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Generalized Musical Intervals and Transformations$

David Lewin

Print publication date: 2007

Print ISBN-13: 9780195317138

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780195317138.001.0001

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(p.245) Appendix A: Melodic and harmonic GIS Structures; Some Notes on the History of Tonal Theory

(p.245) Appendix A: Melodic and harmonic GIS Structures; Some Notes on the History of Tonal Theory

Generalized Musical Intervals and Transformations
Oxford University Press

Chapter 2 surveyed a variety of musical spaces pertinent to theories of Western tonality. In some of these spaces, our intuitions of directed distance or motion from one position to another were measured by steps along some melodic scale, diatonic or chromatic, linear or modular. In other spaces, our intuitions were measured by numbers reflecting harmonic relationships of various kinds, or by moves on a game board derived from harmonic relationships.

The richness of tonal music, and of music in related idioms, is much enhanced by the ways in which a variety of such intuitions come into play. We can review in this connection the Protean meanings of “the interval from F to A♭” in Reflets dans l’eau. We sense a harmonic interval within the D♭-major triad and the Tristan Chord; we sense also a melodic interval moving two steps along a diatonic scale in D♭ major, or along a diatonic hexachord (the cadential descending hexachord B♭A♭G♭FE♭D♭); to some extent we can even hear F-to-A♭ as one melodic Step along the pentatonic scale D♭E♭FA♭B♭, as we listen to the melody at the beginning of the piece; finally we also hear F-to-A♭ as spanning three semitones along a chromatic scale, once the CHROM figure comes onto the scene. A transformational approach enabled us to sidestep these ambiguities in chapter 10, referring there to transformations T and T′ that mapped D♭ to E♭ and A♭ to E♭ respectively; we could conceive T and T′ as transpositions by “intervals” in any-or-all of the conceptual spaces involved; then we could compute a corresponding transformation TȒ1 T′ which mapped A♭ to F in any-or-all of the spaces, a transformation worked out musically in the change from motive X to motive Y.

Such transformational discourse is particularly useful to discuss architectural features of Reflets that obtain no matter what sorts of intervallic (p.246) intuitions one considers. For instance, the tonic of the climactic fortissimo E♭ major is in a T-relation to the tonic of D♭ major no matter what kind of interval, in what kind of melodic or harmonic space, we consider T to be transposing D♭ by. On the other hand, transformational discourse is correspondingly impoverished when it comes to exploring the varieties of spatial and intervallic intuitions at hand, and the ways in which the music brings those intuitions into play, each with the others. In connection with the big climax of Reflets, for instance, the major mode of the fortissimo music in E♭ favors certain kinds of intervallic intuitions over others, when we hear E♭ major, not E♭ minor, in relation to D♭ major. The reader may also recall our discussion of the high C–D♭ in the principal melodic line, and our question: “Is this, or is this not, a T-relation?” That question implicitly involves very broad questions about the premises of the composition: To what extent is the piece diatonic-melodic, so that one scale-step is one scale-step, regardless of its acoustical size? To what extent is the melos chromatic, so that one semitone is something necessarily very different from two semitones, even if both are spanning one diatonic step? To what extent is “the interval” attached to T heard in a harmonic context that gives it a size somewhere between 10/9 and 9/8, but not as small as 16/15? To what extent can techniques of melodic motivic transformation, involving the rhythm and contour of the Z motive in particular, alter our impressions in any or all of these respects? and so on. Exactly these ambiguities must be appreciated, if we are adequately to appreciate the conceptual tensions of the local climax involving C5 and D♭5, beyond its high register and relatively high dynamic level.

We return, then, to the variety of intervallic intuitions surveyed in chapter 2. Pertinent syntheses of these intuitions are essential not only for many occasions in critical listening and analysis, but also for many abstract theoretical purposes. Indeed, such syntheses are among the greatest triumphs in the history of Western music theory, and their neglect or failure has led to some of the more embarrassing moments in that history. Among the latter, we may cite Rameau’s argument that the harmonic intervals of 5/4 and 6/5 may be exchanged in relative register within the harmonic triad, so as to derive the minor triad from the major. This may be allowed, he says, since 5/4 and 6/5 are both “thirds.”1 But at the time he says this, he has not as yet presented us with any scale along which we can measure distances of “three” degrees, and he has assured us very strongly that melody is in any case thoroughly subordinate to harmony.2

(p.247) One might try to replace Rameau’s implicitly melodic argument about the “thirds” by a suitable harmonic argument: The intervals 5/4 and 6/5 are adjacent within the senario; they divide the interval 3/2 = 6/4 harmonically, and so arithmetically when reversed; therefore such a reversal is logical. But one could argue in exactly the same fashion for the intervals 4/3 and 5/4, as they divide 5/3. Would Rameau, then, have accepted the argument that we ought to consider the harmony G4–C5–E5 as functionally equivalent to the harmony G4–B4–E5, by the analogous reasoning? Obviously, he would not have; such reasoning there would violate the principle of the Fundamental Bass. Helmholtz, however, might have been willing to entertain the argument; indeed he actually asserts harmonic equivalence of a certain sort between the six-four position of the major triad and the six-three position of the minor triad. Both those positions comprise the highly consonant verticalities of a fourth, a major third, and a major sixth; having the same vertical-interval content, they are thereby the “most consonant” close positions for their respective pitch-class sets.3

Among the triumphal syntheses mentioned earlier a high position must be reserved for Zarlino’s Istitutioni harmoniche.4 Book 1 discusses intervals as phenomena in a harmonic space. Book 3 discusses intervals all over again as phenomena in melodic space, and synthesizes that approach with the mathematical ideas of book 1. Abstract harmonic ratios are accessible to our perception (as well as our intellect) because they can be filled in by notes of a diatonic series in melodic space; conversely, articulated Segments of a unidirectional diatonic series make sense to our understanding (as well as our perception) because of the harmonic relations obtaining between the boundaries of the segments. This way of interrelating harmonic and melodic space has much in common with central aspects of Schenker’s theories, in particular with Schenker’s understanding of the Zug, and even specifically of the Urlinie. Schenker’s mature theory contains another triumphal synthesis of harmonic and melodic space, understood now in the context of functional tonality.5

Even Zarlino has an embarrassing moment, confusing melodic with harmonic space, when he comes to discuss the minor sixth. He wants to analyze the major and the minor sixths as analogous structures. Specifically, he says that they “are composed … from the fourth plus the major third, or (p.248) the minor third.”6 He might have continued: So, in a major mode such as Ionian, the fourth G3–C4 plus the modal major third C4–E4 yields the major sixth G3–E4, while in a minor mode such as Dorian, the fourth A3–D4 plus the modal minor third D4–F4 yields the minor sixth A3–F4. The modal idea is perfectly clear. Elsewhere, too, Zarlino makes a great point of the modal relation between major and minor thirds in harmonic contexts; indeed he even points to this as a specific resource for harmonic variety, beyond the resources of the senario itself: Some chords have a major third or tenth over the bass, others a minor third or tenth.7 One wishes, then, that he would have produced G3–(C4)–E4 and A3–(D4)–F4, to illustrate a modal analogy between the two sixths as being “composed … from the fourth plus the major third, or the minor third.”

Unfortunately he does not do so. Probably he was not as sensitive as we are to the thirds above the modal tonics C4 and D4 in the harmonic structures above; those thirds are not over the bass notes of the structures. Whatever his motivation, he attempts to realize the analogy of the sixths as a feature of his harmonic space rather than his modal theory, and that leads him into confusion. He has to adjoin the number 8 to the senario in order to get the harmonic ratio 8:5 at hand for the minor sixth, and then he has to argue that the Proportion 8:6:5 is somehow analogous to the proportion 5:4:3 in his harmonic world. He even claims that 6 is a “harmonic mean term” between 8 and 5; this is simply false if “harmonic” is to mean anything at all in the context.8 We may fairly put his argument into modern dress by regarding it as an attempt to draw a direct analogy between the major sixth G3–E4, as divided by C4, and the minor sixth E4–C5, as divided by G5. Of course this does not work. In particular, the conjuction of the minor third E4–G4 below with the fourth G4–C5 above is not at all the same thing as the conjunction of some fourth below with some minor third above, as in the Dorian modal sixth A3–(D4)–F4.

Zarlino could also, of course, have analyzed the sixths as arising by inversion from the thirds. But this approach would have been foreign to his purpose, for then the sixths would not have been primary features of his harmonic space, somehow embedded within the senario. Besides, the sixths that arise from inverting thirds have a very different modal character from the sixths that interest (or should interest) Zarlino. That is exactly the problem (p.249) with his “minor” sixth 8:5 (E4–C5). To the extent that we hear it in a “C-major” modal context as third-degree-to-octave, inverting tonic-to-third-degree = C4–E4 = 5:4, a major third, the sixth itself has a “major” modal character about it, despite its small size. Contrast that with the modal character of A3–F4 in a D-Dorian context, as fifth-degree-(through-tonic-)to-third-degree: This sixth, in its context, has a “minor” modal character as well as a minor absolute size. Similarly, F4–D5 in D-Dorian, inverting D4–F4, has a “minor” modal character despite its large size, while G3–E4 in C-Ionian has both a “major” modal character and a large size.

Zarlino had no Stufen theory that could enable him to make such discriminations. And yet it is quite possible that, even if one had been available to him, he might have rejected it. He would have been uncomfortable making the meaning of his harmonic intervals so dependent on the contextual assignment of a modal tonic. For him this would have weakened the context-free universality of his harmonic theory. Schenker, quite willing to assign structural priority to contextual modal tonics inter alia, uses his Stufen theory to powerful effect in related connections. On the other hand, he finesses certain Problems about the universality of minor harmonic structures which Zarlino attempts to confront, and succeeds in confronting to a remarkable extent.

Hindemith makes an interesting synthesis of melodic and harmonic spaces.9 He tries to show that a chromatic scale from C2 to C3 is filled by those pitches, and only those pitches, which lie in “closest” harmonic relation to C2 within a certain harmonic space. We ignore the overtones of C2; then G2, within the desired scale-segment, is harmonically “close” to C2 because the second partial of G2 is the third partial of C2. F2, within the desired scale-segment, is “close” to C2 since the third partial of F2 is the fourth partial of C2. And so on, casting away harmonic octave-replicates of pitches already generated (which, happily, do not lie within the desired scale-segment). For the most part, this works quite well, though a bit of strain is perceptible in the construction of certain secondary relationships. The essence of Hindemith’s achievement was not just to find pitch classes that can be represented by pitches within a chromatic scale. After all, Zarlino and his forerunners could do that well enough and better. Rather, the achievement was to have shown how pitches within a melodically well-packed region, a chromatic scale from C2 to C3, could be regarded as pitches also within a harmonically well-packed region around the tonic, harmonically well-packed according to Hindemith’s special algorithms for generating harmonic pitch-space. The one pitch with which Hindemith has trouble is A♭2, the minor sixth above the tonic C2. Curiously enough, his troubles resemble Zarlino’s troubies with the minor sixth. Hindemith can generate E2, E♭2, and A2 without using partials of those (p.250) pitches, or of C2, that involve numbers greater than 6. E.g.: The fifth partial of E♭2 is the sixth partial of C2; the third partial of A2 is the fifth partial of C2. But in order to find A♭2 by this method, he would have to have used an eighth-partial relationship: The fifth partial of A♭2 is the eighth partial of C2. This relation would presumably have made A♭2 too “remote” in harmonic space; besides, it might have given rise to awkward questions about seventh-partial relationships. (Zarlino has to deal with the analogs of such questions, when he adjoins the number 8, but not the number 7, to his senario.) Presumably for reasons of these sorts, Hindemith produces not A♭2 but A♭1 by his algorithm; A♭1 is a unique pitch which he generates in this way outside the octave C2–C3. Then, without much explanation, he brings A♭1 up an octave, so that it will lie within his desired scale-segment.10

The foregoing discussion of ways in which some theorists have attempted to integrate harmonic and melodic tonal spaces, or have failed to integrate them, is not meant to be exhaustive or even representative. It is rather intended to show that we do not really have one intuition of something called “musical space.” Instead, we intuit several or many musical spaces at once. GIS structures and transformational systems can help us to explore each one of these intuitions, and to investigate the ways in which they interact, both logically and inside specific musical compositions.


(1) Jean-Philippe Rameau, Traité de l’harmonie réduite à ses principes naturels (Paris: Ballard, 1722). “… la différence du majeur au mineur qui s′y rencontre n′en cause aucune dans le genre de l′intervale qui est toûjours une Tierce de part & d’autre; …” (p. 13).

(2) Ibid. “On divise ordinairement la Musique en Harmonie & en Melodie, quoique celle-cy ne soit qu′une partie de l′autre, & qu′il suffise de connoître l′Harmonie, pour être parfaitement instruit de toutes les proprietez de la Musique, …” (p. 1).

(3) Hermann Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, 2d ed. (Brunswick: Friedrich Vieweg und Sohn, 1865). “…, so folgt hieraus, dass die Quartsextenlage des Duraccords wohllautender ist als die fundamentale, und diese besser als die Sextenlage. Umgekehrt ist die Sextenlage beim Mollaccord besser als die fundamentale, und diese besser als die Quartsextenlage.” (p. 325).

(4) Gioseffo Zarlino, Istitutioni harmoniche, 2d ed. (Venice: Senese, 1573). Facsimile re-publication (Ridgewood, N. J.: Gregg Press, 1966).

(5) Heinrich Schenker, Neue musikalische Theorien und Phantasien, vol. 3, Der freie Satz (Vienna: Universal Edition, 1935).

(6) Zarlino, Ist. harm., book 3, chapter 21. “… sono composte … della Diatesseron et del Ditono, over del Semiditono; …” (p. 193).

(7) Ibid., book 3, chapter 31. “… la varietà dell Harmonia … non consiste solamente nella varietà delle Consonanze, che si trova tra due parti; ma nella varietà anco delle Harmonie, la quale consiste nella positione della chorda, che fà la Terza, over la Decima sopra la parte grave …, overo che sono minori …; overo sono maggiori …” (p. 210).

(8) Ibid., book 1, chapter 16. “… tal proportione tra 8 & 5 termini son capaci di un mezzano termine harmonico, che è il 6; …” (p. 33).

(9) Paul Hindemith, Unterweisung im Tonsatz: Theoretischer Teil (Mainz: B. Schott’s Sönne, 1937), 47–61.

(10) Ibid. “The frequency … of the fourth overtone [of C2] is now divided by 5 … and so generates … A♭1 …, whose second overtone … is inserted in our store of pitches. (Die Schwingungzahl … des vierten Obertones c1 wird nunmehr noch durch 5 … geteilt … und erzeugt so … das 1As …, dessen zweiter Oberton … in unseren Tonvorrat eingereiht wird.)” (p. 54).