’s Little Prelude in
C Major, BWV 924 129
3.22. Schenker’s analysis of Bach’s Prelude in C Minor,
WTC I, BWV 847, mm. 1–18 131
3.23. Alternative analysis of Bach’s Prelude in C Minor,
WTC I, BWV 847, mm. 1–18 132
3.24. Two analyses of the Prelude from Bach’s Partita
No. 3 for Solo Violin 133
3.25. Analysis of Bach, French Suite in D Minor,
BWV 812, Minuet II 135
4.1. Scale membership and tonality 145
4.2. Schenker’s account of mixture 148
4.3. Beethoven, “Heiliger Dankegesang,” String Quartet,
Op. 132 152
4.4. Graph of Beethoven, “Heiliger Dankegesang,”
String Quartet, Op. 132 153
4.5. Brahms, “Vergangen ist mir Gluck und Heil,”
Op. 14, no. 6 155
4.6. Graph of Brahms, “Vergangen ist mir Gluck
und Heil,” Op. 14, no. 6 157
4.7. Chopin, Etude, Op. 10, no. 5 159
4.8. Graph of Debussy, Prélude à “L’Après-midi d’un
4.9. Van den Toorn’s analysis of the opening of
Stravinsky, Petrouchka 162
4.10. Cadences in fifteenth-century music 164
4.11. The Artusi-Monteverdi debate 168
4.12. Renaissance modal polyphony and functional
tonality 169
5.1. Parallel dominant seventh chords 174
5.2. Free dissonances 176
5.3. Non-functional successions. Beethoven, Appassionata
Sonata, Op. 57, 1st movement, mm. 62–87 180
5.4. Extreme chromaticism. Graph of Reger, Piano
Quintet, Op. 64, mm. 1–8 181
5.5. Incomplete transferences of the Ursatz 183
5.6. Interpolations in Debussy, “La sérénade interrompue”
(Préludes, Book 1, no. 9) 184
5.7. Graph of Debussy, “C’est l’extase langoureuse,”
mm. 1–18 188
5.8. Graph of Debussy, “C’est l’extase langoureuse,”
mm. 18–35 189
5.9. Graph of Debussy, “C’est l’extase langoureuse,”
m. 36ff 190
5.10. Evolution of ‘The Sigh Figure’ in Debussy, “C’est
l’extase langoureuse” 191
5.11. Global view of Debussy, “C’est l’extase langoureuse” 192
5.12. Debussy, “La mort des amants,” mm. 1–12 194
5.13. Debussy, “La mort des amants,” mm. 12–18 196
5.14. Debussy, “La mort des amants,” mm. 19–29 198
5.15. Debussy, “La mort des amants,” mm. 30–45 200
5.16. Global view of Debussy, “La mort des amants” 201
5.17. Prolonged dominant-seventh chords 203
5.18. Schubert, “Die Stadt,” Schwanengesang, no. 11 206 5.19. Functional tonality and twentieth-century tonal
practices 208
6.1. Naturalizing music theory 210
6.2. Schenker’s derivation of the major system from
6.3. Lerdahl/Jackendoff’s derivation of Bach, Prelude
in C, WTC I 218
6.4. Learning curve for expert monotonal composition 221 6.5. Sloboda’s “Diagram of typical compositional
resources and processes” 228
6.6. Schenker’s account of expert monotonal composition 231
Few terms in music theory are more profound and more enigmatic than ‘tonality.’ First coined by Alexandre-Étienne Choron in his “Sommaire de l’histoire de la musique” (1810), it was popularized by François-Joseph Fétis in the 1830s and 1840s and has subse-quently remained an essential part of theoretical discourse. Choron originally used the term to denote music in which notes are related functionally to a particular tonic, the tonic triad. This particular brand of tonality is often known as ‘functional tonality’ and is char-acteristic of works written by composers such as Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Schumann, Mendelssohn, Chopin, and Brahms. But, as Choron’s term has gained currency, so it has expanded its meaning consider-ably. Nowadays, the term is often used in a very general sense to denote any music that focuses melodically and/or harmonically on some stable pitch or tonic. This definition covers a broad range of music from many cultures and many time periods, from Medieval plain chant to various twentieth-century idioms.
Of the many attempts to explain the nature of functional tonality, perhaps the most comprehensive was undertaken by Heinrich Schenker (1868–1935).1 In his monumental triptych,
Neue musikalischen Theorien und Phantasien, he systematically inves-tigated the ways in which lines and chords behave in functional tonal contexts. In the first volume, Harmonielehre (1906), he explained how functional harmonies (or Stufen) are organized into progressions (or Stufengang).2 In the second volume, Kontrapunkt
(1910, 1922), he explained the basic properties of tonal voice lead-ing (or Stimmführung).3 And in the final volume, Der freie Satz
(1935), Schenker showed how the principles outlined in the Harmonielehre and Kontrapunkt operate recursively across entire monotonal compositions.4
But what sorts of relationships did Schenker count as tonal or, to be more precise, functionally monotonal? Why do these rela-tionships work in some ways and not others? Why should we prefer Schenker’s theory of functional monotonality to its competitors?
The purpose of this book is to try to answer these questions. The Introduction explores some of the general methodological issues that arise when we try to build, test, and evaluate a plausible theory of tonality. It begins by outlining the main ingredients of such a theory, namely concepts, laws, and procedures, and describes some of the problems that they raise. The Introduction goes on to discuss six criteria that theorists typically use to evaluate the success of their models. These criteria include accuracy, scope, fruitfulness, consistency, simplicity, and coherence. With this broad framework in place, the central portion of the book uses these six criteria to illuminate the foundations of Schenkerian theory. The conclusion describes some of the ways in which Schenkerian theory might develop in the future.
There are several reasons why I have decided to address these issues at the present time. Like many music theorists, I am attracted to Schenker’s work because it offers us not only a powerful model for explaining the tonal system, but also a flexible tool for analyzing the practices of functional monotonal composition. The benefits of such an approach seem clear enough; instead of laboriously labeling each successive harmony with its own Roman numeral, the analyst can study the contrapuntal behavior of those harmonies, both within the local context of an individual phrase and within the global context of the piece as a whole. Although Schenkerian theory deals prima-rily with matters of functional harmony and voice leading, it often leads to important insights about a work’s motivic, rhythmic, and formal structure. This fact is amply demonstrated by Schenker’s best graphic analyses, such as those published in Der Tonwille (1921–24), Das Meisterwerk in der Musik (1925–30), and the Fünf Urlinie-Tafeln (1932).5
Furthermore, since the process of graphing particular pieces often teaches us new ways to hear music and understand the processes of musical composition, Schenkerian analysis can be of great help to performers and composers alike.
At the same time, however, I am also intrigued by the formal properties of Schenker’s model. This fascination is something that I share with many other music theorists. Some, such as Milton Babbitt, Allan Keiler, Fred Lerdahl and Ray Jackendoff, have noted certain parallels between Schenker’s account of tonal relationships and Naom Chomsky’s account of grammatical relationships in
natural language.6
Others, such as Michael Kassler, James Snell, and Stephen Smoliar, have even tried to model Schenkerian theory on the computer.7
I see my work as part of these particular traditions. That being said, Schenkerian theory is still the subject of con-siderable debate. For starters, although Schenker’s ideas are widely discussed in the music theory community, they are still shrouded in mystery. For example, Jonathan Dunsby has recently accused afi-cionados of promoting “a cabalistic image of Schenker” that treats his work as “a secret body of knowledge that can be applied but never fully exposed.”8
To make matters worse, when Schenker’s arguments are scrutinized, they often appear to be disjointed, cryp-tic, and even illogical. William Benjamin has described these short-comings as follows: “While it is true in many instances that the problem lies with Schenker’s way of putting things and not with the formal relationships he has in mind, there are other cases where contradictory lines of reasoning go to the heart of his level-relating style. In these cases the problem results in a conflict between his artistic-compositional side and his formal-theoretic side.”9
Such conflicts are not, however, easy to resolve; as William Rothstein has observed, the ways in which theorists respond to these issues depends as much on their individual interests, tempera-ment, psychological makeup, and their broader sense of the prevailing
Zeitgeist, as on anything Schenker may or may not have written.10
Opinions vary enormously. Some regard Schenker’s work as a theory of musical structure, some as a theory of organic coherence, some as a theory of structural levels, some as theory of voice lead-ing, and some as a theory of tonality.11
I prefer, however, to treat it as a theory of functional monotonal composition. It is this view that I will defend in this book with arguments drawn from analytical philosophy and cognitive science.
Another point of contention is the widely held belief that Schenkerian theory is a closed system, incapable of adaptation. Edward Laufer expresses this view most strongly in his review of Ernst Oster’s English translation of Der freie Satz.12
According to him, “Schenker’s concepts as such are complete: they call for no extensions or modifications.” Elsewhere, he defends the narrow scope of Schenker’s project by noting: “It is ridiculous to demand as a criterion of validity that an approach be applicable to all times and
musics; for it is sufficient that Schenker revealed, in ever new ways, the masterpieces of the classic era: altogether more great music than anyone could hope to come to terms with in several lifetimes.”13
While I am certainly sympathetic to Laufer’s point of view, I would add two important provisos. First, no one has ever offered a convincing argument to show why we should accept Schenker’s concepts as necessary and sufficient for explaining functional tonality. As I see it, such an argument requires us to reconstruct the theory systematically from the ground up. This is precisely what I hope to accomplish in the present volume. Second, although I have no problem accepting that Schenkerian theory is designed to explain functional monotonal pieces of the Common-Practice Period, I acknowledge that such music is part of a broader histori-cal continuum. Eventually, Schenkerians must respond to Joseph Kerman’s complaint that they treat tonal practice as “an absolutely flat plateau flanked by bottomless chasms.”14
I am not suggesting, however, that we should simply revive the approach adopted by Felix Salzer in Structural Hearing; much as I admire Salzer’s broad outlook on music history, I do not think his work provides a satisfactory response to this issue.15
What we need is a way to redirect our think-ing about tonality so that we can explain not only why Schenker’s theory works so well for functional monotonal compositions, but also how this theory enhances our understanding of tonality in general.
Another important area of debate concerns the ways in which Schenkerian analysis interfaces with the processes of hearing and composing. Although I have no doubt that the way in which we graph a piece depends on how we hear that piece, I do not accept the common view that producing a Schenkerian analysis simply means learning to hear music more effectively. For one thing, such a view leaves the theory open to the charge of circularity levelled by Eugene Narmour in his book Beyond Schenkerism.16
According to Narmour, when Schenker tried to demonstrate that a piece could be derived from a given prototype, he always knew in advance what the prototype should be. By bending the piece to fit his preconcep-tion, Schenker was guilty of circular reasoning. From a methodolog-ical standpoint, this is a very serious charge and, so far as I can tell, is one that has never been successfully dismissed by the Schenkerian community.
For another, there is strong evidence from Schenker’s own writings that he was not trying to explain how ordinary people hear a particular piece, but rather to explain how expert com-posers conceive of their music. For example, in his essay “Forset-zung der Urlinie-Betrachtung: I” from Das Meisterwerk I (1925), Schenker insisted that it is the composer’s business to compose out chords and that of the listener and the performer “to retrace” this path from foreground to background.17
Given this claim, I remain unpersuaded by the notion that Schenkerian analyses give us “artistic statements, in music, about music.”18
I firmly believe that, though they may be expressed as tones on a staff, Schenker-ian analyses should be regarded not as pieces of music, but rather as models of music.19
Like any model, these graphs capture some aspects of the music and not others. I would argue that they model an expert composer’s internalized knowledge of functional monotonality.20
My goal, then, is to respond to these various challenges. Since I am mainly motivated by methodological concerns, I have adopted something of a compromise when referring to Schenker’s works. For one thing, Schenker’s views clearly changed over time; his goals in writing the Harmonielehre and Kontrapunkt I were not exactly the same as those of the Fünf Urlinie-Tafeln and Der freie Satz. Having said that, I still believe that there is an underlying continuity to Schenker’s thought. This continuity stems from two basic claims: 1) the laws of tonal voice leading are transformations of the laws of strict counterpoint and are intimately related to certain laws of functional harmony; and 2) complex tonal progressions can be explained as transformations of simple tonal prototypes. Given my interest in these claims and their theoretical implications, I will focus more on the unity of his thought, than on its evolution. Similarly, although one must always be sensitive to nuances in the meaning of particular Schenkerian terms, I have generally quoted from standard English translations. My rationale is simple; while I am aware of the extraordinary problems that arise whenever we try to render Schenker’s often convoluted prose into English, I do not want to become sidetracked with the daunting task of retranslating every passage. Anyone interested in Schenkerian theory must eventually compare the English translation with the original; I have
tried to give citations in such a way that this can be accomplished as easily as possible.
Although this book has been largely written from scratch, it nonetheless draws on material from several published and unpub-lished papers. The Introduction develops ideas originally outlined in two published papers written with Douglas Dempster, “The Scientific Image of Music Theory,” Journal of Music Theory 33/1 (1989), pp. 65–106, and “Evaluating Music Analyses and Theories: Five Perspectives,” Journal of Music Theory 34/2 (1990), pp. 247–79. This work was subsequently updated in two unpublished lectures, “The Scientific Image of Music Theory: Ten Years On,” given at Eastman School of Music in the fall of 1996, and “Choosing between Music Theories: Six Things to Think About” delivered at SUNY Buffalo in the fall of 1998. Chapters 1 and 2 expand arguments first presented in my Ph.D. dissertation, “A Rational Reconstruction of Schenkerian Theory” (Cornell, 1989), and in my paper “Rothstein’s Paradox and Neumeyer’s Fallacies,” Intégral 12 (1998), pp. 95–132. Whereas chapter 3 is entirely new, chapter 4 draws on another unpublished paper, “Schenker and ‘The Myth of Scales,’ ” pre-sented at the Oxford Music Analysis Conference and the annual meeting of the Society for Music Theory in Baltimore in 1988. Chapter 5 borrows from lectures delivered at the University of Texas at Austin in the fall of 2001 and at Oxford University in the spring of 2003. Chapter 6 is mostly new but draws on material that I have been developing with Panayotis Mavromatis.
I would like to take this opportunity to thank several people for helping me along the way. Much of the blame for this book must go to Arnold Whittall for introducing me to Schenkerian theory in my undergraduate years at King’s College, London. Since then, he has been a pillar of support and very kindly read a draft of this man-uscript. This particular project started life in 1983–86, when I was a Junior Fellow at the Society of Fellows, Harvard University. I would like to thank the Society for its unqualified support during those years. Since that time, my thinking about Schenkerian theory has also been shaped by discussions with four other people—Doug Dempster, Dave Headlam, Panayotis Mavromatis, and Bryce Rytting. The fact that I have collaborated with most of them should indi-cate just how much they have taught me. I must also thank all my
graduate students at Eastman for listening to me and for reading chunks of text—Don Traut and Bill Marvin deserve special men-tion—and John Brackett for inviting me to be involved with his dissertation. Special thanks must go to Frank Samarotto for care-fully reading the manuscript. Most recently, I must thank Wayne Alpern and the Mannes Institute for Advanced Studies in Music Theory for inviting me to discuss my work in three workshops in the summer of 2002. I am extremely grateful to the members of my workshop—Kofi Agawu, Richmond Browne, Allan Cadwallader, William Drabkin, Yayoi Everett, Dora Hanninen, Dan Harrison, Peter Kaminsky, Richard Kaplan, Steve Larson, Nicolas Meeus, Margus Partlas, Giogio Sanguinetti, Carl Schachter, and Joe Straus. In terms of the production of the book, I would like to thank Dariusz Terefenko and Ciro Scotto for help in preparing the examples, though I must admit that I was never able to reproduce Schenker’s interlocking slurs to my satisfaction. And I must thank my editors Louise Goldberg and Ralph Locke for their patience and encouragement. Finally, I owe a special thank you to Milton Babbitt for encouraging me to publish my ideas in book form.
I would like to express my gratitude to the following people and publishers for permission to quote from their works and publications:
Cambridge University Press, for permission to use materials from the English translation of Schenker’s Das Meisterwerk in der Musik: (The Masterwork
in Music, ed. William Drabkin, trans. I. Bent, R. Kramer, J. Rothgeb, and
H. Siegel), vols. 1 and 2, © 1994, 1996.
MIT Press, for permission to use the figure showing the derivation of J. S. Bach’s Prelude in C Major, WTC 1, from Fred Lerdahl and Ray Jackendoff, A
Gen-erative Theory of Tonal Music, © 1983.
John Rothgeb, for permission to use materials from his edition/translation of Schenker’s Kontrapunkt I & II (Counterpoint I and II, ed. John Rothgeb, trans. John Rothgeb and Jürgen Thym, 2 vols., [New York, Schirmer, 1987]). Universal Edition, for permission to use materials from Schenker’s Harmonielehre (1906) and from Schenker’s Der freie Satz (1935), copyright © Universal Edition A.G. Vienna.
Theoretical and Meta-Theoretical
Issues
What should we expect from a successful theory of tonality? Why should we prefer one theory of tonality over another? To what extent do theories of tonality pose the same methodological prob-lems as theories in other domains? Although these are surely basic questions for any music theorist to ask, they are by no means easy ones to answer. In part, the difficulties stem from the fact that the term ‘tonality’ has come to mean different things to different peo-ple; as mentioned in the preface, some theorists use it very generally to denote music that centers on a stable pitch or tonic, whereas others use the term more restrictively to denote music that centers functionally on a particular tonic triad. But difficulties also arise because theorists often disagree about what they take to be the goals of their work. Once again, opinions differ widely. Some believe that theory building is an explanatory pursuit akin to the natural and social sciences, whereas others believe that it is a critical activity, analogous to art criticism or literary theory. As a result, some theo-rists deal exclusively with the internal properties of tonal music, whereas others insist that these properties cannot be studied apart from their cognitive, aesthetic, historical, and ideological context.
The purpose of this Introduction is not to address these issues in a systematic manner, but rather to pinpoint some of the method-ological concerns that shape my own particular views about tonal theory. I will proceed from the assumption that, before we can assess the cognitive, aesthetic, historical, and ideological implications of a particular theory, we must first see how that theory explains why tonal music behaves in some ways and not others. Since I believe that, at some level, we process our knowledge of music separately from our knowledge of other domains, I find it useful to treat music
theory autonomously from other disciplines. Furthermore, since I also believe that tonality is basically a general property of voice leading and harmony, I will focus my attention on explaining these phenomena. This does not mean, however, that I am uninterested in thematic, rhythmic, or formal relationships. Rather, I do so because I am concerned with explaining the tonality of music rather than explaining tonal music per se. In fact, I readily accept that there is much more to understanding the latter than simply explaining its tonality. My discussion has three main parts.
Part 1 outlines what I take to be the main goals of any theory of tonality: 1) to develop a vocabulary of concepts for describing what relationships count as tonal; 2) to discover a set of covering laws for explaining why these relationships work in some ways in ways and not others, and 3) to devise procedures for explaining how to produce specific tonal relationships. Next, part 2 discusses some of the issues that arise in testing a particular theory of tonality. Finally, part 3 discusses six criteria that theorists frequently use to pick one theory of tonality over another: accuracy, scope, consistency, sim-plicity, fruitfulness, and coherence.
Basic Goals and Assumptions
Although music theorists formulate theories of tonality for a vari-ety of reasons, three seem to be especially important. The first is to develop a vocabulary of concepts for describing what relationships count as tonal or, more specifically, functionally tonal. Concepts are terms we use to categorize our observations into broad types. According to ‘The Classical Theory of Concepts,’ defining a con-cept involves establishing a set of necessary and sufficient condi-tions that something must satisfy if it is to fall under that concept.1
Music theorists have traditionally expended considerable effort on developing concepts to describe a wide range of tonal relationships. Many of these concepts allow us to describe how notes behave lin-early. For example, when describing the tonal properties of the music shown in figure I.1a (Beethoven, Six Variations, WoO 70), we might observe that the thirty-second note B in the treble clef,
m. 13, passes between A and C. This account presupposes that the concept ‘passing tone’ can be defined as an unaccented dissonance that moves by step between two consonances a third apart. Such a definition conveys the necessary condition that passing tones move by step and is sufficiently precise to differentiate passing tones from other dissonances, such as neighbor tones, cambiatas, and suspen-sions. Alternatively, we might use quite different concepts to describe how notes behave harmonically. In the case of figure I.1a, we might observe that the passage begins and ends on a G triad with the root in the bass. This description presumes that the con-cept ‘triad’ can be defined as a chord with a root and two other members a third and a fifth above. The latter definition conveys the necessary condition that triads are built from thirds and fifths, and yet is sufficiently broad to encompass major, minor, augmented, and diminished triads.
It goes without saying that both of these descriptions tell us sig-nificant things about the tonal properties of figure I.1a. And yet,
Figure I.1. Explaining tonality.
a. Beethoven, Six Variations, WoO 70, mm. 13–16.
neither one actually explains why Beethoven’s music is ‘in’ G major. The problem is that to explain why the passage is tonal, it is not enough to describe what melodic tones and triads are present; we must also say why they are related to each other in some ways and not others. To do so, music theorists invoke various laws of harmony and voice leading. These laws are general claims about the ways in which melodic tones and triads usually behave in tonal contexts.2
They are often, but not always, expressed in a conditional form: if X occurs in context Y, then Z will happen. For example, to explain why the passage in figure I.1a establishes the key of G major, we might invoke the following law: “If the leading tone appears in tonal contexts, then it normally ascends by half step onto the tonic.” This law explains why the melody rises from F to G in mm. 13–14, why the alto part follows suit in mm. 14–15, and even why the same theme is in B-flat major when Beethoven transposes it up a minor third in his piano sonata, Op. 22 (see figure I.1b). Having said this, it is important to note that the law governing leading tones is not true all of the time. In figure I.1a, for example, the alto F in mm. 15–16 moves to B3 and not G4, presumably to avoid doubling the soprano part. Since most laws of tonality are generally but not uni-versally true, they are best classified as law-like generalizations.
Besides introducing concepts to describe what relationships are tonal and invoking law-like generalizations to explain why tonal music behaves in some ways and not others, theorists often have yet another important goal: to explain how specific tonal relationships are produced. This task requires them to develop a set of procedures. Procedures consist of strings of commands that are usually expressed in conditional form: to produce X, do Y, then Z, and so on. Over the centuries, tonal theorists have developed procedures for accomplish-ing a variety of tasks from harmonizaccomplish-ing a scale to composaccomplish-ing a prel-ude from a given figured bass. Take, for example, figure I.2 (A procedure for composing typical tonal melodies). This procedure has six basic steps. First, pick a final tonic for the melody as a whole (fig-ure I.2a). Second, begin the melody on a member of the tonic triad and end with a stepwise descent onto the tonic (figure I.2b). Third, pick a climax note midway through the melody and not more than an octave above the tonic (figure I.2c). Fourth, reinforce the tonic at the opening (figure I.2d). Fifth, join the opening to the climax and
the climax to the cadence (figure I.2e). Sixth, fill in any details and check to see that the melody has a good overall shape and satisfies any general laws of tonal voice leading, for example, that leading tones normally ascend by half step onto the tonic (figure I.2f).
The preceding discussion has highlighted the central role con-cepts, laws, and procedures have traditionally played in tonal theory, but it is important to realize that these components are a lot more difficult to deal with than we might initially suppose. Take, for example, concepts. While it is certainly possible to find necessary and sufficient conditions for many concepts, cognitive scientists have found that certain concepts cannot be defined in this manner. Instead, they tend to define such concepts by appealing to the notion of prototypes.3
As Alvin Goldman explains:
Concepts are represented in terms of properties that need not be strictly necessary but are frequently present in instances of the concept. These
a. Pick a final tonic for the melody as a whole.
1 b. Begin the melody on 8, 5, 3, or 1 and end with a stepwise
descent onto the tonic.
Cadence
3 2 1
c. Pick a climax about two thirds through the melody and not more than an octave above the tonic.
Climax Cadence
3 4 2 1
d. Reinforce the tonic at the opening.
Climax Cadence
3 4 7 1 4 2 1
e. Join the opening to the climax and the climax to the cadence. Climax Cadence
3 4 7 1 2 3 4 2 1
f. Fill in any details and check to see that the melody has a good over-all shape and that it satisfies any general laws of melodic motion.
Climax Cadence
3 2 3 4 6 7 1 2 3 4 21
properties are weighted by their frequency or by their perceptual salience. A collection of such properties is called a prototype.4
He adds: “Under the prototype view, an object is categorized as an instance of a concept if it is sufficiently similar to the prototype, similarity being determined (in part) by the number of properties in the prototype possessed by the instance and by the sum of their weights.”5
Although Goldman does not explicitly say so, the ‘perfect’ prototype may not actually ‘exist’ in the world at all; it may be an idealization that combines features from many different individuals. We can illustrate these points by reconsidering our definition of passing tones (see figure I.3, Five forms of passing tone). Although we defined passing tones as unaccented dissonances that move by step between two consonances a third apart, some passing tones do not satisfy this definition. In figure I.3a, for exam-ple, the pitch B in m. 1 seems to behave as a passing tone, even though it is consonant, and in figure I.3b the notes F and E in m. 1 are both dissonant and seem to connect two consonances a fourth, not a third, apart. More remarkably, figure I.3c contains an accented passing tone, figure I.3d includes a chromatic passing tone, and, if you believe Schenker, figure I.3e contains a leaping passing tone (or springender Durchgang)!6
In other words, it is much easier to think about passing tones in terms of prototypes and variants, than it is to provide a necessary and sufficient definition that works in all cases.7
Laws, too, pose their own problems.8
To begin with, not all gen-eralizations are law-like.9
Take, for example, the claim that all pieces in G major have a key signature of one sharp. Even if true, which it is not, this generalization does not stand up as a law because it does not explain why there is any connection between having a signature of one sharp and establishing the key of G. To ensure that particular
generalizations are law-like, Nelson Goodman and others have suggested they should support so-called counterfactual condition-als.10
Counterfactual conditionals are hypothetical statements that suggest what would have been the case had things occurred differ-ently (see figure I.4, Counterfactual conditionals). For example, when explaining why the cadence in figure I.4a establishes the key of G major, we might invoke our law governing leading tones. In this case, the F in the first chord ascends by half step to the G in the final sonority. We might support this law-like generalization by not-ing that if the phrase had been in F major, then the F in the first chord would have descended to E, before moving back onto F for the final chord (see figure I.4b). Since we know that the piece in question is in G major, our remarks about what might have hap-pened if the piece were in F major are known as counterfactual con-ditionals. While counterfactual conditionals have proved very useful in helping us determine whether a particular generalization is indeed law-like, they nonetheless raise their own sets of questions; it
is unclear not only how to ensure that they are relevant in any given context, but also that they can be used to support all law-like gener-alizations.
It is also debatable whether law-like generalizations are always necessary and sufficient for explanations. Certainly, many experts believe that scientific research is fundamentally law seeking or
nomo-thetic.11 This prompted Carl Hempel and Paul Oppenheim to
advance ‘The Covering-Law Model’ of explanation.12According to
them, explanations are arguments in which the premises are sets of covering laws and initial conditions, and the conclusion is some statement about the phenomena to be explained (see figure I.5, ‘The Covering-Law Model’). If the laws are universal and the argu-ments are deductively valid, then the result fits ‘The Deductive-Nomological Model,’ and if the laws are not universal and the arguments are only inductively valid, then they conform to ‘The Inductive-Statistical Model.’ Figure I.6 (Explaining suspensions) illustrates what Hempel and Oppenheim had in mind. Suppose, for example, that we want to explain why a particular suspension C resolves by step to B (see figure I.6a). We might do so by invoking a simple law of tonal voice leading: namely, that suspensions normally resolve down by step onto consonances (see figure I.6b). Given the initial conditions that the seventh C–D on the down beat of m. 2 is dissonant and that the dissonance is a suspension, this law-like gen-eralization allows us to deduce that the dissonant tone C on the down beat of m. 2 will resolve down by step onto the consonant tone B in m. 2. This is a perfectly acceptable explanation.
Although ‘The Covering-Law Model’ certainly produces acceptable explanations, it is unclear whether covering laws are absolutely necessary for all plausible explanations. In particular,
‘The Covering-Law Model’ Statement of initial conditions Statement of covering laws Statement about phenomena to be explained.
critics have suggested that some types of explanation, such as so-called functional explanations used in biology, or narrative explanations found in history, do not necessarily involve covering laws, at least not in any explicit way.13 Functional explanations
explain how particular parts of a complex system help to reinforce the system as a whole. For example, when explaining the tonal motion of Beethoven’s “Waldstein” Sonata, we might note that, in the first movement, the function of the recapitulation is to recom-pose the tonal motion of the exposition, that the function of the exposition is to modulate from the first key (C major) to the second key (E major), and that the function of this modulation is to create a pattern of tonal tension, and so on. Though this explanation seems plausible enough, it is unclear what covering laws it uses. Historical narratives often proceed on similar lines. For example, when explaining why the climax of a given aria appears on a high ‘C’ we might note that this aria was written for a particular tenor to sing at La Scala, and that high ‘C’ was his top note. Although this explanation seems to make historical sense, we would never sug-gest that, as a general rule, composers always make sure that the climax of an aria necessarily corresponds to the highest note in the singer’s register.14
Figure I.6. Explaining suspensions.
b. c.
Initial The seventh C-D on the down The sixth B-D on the weak Conditions beat of m. 2 is dissonant beat of m. 2 is consonant
This dissonance is a suspension This consonance is a resolution. Covering Suspensions generally resolve Suspensions generally resolve Laws down by step consonances down by step onto onto consonances Explanation Resolution on weak beat Suspension on down beat
By the same token, Sylvain Bromberger and others have raised doubts about whether covering laws are sufficient for all explana-tions.15
We can paraphrase their point by comparing the explana-tion given in figure I.6b with the one shown in figure I.6c. In figure I.6b, we explained why the dissonant note C on the down beat of m. 2 resolved down by step to a consonant note B by invoking the law-like generalization that suspensions normally resolve down by step. However, we can also use the same covering law in a quite dif-ferent way. This time we might start with the initial conditions that the sixth B–D on the weak beat of m. 2 is consonant and that it resolves a suspension on the preceding down beat. Since our cover-ing law states that suspensions generally resolve down by step, we can deduce that the consonant note B is preceded by a dissonant note C. Although this argument is logically consistent, it does not carry the same weight as the argument given in figure I.6b. The problem is that the argument in figure I.6b explains the causal con-nections between the suspension and the resolution, whereas the one in figure I.6c does not. This, in turn, suggests that it is not enough to invoke covering laws in our explanations; our explana-tions must also be able to explain how one event causes another.
One way to guarantee such casual connections is by reformat-ting our covering laws in procedural form.16
As mentioned earlier, procedures are strings of commands that we express in conditional form: to produce X, do Y, then Z, and so on. This point is illustrated in figure I.7, (A procedure for generating 7–6 suspensions). This procedure involves three distinct steps:
1) take an upper voice that descends by step C to B (figure I.7a); 2) add a lower voice that moves in parallel sixths below, E to D
(figure I.7b); and
3) displace the first note of the upper voice over the second note of the lower voice (figure I.7c). This step produces the 7–6 suspension.
Significantly, this procedure implies all of the same knowledge as the explanation given in figure I.6b. In particular, it implies that suspensions generally resolve down by step. And yet, the procedure adds something extra: it also indicates that suspensions are caused by displacing the upper voice over the lower voice.
Whatever advantages procedural explanations may give us, they can, however, be slippery things to deal with. Reconsider, for a moment, the procedure given in figure I.2. This strategy identified six basic steps for composing a typical tonal melody. In order for us to determine whether the procedure is successful or not, we must decide whether or not our new melody resembles the melody in fig-ure I.1. But what does it mean to say that two melodies resemble one another? To answer this question we must invoke some notion of similarity, but it is by no means obvious what this step involves. As the philosophers Quine and Ullian explain: “Everything is sim-ilar to everything in some respect. Any two things share as many traits as any other two, if we are undiscriminating about what to call a trait; things can be grouped in no end of arbitrary ways.”17
In other words, we can judge similarity in different ways, depending upon what examples we take as prototypical and on how we decide to measure similarity.
It seems, then, that in describing what relationships are tonal, explaining why these relationships create music that behaves in some ways and not others, and explaining how to produce specific tonal relationships, music theorists draw on a rich assortment of
Figure I.7. A procedure for generating 7–6 suspensions.
a. Take an upper voice that descends by step from C to B.
b. Add a lower voice that moves in parallel sixths below, E to D.
c. Displace the first note of the upper voice over the second note of the lower voice to cre-ate a 7–6 suspension.
concepts, laws, and procedures. Although each component raises its own methodological issues, a given theory of tonality takes a specific cluster of concepts, laws, and procedures, and structures it in a par-ticular way. If these theories are successful, then we normally expect that these knowledge structures will give us explanations and pre-dictions that are coherent, reliable, and capable of being tested empirically by other theorists. And they should work for all and only all tonal music, or at least all and only all functional pieces. But how do we go about building such a theory? How, in fact, do we check to see that it actually covers all and only all music that we classify as tonal? Let us see how we might answer these new questions.
Building and Testing Theories
According to conventional wisdom, theorists use a very simple strat-egy for coming up with their theories: they make guesses and then they test them.18
This approach is summarized in figure I.8 (‘The Hypothetico-Deductive Method’).19
In fact we can subdivide ‘The Hypothetico-Deductive Method’ (or H-D) into four basic steps:
First, observe some distinct phenomenon in a well-defined test sample (figure I.8a).
Second, guess some laws that seem to explain these observations (figure I.8b).
Third, deduce some predictable consequences that are implied if these laws are correct (figure I.8c).
Fourth, see if these predictions are confirmed by further obser-vations (figure I.8d).
If the predictions are indeed confirmed, then theorists carry on using their laws; if they aren’t, then they must either modify them, or they must replace them with new laws and start the procedure all over again. This final step is crucial to the entire process; ideally, it implies not only that the predictions are testable, but also that they are testable by someone else and under different conditions.
Now, if we want to make a theory to explain the behavior of tonal voice leading and harmony, then we might adopt the follow-ing plan. We might begin by usfollow-ing certain familiar concepts to study a specific corpus of pieces that a given community regards as
quintessentially tonal. Chances are we would pick pieces by such composers as Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Schumann, Mendelssohn, Chopin, and Brahms.20Next, we might develop general laws that cover the
behavior of these concepts. For example, we might generalize about how lines and chords behave in specific contexts. We could then predict how the lines and chords might behave in other contexts, preferably those that are slightly different from the ones used to make the original theory. If our predictions are confirmed, then we will keep on using our theory to explain the behavior of similar works from the same corpus; if, however, they are disconfirmed, then we must either modify the theory or find an alternative one that does work.
So much for conventional wisdom; we find it perpetuated in many introductions to science. But when we look at how theorists actually work, we soon find that the process of building and testing theories is a good deal more complex than figure I.8 suggests. To begin with, it is very unlikely that any music theorists would actually build a new theory of tonality from scratch. Instead, they are more likely to start by taking some preexisting model and seeing if indi-vidual laws stand up to close scrutiny. Once they encounter a prob-lem, then they will propose new covering laws. But disconfirming existing laws and confirming new laws is no easy task; on the con-trary, these activities are riddled with problems and inconsistencies.21
These difficulties stem from the fact that, as David Hume famously remarked, “all inferences from experience, therefore, are effects of
a. Observe phenomenon in some well-defined test sample.
b. Guess laws to explain these observations.
c. Deduce some consequences that are implied if the laws are correct.
d. See if these predictions are confirmed by further observations. If they are, then keep using the new laws, if they aren’t, then modify them or replace with some new laws and start procedure over.
custom, not of [logical] reasoning.”22
Hume’s claim does not mean that inferences from experience are necessarily untrue; rather it sug-gests that they always fall short of certainty. As a result, our theories will always be fallible, though we may not always know where they will fail.
Besides recognizing the general problems noted by Hume, philosophers have discussed several other difficulties. One of the most famous is ‘The Raven Paradox.’23
This paradox arises because law statements of the form “for all x, if x is F, then x is G” are log-ically equivalent to those of the form “for all x, if x is not G, then x is not F.” If x stands for piece of music, F stands for Beethoven and G stands for tonal, then the first law-statement “all pieces of music by Beethoven are tonal” is equivalent to the observation that a particular non-tonal piece, say Babbitt’s Philomel, is indeed not by Beethoven. What is paradoxical is that an atonal piece by Babbitt should count as evidence that confirms the generalization that all pieces by Beethoven are tonal; after all, Babbitt’s music appears to have little in common with Beethoven’s and it hardly seems relevant to any claims about whether the latter is tonal or not.
Relevance also features prominently in another paradox known as ‘The Grue Paradox.’ This paradox was first discussed by Nelson Goodman.24
According to him, the issue of whether a generalization is supported by its instances depends on the nature of the properties that appear in that generalization. Paraphrasing Goodman, let us imagine a new property, ‘gronality’ which we define as follows: a piece is ‘gronal’ if it is classified as tonal before the year 2010 and atonal after that point. Now consider the following generalizations: 1) all works by Burt Bacharach are tonal and 2) all pieces by Burt Bacharach are ‘gronal.’ All works examined before the year 2010 will support not only the first generalization, but also the second one. This result is problematic because we want to use our general-izations to predict what will happen at some later date; as it stands, we have no basis for knowing whether the piece will be tonal or ‘gronal.’ To resolve this paradox, Goodman proposed that the law-like status of a generalization is a matter of entrenchment and projectibility. According to him, a predicate is entrenched if it is true as a matter of historical fact and has been used to formulate
true predictions.25
He suggests that this property is the only one that allows us to project what will happen in the future. Tonality is just such a predicate; it is a trait that we naturally project from past observation to future expectation. ‘Gronality’ is not, however, because we have no reason to suppose that Burt Bacharach wrote music that can been classified as tonal at one point in time and atonal at some later date.26
‘The Grue Paradox’ leads to a more general problem in confir-mation; even if we agree on the same body of evidence, there is no reason to suppose that this data can be explained by only one theory; as we have seen, we can always invent new predicates, such as ‘grue-ness’ or ‘gronality,’ that capture some aspect of the piece. This means that, in principle at least, the evidence always underde-termines theories; there are always a variety of theories that will accommodate any given set of data. Pierre Duhem and W. V. Quine have gone even further to claim that, taken on its own, a particular piece of experimental evidence is seldom used to falsify an entire theory, because each element of the theory is somehow related to another element in the theory. As Quine puts it, “our statements about the external world face the tribunal of sense experience not individually but only as a corporate body.”27
In other words, “any seemingly disconfirming observational evidence can always be accommodated to any theory.”28
This claim is usually known as ‘The Duhem-Quine Thesis.’
Although extremely controversial, ‘The Duhem-Quine Thesis’ is significant because it threatens to undermine the most famous alternative to H-D. Given the many paradoxes of confirmation, Karl Popper and others have suggested that, instead of defending their theories by finding more and more supporting evidence, scientists should actually spend their time trying to show that some hypotheses are false.29
In this sense, the guiding principle of test-ability is not confirmation but falsification. The rationale behind Popper’s thinking is simple enough and is apparent from the argu-ments given in figure I.9 (The logic of falsification). According to Popper, H-D seems to follow the plan given in figure I.9a. Let us assume that, if a particular explanation E is valid, then it will make a given prediction P. When researchers test this prediction and find that it is indeed accurate, they regard this as confirmation of their
explanation. But, according to Popper, this line of reasoning is invalid. Since explanation E was used to come up with prediction P, prediction P cannot then be used to confirm explanation E. That would commit the fallacy of affirming the antecedent (see figure I.9b). To avoid this problem, Popper insists that H-D should be used, not to confirm, but to falsify a theory. This means that, given explanation E and prediction P, knowing that P is false allows us to deduce that E is false as well (see figure I.9c). Such an argument follows the principle of modus tollens given in figure I.9d.
Popper used the notion that H-D can be used to falsify an explanation to reach two important conclusions. First, he proposed that even our best knowledge is fallible or conjectural. To quote him, “we cannot reach certainty . . . all we can do is to criticize [our theories], and to test them, as severely as our ingenuity per-mits.”30
Second, Popper decided that “the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.”31
He even claimed that falsifiability serves as a criterion for making demarcations between science and non-science; whereas scientific theories must always have discrete boundaries and can be falsified, non-science need not have such boundaries and cannot be falsified. Critics, however, have responded that strict falsification is hard to uphold in light of ‘The Duhem-Quine Thesis.’ If, as Duhem and Quine insist, “any seemingly disconfirming observational evidence can always be accommodated to any theory,” then it is hard to see how we can decisively falsify any theory. Each time we come up with a counter example, we can simply adjust our theory to make it
a. If Explanation E is valid, b. If X→ Y then Prediction P is true.
Prediction P is true Y
∴ Explanation E is valid ∴ X (invalid) c. If Explanation E is valid, d. modus tollens
then Prediction P is true. If X→ Y Prediction P is false ⫺Y
∴ Explanation E is false ∴ ⫺X (valid)
fit. If we cannot use specific observations to falsify a given theory, then we cannot pick one theory over another on purely evidential grounds. Popper’s claim that falsifiability provides us with a defini-tive means of discriminating science from non-science seems, therefore, too strong.32
As it happens, Popper’s views have also been challenged by recent findings in cognitive science. Research by Tweney, Doherty, Mynatt, and others has suggested that scientists do not usually set out simply to falsify existing theories; on the contrary, they nor-mally start out by seeking confirmatory data; only when this data has been obtained does it make sense to engage in rigorous falsifi-cation.33
Thus, while it might be true that successful theories are initially conjectural, the accumulation of supporting evidence will eventually move them beyond that status.34
Most people do, in fact, believe that theories become more strongly confirmed the more supporting evidence has been amassed. This point suggests that our understanding of what makes a successful music theory must eventually take account of the ways in which music theorists actu-ally work, rather than simply relying on their logical or empirical content.
All in all, just as ‘The Covering Law Model’ provides an idealized picture of explanation, so ‘The Hypothetico-Deductive Method’ presents an idealized account of how music theorists confirm or refute a particular theory. While the latter conveys many aspects of how music theorists work, the process of building and testing theories involves a far more complex interplay between confirmation and falsification. As we have seen, this process is always open ended; music theorists do not begin with a blank slate, they do not have foolproof methods, and they do not reach definitive solutions. Instead, they plunge in medias res. They start working within the context of an existing music theory, even if they know some portions of that theory are surely wrong. They then try to overcome certain specific problems, using the rest of the theory to support their work. To borrow an image from Neu-rath and Quine, this situation is like that facing sailors at sea on a leaking boat.35
Unable to rebuild their vessel from the keel up in a dry dock, the crew is forced to fix the leaks while adrift on the open water. As they work on leaks in one area of the boat, the
sailors rely on the remaining timbers to keep the craft afloat. But as one leak is patched so another appears; bit-by-bit the boat becomes transformed into something new. In fixing the leaks, music theorists typically try to balance what Quine has described as “the drive for evidence and the drive for system.”36 According
to him, the former demands that “theoretical terms should be sub-ject to observable criteria, the more the better, the more directly the better, other things being equal” while the latter insists that these terms “should lend themselves to systematic laws, the sim-pler the better, other things being equal.” Quine adds, “If either of these drives were unchecked by the other, it would issue in some-thing unworthy of the name scientific theory: in the one case a mere record of observations, and on the other a myth without foundation.”37
Six Criteria for Evaluating Theories
It should be clear by now that the task of building and testing music theories is not only a lot messier than we might suppose, but it is also plagued by many of the same methodological problems as the-ories in other disciplines, especially the natural and social sciences. In very general terms, we have seen that these problems often involve finding effective ways to balance the drive for evidence with the drive for system. Although this all sounds reasonable enough, we can spell out more clearly how such a balance might be achieved. Following, Kuhn, Quine and others, we can invoke sev-eral concrete criteria for doing so: figure I.10 (Six criteria for evalu-ating theories) includes the notions of accuracy, scope, fruitfulness, consistency, simplicity, and coherence.38
The list is by no means exhaustive; other criteria, such as completeness, elegance, or even ‘coolness,’ could easily be added. Figure I.10, however, gives us a good place to start our inquiry. For convenience, the six criteria are divided into two types—those that relate to the evidential basis of theories and those which relate to the systematic aspects of a model. The horizontal arrow at the top of the figure suggests that there may be inherent conflicts between the evidential concerns (accuracy, scope, and fruitfulness) and the systematic concerns
(consistency, simplicity, and coherence). The vertical arrows along the sides suggest that there may be a similar tension between mem-bers of the same type.
The first criterion on our list is accuracy. Since a successful theory should explain why certain phenomena behave the way they do and predict what will happen in new situations, we will surely want the most exact explanations and predictions possible. Although our theories can never be completely accurate, we gener-ally see increased precision as a virtue, so that, given two theories, we usually prefer the one that is more accurate, other things being equal. And yet, it is by no means obvious how to measure the accu-racy of competing models. Norwood Hanson, Thomas Kuhn, Paul Feyerabend, and others have insisted that since our observations about the world may be theory-laden, decisions about what consti-tutes evidence will be determined by our theoretical prejudices.39
Indeed, to paraphrase Quine, “Our judgments about what there is are always embedded in some sort of theory; we can substitute one theory for another but we cannot detach ourselves from theory altogether and see the world unclouded by any preconception of it.”40
If competing theories reflect widely different values, there may be no neutral grounds for comparing their accuracy. In such cases the two theories are said to be incommensurate. Critics, however, have countered that the problems of theory-ladenness and incommensurability are greatly exaggerated. While it may be true that observations tend to be theory laden, this doesn’t mean that we can never distinguish observational terms from theo-retical terms. Indeed, as Quine points out, “theotheo-retical sentences grade off to observation sentences”; some observations come with
Evidence Accuracy Scope Fruitfulness System Consistency Simplicity Coherence
negligible theoretical baggage, while others come with a lot.41
Compare, for example, the observation that the first movement of the “Eroica” Symphony begins with an E triad with the claim that the movement is “in” E. Whereas the former involves few theoretical assumptions and is readily apparent to most listeners, the latter is highly theory laden and presupposes an elaborate the-ory of key relations.42
But even though many of our observations are biased, theo-rists working with different frameworks are still able to reach some degree of consensus in specific cases. In this respect, the main issue is that of intersubjective testability rather than of objectivity per se. This notion of intersubjective testability through bias is most remarkable when the various biases are not only different, but also contradictory. I refer to this as ‘The Hostile Witness Prin-ciple.’ Very simply, this principle suggests that a particular claim gains force when it is confirmed by theories that are directly opposed to one another. Such situations often arise because, as Richard Boyd points out, “A particular experiment can be con-ducted on the basis of a methodology that—however theory-dependent—is not committed to either of the two contesting theories.”43
The second criterion in figure I.10 is scope. Just as we want our theories to be as accurate as possible, so we also put a premium on their breadth of coverage. This means that, given two theories, we normally prefer the one that covers the larger array of pieces or wider range of properties, other things being equal. Perhaps the most common way to expand the scope of our theories is by sub-suming hitherto separate theories under a single scheme. This is known as ‘Theory Reduction.’44
For example, if we proposed a the-ory of functional monotonality that subsumes the thethe-ory of tonal voice leading with the theory of functional monotonal harmony, then we should prefer it to a rival theory that explains only the behavior of tonal voice leading or that explains only the behavior of functional monotonal harmony. This does not mean, however, that generality is always a good thing; on the contrary, some theories are so general that they lose their explanatory force. That’s the snag with Anne Elk’s theory of the brontosaurus.45
While it may be true that the only thing common to all brontosauruses is that they “are
thin at one end, much, much thicker in the middle and then thin again at the far end,” this account is so general that it is trivial. The notion of ‘Theory Reduction’ has likewise been questioned. While there are certainly situations in which the model seems to apply, it does not explain every option. Kuhn, for example, has suggested that explanatory scope can expand through conceptual innovations or paradigm shifts, rather than the addition of new laws or the reduction of one theory into another.46
To overcome these difficul-ties Philip Kitcher and others have advocated the notion of ‘Theo-retical Unification.’ According to Kitcher, the success of theories depends on “minimizing the number or patterns of derivation employed and maximizing the number of conclusions generated.”47
When evaluating the success of our theories, we do not simply want to keep duplicating results in familiar pieces; we also want to use our concepts, laws, and procedures to predict how things will behave in other, perhaps novel, works and disclose new phenomena or previously unnoted relationships among those already known.48
To do this, we must be able to predict every consequence and not merely a smattering of special cases.49
This idea represents the third criterion in figure I.10, namely fruitfulness. Very simply, given two theories of functional monotonality, we prefer the one that makes the more fruitful predictions, other things being equal. According to Kuhn, the criterion of fruitfulness “deserves more emphasis than it has yet received.”50
Just as it is hard to measure the accuracy of rival theories, it is also difficult to assess their fruitfulness, especially if the theories draw on widely different bodies of empirical data. This issue is troubling because successful theories often evolve considerably over time; it may take a long while for theorists to appreciate just how fruitful a theory may be and even longer to con-sider all of its ramifications. As a result, fruitfulness may not play a significant role when a theory is originally presented to the world but will become more significant as that theory matures.
Whereas our first three criteria concern the drive for evidence, our fourth criterion concerns the drive for system. When formulat-ing a music theory, we will want it to be as internally consistent as possible, other things being equal. Inconsistencies are bad because they prevent us from making concrete predictions; if we cannot make concrete predictions, then we cannot subject our work to
