Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
1 Chapter 1: Introduction to Pairwise Well-Formed Scales This dissertation arises from my efforts during the past decade to understand how it is that certain collections in the pitch domain have supported vast musical repertories, across cultures and historical eras. In these attempts, I have regarded the underlying pitch collections as the embodiments of certain relationships which composers, or perhaps even entire cultures, may come to exploit. The first stage of this project has been to characterize some of the structural features of these collections. I emphasize that only some of these features have been singled out for investigation: the most important musical entities, such as the diatonic scale and the harmonic triad, seem to be overdetermined, supported by multiple rationales. Some of these rationales are mutually implicated, some are incommensurable with one another, and others are in conflict. The immediate inspiration and technical basis for the research presented here is the collaborative work I have done with my colleague Norman Carey, first and foremost, and some of the ideas that arose during the Working Group in Music Theory held at SUNY Buffalo, July 28-29, 1993, a symposium organized by John Clough to discuss a relation suggested by Richard Cohn, the P-relation.1 In this introductory section, I wish 1The symposium was attended by Norman Carey, David Clampitt, John Clough, Richard Cohn, Jack Douthett, Dan Harrison, Martha Hyde, Carol Krumhansl, David Lewin, and Charles Smith. Cohn’s initial notions were presented in “Generalized Cycles of Fifths, Some Late-Nineteenth Century Applications, and Some Extensions to Microtonal and Beat-Class Spaces,” the keynote address for the Fifth Annual Meeting of Music Theory Midwest/Eighth Biennial Symposium of the Indiana University Graduate Theory Association, Bloomington, Indiana, May 14, 1994. Some of Cohn’s ideas are contained in “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions,” Music Analysis vol. 15, no. 1 (1996): 9-40. David Lewin’s “Cohn Functions,” submitted to Journal of Music Theory, elaborates work Lewin presented at the symposium. 2 to present a thumbnail sketch of the proximate history of the development of my ideas, and to acknowledge my immediate intellectual debts to others. The more general background, in the form of a review of the relevant literature, is a part of Chapter 2. Norman Carey and I took as our point of departure commonalities between diatonic and pentatonic scales. We investigated the mathematical conditions underlying these commonalities, and derived the set of equivalences and implications that constitute the theory of well-formed scales, as we call them. The category of well-formed scales was described by Carey and me in our 1989 paper.2 We generally speak of scales rather than sets because our formulation has the strongest claim to musical validity when musical intuition supports the notion of a generic scale-step interval. A scale is wellformed if it is generated by a single interval that spans the same number of scale-steps whenever the interval occurs. For example, the usual pentatonic is generated by the perfect fifth, and all perfect “fifths” within the pentatonic span three pentatonic scalesteps. The pentatonic scale is considered a non-degenerate well-formed scale, because it does not simply divide the octave into equal parts; that is, there is a remainder interval, also spanning three pentatonic scale-steps, but not of the same size as the perfect fifth. The theory of well-formed scales is a prerequisite for the theory of pairwise well-formed scales, and the necessary foundation will be provided in Chapter 2, along with a review of the pertinent diatonic set theory literature. Where proofs of statements are in published material, the references will be provided. Otherwise, proofs will be offered in this dissertation. The material on well-formed scales, however, unless otherwise stated, is attributable to both Carey and myself. 2Norman Carey and David Clampitt, “Aspects of Well-Formed Scales,” Music Theory Spectrum vol. 11, no. 2 (1989): 187-206. Further development of the theory is to be found in sections V and VI of our paper, “Regions: A Theory of Tonal Spaces in Early Medieval Treatises,” Journal of Music Theory vol. 40, no. 1 (1996): 113-47; and in our paper “Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues,” to appear in Perspectives of New Music. 3 The notion of a pairwise well-formed scale is my own extension of the theory to a class that includes (but is not limited to) some scales that incorporate the step interval of the augmented second. These scales are prominent in world musics, and have been appropriated by composers of Western art music, especially for exotic effects. A pairwise well-formed scale is one with three distinct step-interval sizes which is well-formed when any two step-interval sizes are taken to be equivalent. Examples of such scales include the Japanese hemitonic pentatonics such as hira-joshi, e.g., E F A B C (E); the octatonicminus-one-note or Romanian major, e.g., C D# E F# G A Bb (C); and the so-called Hungarian or Gypsy minor scale, modes of which are also prominent in Indian, Arabic, and Jewish music, e.g., C D Eb F# G Ab B (C). Cohn’s point of departure was to observe a feature shared by diatonic and pentatonic scales and by harmonic triads, considered as pitch-class sets. In each case, it is possible to form a non-trivial cycle of sets all of the same set class such that adjacent sets in the cycle have all but one pitch class in common and the remaining pitch classes differ by interval class 1. The adjacent sets are said to be P-related, and the cycle is called a Pcycle (assuming it is non-trivial, i.e., embraces at least 3 distinct sets).3 The three conditions for the P-relation are thus: (1) preservation of set class; (2) maximal retention of common tones (3) minimal motion by the moving pitch class. In the case of the diatonic and pentatonic sets, the P-cycle simply follows the familiar circle of fifths, while cycles of six harmonic triads move alternately according to the Parallel transform and the Leittonwechsel transform, the two mode-altering quasi-Riemannian transformations that move a single note by a semitone. Both can be thought of as contextually defined inversions, the former inverting a triad by altering its third, the latter inverting a triad by altering its root or fifth, respectively, depending on whether the move is from a major triad to a minor triad or the reverse.4 3The P-cycle is called a “maximally smooth cycle” in Cohn’s 1996 article. addition to Cohn, “Maximally Smooth Cycles,” modern applications of these and other Riemannian transformations are presented in David Lewin, Generalized Musical 4In 4 There are number of differences between the pentatonic/diatonic situations and the case of the harmonic triad, some of them noted by Cohn himself, although he wished to emphasize the unification he had achieved under the aegis of the P-cycle. Cohn called the pentatonic/diatonic type unidirectional, because the sets in the cycle are transposed (or inverted) by means of pitch classes moving always in the same direction as one moves through the cycle. The triad-type he referred to as toggling, because the sets are inverted (not transposed) by moving pitch classes in alternating directions as one proceeds through the cycle. For example, a C-major triad moves to a C-minor triad under the Parallel transform (P), followed by a move to an Ab-major triad under the Leittonwechsel transform (L); thus E descends to Eb and G ascends to Ab. The directions the pitch classes move turns out not to be essential, but the distinction between cycles of transpositions and cycles where adjacent sets are inverted does prove essential. I noted that, although in the modulo 12 system all the variable notes moved by the same interval, in other tuning systems, including open tuning systems such as the Pythagorean or 1 4 -comma meantone temperament, what one may call the voice-leading interval differs according to the case. In the diatonic case, the voice-leading motion is by a chromatic semitone; in the pentatonic, the variable note moves by a diatonic semitone; while the harmonic triad involves two distinct voice-leading motions, differing not only in direction but in size: the diatonic semitone for L and the chromatic semitone for P. If one considers the line of fifths, the diatonic semitone descent is 5 fifths and the chromatic semitone ascent is 7 fifths in the same direction on the line: . . . -3 -2 -1 0 1 2 3 4 5 6 7 8 9 . . . Ab Eb Bb F C G D A E B F# C# G# Intervals and Transformations (New Haven and London: Yale University Press, 1987); Lewin, “Amfortas’s Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic Cb/B,” 19th-Century Music vol. 7, no. 3 (1984): 336-49; Brian Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory vol. 39, no. 1 (1995): 101-38; and Henry Klumpenhouwer, “Some Remarks on the Use of Riemann Transformations,” Music Theory Online vol. 0, no. 9 (1994). 5 F C C A A E A E C# Of course, these motions are always in different directions, and only of the same magnitude when -5 7, that is, in a modulo 12 system. It does not matter whether the unit of measurement is fifths or semitones, in any case this equivalence means equal temperament. P-relations are defined only in chromatic universes (or systems) modulo N. What this suggested to me was that the condition that restricted the voice-leading interval to interval class 1 was unnecessary, that the P-relation might be better thought of as a special case of a more general Q-relation in which the variable pitch class was free to move by any interval class, keeping the other conditions for a P-relation: preservation of set class and maximal retention of common tones. Pending further clarifications below, a Q-cycle is defined analogously to a P-cycle, that is, a cycle embracing at least three distinct sets where adjacent sets in the cycle are Q-related.5 Since Q is the weaker relation, it would be logical to shelter P-cycles under the Q-cycle umbrella, but for a while I will preserve the distinction and reserve the term “Q-cycle” for a cycle where the interval of motion must at some point be greater than 1 semitone. In later chapters the term Q-cycle will be used to include P-cycles. There are aspects of the P-relation that give it a privileged status: on practical musical grounds because of voice-leading norms (Schoenberg’s Law of the Shortest Path) and formally, among other reasons, because the restriction to interval class 1 makes unnecessary a further condition that otherwise must be stipulated. If the moving note can only move by interval class 1, it necessarily must slide into an open position, rather than leap over stationary pitch classes. This turns out to be a necessary formal constraint, so to 5Lewin treats a similar generalization. In “Cohn Functions,” Lewin credits me with identifying the set classes that support what I am calling Q-cycles. Other generalizations have been studied extensively by Jack Douthett, in work that has been disseminated informally. 6 preserve it I insist that the moving pitch class not be permitted to leap over one or more stationary pitch classes. If there is such a leap, I will call it a Q*-relation. Thus, in a proper Q-relation, the move effects an exchange of adjacent intervals in the descriptions of the sets as a sequence of step intervals. For example, in Forte’s 5-7:{01267} set class, {01267} or <11415> and {01567} or <14115> are Q-related, while {01267} or <11415> and {0167e} or <15141> are Q*-related.6 Similarly, the cycle of six 7-33 (whole-toneplus-one) sets below is a Q*-cycle: {02468te} {012468t} {023468t} {024568t} {024678t} {024689t} ({02468te}) In this case the whole-tone motion consistently involves a leap over a stationary pitch class. Both of these examples of Q*-relations have their own interest, and the former will be put in play in an analytical example in Chapter 6. Another formal point that must be taken up in establishing the proper definition of a Q-cycle involves trichords. We will find that some trichordal sets can support P- or Qcycles in several different ways, but that all larger sets that support P- or Q-cycles can do so in only one way. For trichords, I will insist that for each Q-cycle, at most two types of voice-leading moves be permitted. For example, in the case of the harmonic triad, one may form a cycle by alternating the Parallel transform with the Relative transform, but not by mixing Parallel, Relative, and Leittonwechsel in a single cycle.7 The connection to well-formed and pairwise well-formed scales is suggested by the fact that diatonic and pentatonic scales are the examples par excellence of wellformedness, while the harmonic triad is, albeit somewhat trivially, an example of 6Pitch-class sets are labeled according to Forte’s system. See Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973). 7Here and elsewhere, I intend by the terms Parallel and Relative the meanings they have in English parlance, not the reversed meanings they have in German theory. 7 pairwise well-formedness. One is unlikely to consider the notion of pairwise well- formedness on the basis of the triad alone, but when one generalizes to Q-cycles, involving sets like the 5-7 set mentioned above, one finds pairwise well-formed scales among the sets that support Q-cycles. The situation turns out to be more complicated than a simple equivalence, but as this dissertation will show, the (mutually exclusive) properties well-formedness and pairwise well-formedness interact in definite and interesting ways with the ability to support P- and Q-cycles (or infinite P- or Q-chains, in the case of open tuning systems such as just intonation, Pythagorean tuning, etc.). Indeed, all and only non-degenerate well-formed sets are able to support one type of P- or Q-cycle (or infinite chain), while pairwise well-formedness is all but equivalent, in a well-defined way, to support for another type of P- or Q-cycle. Furthermore, as I will show, the pairwise well-formed sets that comprise a P- or Q-cycle can be construed as a Lewinian Generalized Interval System (GIS) in multiple ways.8 The cycle of pairwise well-formed sets can be understood as a commutative GIS with an associated cyclic group, and an interval function that measures intervallic distance in terms of distance along the P- or Q-cycle; or as a (generally) non-commutative GIS, where the associated group that operates on the sets in a simply transitive way may be either a subgroup of the Tn/In group, or a related group that is generated by the contextual inversions that form the cycle. These GIS constructions will be introduced later in Chapter 1, discussed further in Chapter 5, and made use of in analytical situations in Chapter 6. This concludes the evolutionary sketch I wished to provide. The remainder of Chapter 1 will further motivate the notion of pairwise well-formedness through a series of examples, state some of the definitions, theorems, and corollaries, but without proofs or undue formalism, and provide a guide to the rest of the dissertation. Chapter 2 will, as stated above, provide a review of the pertinent literature, especially the literature of 8Lewin, Generalized Musical Intervals and Transformations, 26. 8 diatonic theory, and in particular will develop the required material from the theory of well-formed scales and the necessary formal apparatus towards the treatment of pairwise well-formed scales. Chapter 3 will give that formal treatment, providing proofs of the assertions made in this chapter concerning the structure of pairwise well-formed scales. Chapter 4 develops the transformational theory of well-formed and pairwise well-formed scales, while Chapter 5 is devoted to Generalized Interval Systems arising from pairwise well-formed scales, and in particular elaborates the multiple descriptions available through commutative and non-commutative GISs. Chapter 6 is a series of analytical applications of some of the material developed. *** Since pairwise well-formedness is a generalization of well-formedness, part of the motivation for considering the more general category depends upon an understanding of the musical significance of the notion of a well-formed scale. This significance is suggested by the list of scales or sets generated by the perfect fifth that are well-formed: two notes a fifth apart, three notes that are in the same relation as are tonic, subdominant, and dominant in a major or minor diatonic scale, the usual pentatonic scale, the usual diatonic scale, and the chromatic scale. All equal divisions of the octave trivially satisfy the definition of well-formedness, and are called degenerate well-formed scales, but if the perfect fifth is the interval given by the harmonic series, with frequency ratio 3/2, then the chromatic scale is a non-degenerate well-formed scale, with two different types of step intervals, diatonic and chromatic semitones. The next scale in the hierarchy also has a place in the history of theory, the 17-note division of the octave discussed in Arabic theory by Al-F a r a b i and in Renaissance theories of chromaticism such as that of Prosdocimo de’ Beldomandi.9 Although the perfect fourth/fifth is the generator for these 9Curt Sachs, The Rise of Music in the Ancient World (New York: W. W. Norton & Co., 1943), 177; Jan Herlinger, “Fractional Divisions of the Whole Tone,” Music Theory Spectrum vol. 3 (1981): 74-83; idem, ed. Prosdocimo de’ Beldomandi Brevis summula proportionum . . . (Lincoln, Nebraska, and London: University of Nebraska Press, 1987), 139-47. 9 significant examples, well-formed sets generated by other intervals are possible: in the usual 12-note chromatic, there are 21 non-degenerate well-formed sets in all, 17 of which are generated by interval classes 1 to 4. Microtonal well-formed scales are also a possible compositional resource: examples include the underlying sets in some of the microtonal études by Blackwood, and theoretical constructions by Mandelbaum, Balzano, Gamer, and others. Finally, well-formed scales may be interpreted as cyclic or quasi-cyclic rhythmic patterns, as discussed in Carey’s and my paper “Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues,” as well as in Marc Wooldridge’s dissertation.10 Most work in diatonic set theory takes seriously the distinction between generic and specific measure: generic measure counts the number of scale-steps that the interval spans, while specific measure is the actual size (quality) of the interval. Recall that in the definition of a well-formed scale, a specific interval, the generating interval, always has a constant generic description, spanning the same number of scale-steps whenever it occurs in the set. In the context of well-formed scales, specific measure is expressed as an integer multiple of the generating interval—the fifth, or its replacement, the formal fifth—reduced modulo the octave—or some interval replacing it in its role of determining octave equivalence. For example, in the usual diatonic set, 2 fifths modulo the octave is the major second, and -5 fifths is the minor second. The musical significance of wellformedness lies principally in its systematic distinction between steps and leaps, and generally between intervals of different generic measure: in a well-formed set, not only does the fifth (or formal fifth, the generating interval) always retain its generic identity, by definition, but also, the interval determined by n fifths always retains its generic identity, no matter where in the set it may be located. In particular, this means that if n fifths is a step interval somewhere in the set, it remains unambiguously a step everywhere in the system. In the hexachordal scale that is generated by perfect fifths, C D E F G A (C), the 10Marc Wooldridge, Rhythmic Implications of Diatonic Theory: A Study of Scott Joplin’s Ragtime Piano Works, Ph.D. diss. State University of New York at Buffalo, 1992. 10 minor third (-3 fifths) is sometimes a step (at A to C) and sometimes a skip (at D to F and E to G). Of course, in this set some of the perfect fifths span 4 step intervals, such as C to G, and others, such as G to D, span only three step intervals, so the set is not well-formed. As Carey and I pointed out in “Regions,” this distinction between the well-formed diatonic and pentatonic and the hexachord was drawn by informally by Dahlhaus: “The pentatonic and heptatonic scales are systems. In comparison, the hexachord is a mere auxiliary construction. As a system it would be self-contradictory. . . . would . . . lead to the absurd consequence that the listener would have to alternate between the idea of the minor third as a ‘step’ and as a ‘leap.’”11 This coordination between generic and specific intervals in well-formed scales entails other consequences that signify an efficient information system. A property defined by Clough and Myerson, called cardinality equals variety for lines, holds for all non-degenerate well-formed scales, limiting the number of possible types of melodic fragments of a given generic description to the number of distinct pitch classes that participate in the line. Thus, in particular, all non-zero generic intervals (two-note lines) come in two specific varieties. Clough and Myerson call this special case Myhill’s Property.12 This will be discussed more precisely in Chapter 2, but for now it suffices to observe, impressionistically at least, that melodic constructions in well-formed scales are not too boring, not too exciting, which may be a good thing in systems of communication as well as in modes of political organization. Along related lines is the property of self-similarity, which is one of several ways of characterizing well-formed sets. The diatonic scale is self-similar in the following respect: the distribution of semitones within any diatonic interval is approximately equal to the overall distribution of semitones within the octave, namely 2 in 7. Consider two 11Carl Dalhaus, Studies on the Origin of Harmonic Tonality, Robert Gjerdingen, trans. (Princeton: Princeton University Press, 1990), 72. 12John Clough and Gerald Myerson, “Variety and Multiplicity in Diatonic Systems,” Journal of Music Theory vol. 29, no. 2 (1985): 249-70. 11 diatonic segments, for example C D E F G A and E F G A B C, spanning major and minor sixths, respectively. Of the five steps in the major sixth, one is a half step, while the ratio is two in five in the case of the minor sixth. Now 1/5 < 2/7 < 2/5, and 1/5 and 2/5 are, in fact, the closest approximations to 2/7 with denominator 5. The same holds true for seconds, thirds, fourths, fifths, and sevenths as well. The self-similarity property balances diversity and uniformity, amplifying the discussion of the dialectic of pattern matching and position finding by Richmond Browne. The property is also suggestive of the maximal evenness property defined by Clough and Douthett, to be discussed in Chapter 2.13 It is useful to have a procedure for determining whether a given scale is wellformed or not, given its sequence of step intervals. By virtue of octave equivalence, (or whatever plays the role of octave equivalence in a given context), the structure of a scale can be determined by Richard Crisman’s “successive-interval array” or Robert Morris’s CINT1, what I will informally call its sequence of step intervals.14 Typically, I will consider all modes and inversions of a scale to be equivalent, thus a sequence of step intervals is determined only up to rotation and retrogression. For example, a diatonic scale is characterized by a sequence of step intervals in which semitones are separated alternately by two and three whole tones; thus, any rotation of <baabaaa> represents a diatonic scale, where b represents a semitone interval, a a whole tone interval. This example points up another level of abstraction that will frequently be invoked: it is often 13Richmond Browne, “Tonal Implications of the Diatonic Set,” In Theory Only vol. 5, nos. 6-7 (1981): 3-21. John Clough and Jack Douthett, “Maximally Even Sets,” Journal of Music Theory vol. 35, nos. 1-2 (1991): 93-173. These articles will be summarized in the review of the literature in Chapter 2. 14Richard Chrisman, “Identification and Correlation of Pitch-Sets,” Journal of Music Theory vol. 15 (1971): 58-83; idem, “Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays,” Journal of Music Theory vol. 21, no. 1 (1977): 1-28. Robert Morris, Composition with Pitch-Classes (New Haven and London: Yale University Press, 1987), 107. See also, Eric Regener, “On Allen Forte’s Theory of Chords,” Perspectives of New Music vol. 13, no. 1 (1974): 191-214. 12 of little structural moment to specify what the tokens, here a and b, represent. Many of the abstract or combinatorial features of the set depend only upon the pattern. Among the prerequisite results developed in Chapter 2 is one that allows the determination of whether a scale or set is well-formed or not by examination of its sequence of step intervals: in a well-formed scale of cardinality N, the number of step intervals spanned by the generating interval is the multiplicative inverse of the multiplicity one of the two step intervals. (That there are necessarily exactly two distinct step interval types is a special case of Myhill’s Property.) For example, consider the diatonic scale pattern, <baabaaa>. Count one of the step intervals; this is the multiplicity g of that step interval. In this case, for example, there are five as. If N is the cardinality of the scale, find the multiplicative inverse mod N of the step-interval multiplicity, that is, solve the congruence gx 1mod N. In this instance, that means solve the congruence 5x 1mod 7; thus, x=3. If it is impossible to find such an inverse, because the multiplicity is not relatively prime to N, then the scale is not well-formed. If it is possible to solve the congruence, then check the intervals that span x steps. If N-1 of them are all of the same size (i.e., enclose the same number of as and bs), then the scale is generated by an interval of constant span, hence is well-formed; if not, the scale is not well-formed. In the case of the diatonic pattern, the first six fourths all enclose two as and one b, that is, they are “perfect fourths.” Therefore, the scale is generated by a single interval that spans the same number of scale-steps whenever the interval occurs, thus, by definition, the scale is well-formed. Of course, we could have started by setting g=2, the number of half steps. Then we would have determined that the diatonic set is also generated by the perfect fifth. In Chapter 2 I will present an algorithm for determining all possible well-formed scales for a given generating interval, from a theorem attributable to Carey and Clampitt and stated in our 1989 article. As I will prove in Chapter 3, however, pairwise wellformed scales are (with a well-defined exception) not generated sets. The procedure 13 outlined above will nevertheless permit us to determine whether or not any given sequence of step intervals is a pairwise well-formed scale. The formal definition of pairwise well-formedness is as follows: Definition 1.1: A scale is pairwise well-formed if, when any pair of step intervals is equivalenced, the resulting pattern is that of a non-degenerate well-formed scale. From the definition it is immediate that there are at least three step-interval sizes: any scale with two step interval sizes is reduced to a degenerate well-formed scale when its step intervals are equivalenced. It turns out that in a pairwise well-formed scale, there are also at most three step-interval sizes; again, this is a formal result in Chapter 3. Thus, if a , b , c are the step sizes, considering a b with c , a c with b , and b c with a yields a well-formed set in each case. For example, given a scale with a pattern of three step intervals, say, <abcabac>, one can determine whether it is pairwise well-formed by equivalencing each pair of step intervals and checking whether the resulting scale is well-formed, using the procedure described above, if necessary. In this case, equivalencing a and b, the resulting pattern is <aacaaac>, a diatonic pattern, which is well-formed. Equivalencing a and c, the resulting pattern is <abaabaa>; again the pattern is a diatonic pattern and well-formed. Finally, equivalencing b and c, the resulting pattern is <abbabab>, which is again wellformed: there are 4 b steps, 2 is the multiplicative inverse of 4 mod 7, and the interval that spans 2 steps and encloses 1 a step and 1 b step can be seen to be the generating interval for the scale, so the scale is pairwise well-formed.15 15The generator by definition has multiplicity N-1. In this case the six ab (or ba) intervals from <abbabab> can be found comprising order positions 1 and 2, 3 and 4, 4 and 5, 5 and 6, 6 and 7, and 7 and 1. 14 Pairwise well-formedness is a second-order construction, undeniably rather abstract. I am not saying that one experiences, perceptually, any of these equivalences, only that if one performs these mental operations, one obtains the formal result. As in the case of simple well-formedness, what is of interest is the logical structure of the set of properties possessed by this class of scales. In particular, the transformational properties of almost all of these scales are of more immediate musical salience. One of the structural properties of pairwise well-formed scales is trivalence: not only do steps come in three sizes, but all generic intervals—thirds, fourths, and so on— are of three types. This is reminiscent of Myhill’s Property defined by Clough and Myerson, wherein all generic intervals come in two sizes. Since, as we will see in Chapter 2, Myhill’s Property is actually equivalent to non-degenerate well-formedness, one might conjecture that trivalence is equivalent to pairwise well-formedness. In the present context, however, trivalence is by no means so strong a property as Myhill’s Property: there are many pitch-class sets with trivalence that are not pairwise wellformed, so the implication goes just one way. An example of a trivalent set that is not pairwise well-formed is the pentachord 5-22:{01478}, Anatol Vieru’s Bacovia mode.16 If, however, N is the cardinality of a pairwise well-formed scale, and d and N are relatively prime, we will see that the interval cycles of generic intervals spanning d steps (cycles of thirds, fourths, etc.) themselves form pairwise well-formed patterns. In this respect, the property of well-formed scales carries over to all pairwise well-formed scales, mutatis mutandi. The transformational properties of pairwise well-formed scales involve the P- and Q-relations discussed in the introduction, that have been studied by Richard Cohn and David Lewin in various musical contexts. My treatment relates the categories well- 16Anatol Vieru, The Book of Modes, Yvonne Petrescu, trans. (Bucharest: Editura Muzicala, 1993). 15 formed and pairwise well-formed to P- and Q-cycles and -chains. The relevant definitions follow. Def. 1.2. P-relation (Cohn): Two pitch-class sets are in the P-relation if there exists a Tn or In mapping one set onto the other that leaves all but one pitch class of the set invariant and moves the remaining pitch class by interval class 1. Only defined in chromatic universes mod N. Def. 1.3. P-cycle (Cohn): A cycle of length greater than 2 in which adjacent sets are P-related. Def. 1.4. Q-relation: Two pitch-class sets are in a Q-relation if there exists a Tn or In mapping one set onto the other that leaves all but one pitch class of the set invariant and moves the remaining pitch class by any interval class, where the moving pitch class slides between frozen pitch classes, rather than leaping over them. For example, {01267} and {01567} are Qrelated (2 slides to 5), whereas {01267} and {01267e} are not (2 leaps over stationary pitch classes to 11). Def. 1.5. Q-cycle: A cycle of length greater than 2 in which adjacent sets are Q-related. For trichords, in addition, it is required that only one or two of the three possible interval exchanges take place in the course of the cycle. That is, if the trichord has sequence of step intervals <abc>, and the cycle proceeds by moving one pitch class to effect the exchange of a and b, and then by moving another to effect the exchange of b and c, it may not also proceed by exchanging a and c. 16 Among the interesting aspects of well-formedness and pairwise well-formedness are the ways these properties interact with support for P- and Q-cycles (or infinite chains of P- or Q-related sets). Table 1.1 lists the set classes in the usual 12-note universe that are well-formed (WF) or pairwise well-formed (PWWF) and/or support P- or Q-cycles. 17 Table 1.1 The one- and two-note set classes are trivial examples. All are trivially wellformed (WF) and support Q-cycles; the singleton class trivially supports P- or Q-cycles. 3-1, 3-6, 3-9, 3-10 are WF and support Q-cycles, and 3-2, 3-3, 3-4, 3-5, 3-7, and 3-8 are trivially pairwise well-formed (PWWF) and support Q-cycles. 3-11 is PWWF and supports a P- cycle and Q-cycles. (3-12 is degenerate well-formed.) Label Prime Form 4-1 4-21 0,1,2,3 0,2,4,6 1119 2226 WF, Q-cyclic WF, Q-cyclic 5-1 5-3 5-6 5-7 5-10 5-20 5-21 5-23 5-32 5-33 5-35 0,1,2,3,4 0,1,2,4,5 0,1,2,5,6 0,1,2,6,7 0,1,3,4,6 0,1,3,7,8 0,1,4,5,8 0,2,3,5,7 0,1,4,6,9 0,2,4,6,8 0,2,4,7,9 11118 11217 11316 11415 12126 12414 13134 21225 13233 22224 22323 WF, Q-cyclic PWWF, Q-cyclic PWWF, Q-cycic PWWF, Q-cyclic Webern op. 5/4 PWWF, Q-cyclic PWWF, Q-cyclic Japanese In-scale PWWF, Q-cyclic PWWF, Q-cyclic PWWF, Q-cyclic WF, Q-cyclic whole-tone minus 1 WF, P-cyclic usual pentatonic 6-1 6-Z44 0,1,2,3,4,5 0,1,2,5,6,9 111117 113133 WF, Q-cyclic Q-cyclic Schoenberg signature 7-1 7-5 7-7 7-22 7-31 7-35 7-Z37 0,1,2,3,4,5,6 0,1,2,3,5,6,7 0,1,2,3,6,7,8 0,1,2,5,6,8,9 0,1,3,4,6,7,9 0,1,3,5,6,8,t 0,1,3,4,5,7,8 1111116 1112115 1113114 1131213 1212123 1221222 1211214 WF, Q-cyclic PWWF, Q-cyclic PWWF, Q-cyclic Webern op.5/4 PWWF Hungarian (gypsy) minor PWWF, Q-cyclic folk scale WF, P-cyclic usual diatonic PWWF 8-1 0,1,2,3,4,5,6,7 11111115 WF, Q-cyclic 9-1 9-5 9-11 0,1,2,3,4,5,6,7,8 0,1,2,3,4,6,7,8,9 0,1,2,3,5,6,7,9,t 111111114 111121113 111211212 WF, Q-cyclic PWWF, Q-cyclic P-cyclic harmonic triad complement 1111111113 WF, Q-cyclic 10-1 0,1,2,3,4,5,6,7,8,9 Interval Form Properties 11-1 0,1,2,3,4,5,6,7,8,9,t 11111111112 WF, P-cyclic 18 Non-degenerate WF, PWWF, and P- and Q-cyclic set classes Of course, all these objects are defined for chromatic universes of any cardinality. Moreover, it is not necessary to conceive of scales as embedded in chromatic universes. Thus, the diatonic scale in Pythagorean tuning is well-formed under the definition, but instead of a P-cycle it supports an infinite Q-chain, where the variable pitch class moves consistently by a Pythagorean chromatic semitone (C major to G major is pitch class F moving to F#). The diatonic scale in just intonation is pairwise well-formed and supports an infinite Q-chain. Here the intervals of motion are alternately syntonic commas and larger limmas. The just major scale has step intervals (in frequency ratios) as follows: do re 9 8 mi 10 9 fa 16 15 sol 9 8 la 10 9 ti 9 8 (do) 16 15 If re is lowered by a syntonic comma (81/80), this produces an inverted form of 9 80 10 the scale: <10/9, 9/8, 16/15, 9/8, 10/9, 9/8, 16/15>. (Because . Since the 8 81 9 intervals are expressed here in frequency ratios, “subtracting a syntonic comma” is expressed as division by 81/80, or multiplication by its inverse.) If we follow this operation by lowering ti by a larger limma (multiplication by 128/135), the result is an inversion again. The composition of the two operations is a transposition by a just (Pythagorean) perfect fourth. Alternating these two operations forms an infinite Q-chain. The just scale is pairwise well-formed, because equivalencing the 9/8 and 10/9 whole steps, the resulting scale is of the same formal type as the usual diatonic, with 5 whole steps and 2 half steps; similarly if 9/8 and 16/15 are equivalenced, and if 10/9 and 16/15 are equivalenced, again a well-formed scale results, but in this case the generating interval is a third (or a sixth). Note that we need not have checked for pairwise well- 19 formedness in this particular case, because the just scale is a realization of the abstract step-interval sequence studied above, <abcabac>. Because the just scale is pairwise well-formed, and because Easley Blackwood in his microtonal études chooses subsets of the equal-tempered divisions of the octave of 13 to 24 notes that best approximate the (just) diatonic scale, some of his études are based upon sets that are pairwise well-formed. One of the Indian forms of the diatonic, the sa-grama, is also pairwise wellformed. Presenting it as a partition of 22 srutis (which may be an artificial imposition of the modulo 22 system)17 the sa-grama is: 3 sa 2 ri 4 ga 3 ma 4 pa 2 dha ni 4 (sa) As in the case of the just scale, one can invert sa-grama either by lowering sa by one sruti, or by lowering ga by two srutis. By alternating these two moves, a Q-cycle is formed. Again, the scale belongs to the class represented abstractly by the step-interval sequence <abcabac>. Other pairwise well-formed scales are prominent in world musics. One is a Japanese hemitonic pentatonic, the In-scale, which may be notated as E F A B C (E). Like the usual pentatonic, it can be considered a connected segment of the diatonic cycle of fifths, but unlike the usual pentatonic, this subset embraces the diminished fifth as well: A E B F C. Its cyclic step-interval sequence modulo 12 is <1 4 2 1 4>. It is possible to modulate to an inverted form of the scale either by moving one note down a (chromatic) semitone, or by moving one note up a tone, and if one assumes 12-note equal temperament, through all 24 members of the set class 5-20: E F A B C E F A Bb C E F A Bb D . . . B C E F# G B C E F G (B C E F A). Here the 17See the discussion of this representation of the srutis in John Clough, Jack Douthett, N. Ramanathan, Lewis Rowell, “Early Indian Heptatonic Scales and Recent Diatonic Theory,” Music Theory Spectrum vol. 15, no. 1 (1993): 36-58. 20 product of two succesive inversions is a T5-transpose of the original set. As an exercise, the reader may readily verify that this Japanese pentatonic scale satisfies the definition as a pairwise well-formed scale. The set classes 5-7:{01267} and 7-7:{0123678} are both pairwise well-formed. The step-interval sequences for these sets are <11415> and <1113114>. Both support Qcycles, where the product of two successive inversions is T1/T11. The Q-relation cycles for 5-5 and 7-7, as well as for 5-20, exhaust their respective set classes (since 1, 11, 5 and 7 are units mod 12, i.e. relatively prime to 12). The cycle through 7-7 is shown below: Q Q Q Q {0123678} {0125678} {012567e} {014567e} . . . . . . Q {1236789} ({0123678}) As table 1.1 shows, 5-7 and 7-7 are the only pentachord/heptachord complements in the usual 12-note chromatic universe that are pairwise well-formed. Q-relation chains of these sets figure in an analysis of Webern op. 5, no. 4, in Chapter 6. The octatonic-minus-one set 7-31: {0, 1, 3, 4, 6, 7, 9} is also pairwise wellformed, and supports a Q-cycle of length 8. In this cycle, the voice-leading moves are alternately by whole steps and semitones, that is, the sets are alternately Q- and P-related. The three distinct Q-cycles partition the 7-31 set class: each of the cycles embraces the 8 seven-note subsets of each of the three octatonic pitch-class sets, as shown below. {0134679} {t134679} {t034679} {t014679} Q P Q P {t013679} {t013479} {t013469} {t013467} Q P {0134679} Q P 21 Although the set figures in much of Stravinsky’s octatonic music, it is especially prominent in Les Noces. The set also appears in some Eastern European folk music; Vieru calls one mode of it Romanian major. As table 1.1 suggests, the properties well-formedness and pairwise wellformedness and P- or Q-cyclicity are closely related, but in complicated ways. In the type of P- or Q-cycle that Cohn calls unidirectional, closely related to what Lewin calls a twoplace generated Cohn function, adjacent sets in the cycle are transpositionally related (e.g., the diatonic set). One of the formal results proven in this dissertation is that a set is non-degenerate well-formed if and only if it supports a unidirectional P- or Q-cycle or infinite chain. This is another characterization of well-formedness. There is a type of P- or Q-cycle where the participating set is neither well-formed nor pairwise well-formed. The only examples in the 12-note universe are the Schoenberg signature hexachord, 6Z-44, and the complement of the harmonic triad, 9-11. This supports a P-cycle, but is neither well-formed nor pairwise well-formed. Lewin calls this type a two-place antithesis-pair Cohn function. This class of Q-cyclic sets will not be analyzed in this dissertation. There is a another type of P- or Q-cycle, affiliated with what Lewin calls a generated 3-place Cohn function, that is all but equivalent to pairwise well-formedness.18 That is, all of the sets that support this type of P- or Q-cycle are pairwise well-formed, but there is one class of sets—it is not too much to say that there is essentially only one set— that is pairwise well-formed but does not support a P- or Q-cycle. This set has cardinality 7, and has the interval pattern <abacaba>. As unlikely as it might seem, this is the only 18Since Lewin’s study has not appeared in print as of this writing, I will mention that well-formedness and pairwise well-formedness are not categories that he addresses. Thus, we often consider similar objects of thought, but approach them from different points of view. Cohn cycles and Q-cycles are closely linked, but are distinct in that Cohn cycles take place at the level of the step-interval sequence, involving rotations or retrogressions of the sequence, whereas Q-cycles involve the actual transpositions or inversions of the set. There will be more about this in Chapter 4. 22 set type, of whatever cardinality, that is pairwise well-formed but is frozen in terms of Cohn-type voice-leading moves. Alone among pairwise well-formed scales, scales of this type do not have distinct inversions (that is, are symmetric under inversion). To check the pairwise well-formedness of such scales, note that equivalencing a and b results in the pattern <aaaaaac>, which is obviously well-formed, following the pattern of the diatonic cycle of fifths; equivalencing a and c results in the pattern <abaaaba>, the pattern of the diatonic scale itself; and equivalencing b and c results in the pattern <abababa>, the pattern we have shown before to be well-formed, and which follows the pattern of the diatonic cycle of thirds. These are all of the essentially different well-formed patterns for scales of cardinality 7. As table 1.1 shows, there are two set-class representatives of this type in the universe of twelve pitch classes: 7-22 and 7Z-37. Of these, 7-22 is the more interesting, since the step intervals differ by one or two semitones, thus better satisfying our intuitive notion of what a “scale” should be. It may be construed as a harmonic minor scale with raised fourth degree, the so-called Hungarian or Gypsy minor.19 The first eight measures at the solo entrance in the first movement of the Brahms Violin Concerto, for example, use all and only notes of this scale. (Interestingly, it is usually the last movement of the Concerto that is cited in discussions of the Hungarian flavor of the piece and the reference to Joachim’s origins.) Similarly, the first two measures of Mozart’s celebrated C-minor Fantasy, K. 475, employ all and only notes of this scale. The same collection and mode, C D Eb F# G Ab B (C), is the South Indian me la Si m hendramadhyama , the scale for Ra ga Si m hendramadhyama . Another mode, a North Indian that, is called bhairav: C Db E F G Ab B (C). It is also the Afghan mode Beiru, a word related to the Hindustani bhairav. Bartók’s “Arabian Song,” no. 42 of his 44 Duos for Two Violins, is in this mode, withholding the modal center until almost nine measures have been played. Undoubtedly the symmetric disposition of this mode about its modal center appealed to 19According to Bartók, this scale is unknown in traditional Hungarian music. 23 Bartók. It is one of the Arabic maqams, though an unusual one, but perhaps was once more common, because it is also a Spanish type, most likely due to Moorish influence. Examples in Spanish-flavored European art music are found in Ravel’s Rapsodie Espagnole and Debussy’s La Soirée dans Grenade. Hungarian minor is also one variant of a Jewish mode called Avannah Roboah, and arises in much twentieth-century and some nineteenth-century repertory by composers affecting an Hebraic style. The scale is prominent in the music of Ernest Bloch, and has been used evocatively by Shulamit Ran. A curious harmonic occurence of the pitch-class set 7-22 is in Le Sacre du printemps: the string harmonics chord at the end of the 4-measure Kiss of the Earth section, 1 before R72. The chord fascinated Ernest Ansermet, for whom it represented “sans doute le maximum de tension harmonique que puisse se signifier la conscience musicale.” It bothered Allen Forte, because it was a prominent harmonic entity that did not fit any of his set-complexes for the work. Richard Taruskin, in his review of Forte’s book (p. 124), pointed out that the chord could be realized as a stack of major and minor thirds: Ab C Eb G B D F# (Ab). That even F#-Ab looks like a third is an artifact of our notational system, but in fact these are all generic thirds in this scale.20 Sets of this type, what I will call the class of singular pairwise well-formed scales, have the property that all of their generic interval cycles are formally identical. For example, 7-22 has the step-interval sequence <1131213>, while its cycle of thirds is <2434434> and its cycle of fourths is <5456556>, all equivalent up to rotation to the abstract cyclic form <aabacab>. Note that in each case, the multiplicities of the three specific varieties of intervals of a given generic length are 1, 2, and 4. In few of the examples of the use of this scale does the knowledge that it is a pairwise well-formed scale—and one with special properties—lead to analytical insight, 20Ernest Ansermet, Les Fondements de la Musique dans la Conscience Humaine (Neuchâtel: Editions de la Baconnière, 1961), 2:187; Allen Forte, The Harmonic Organization of the Rite of Spring (New Haven: Yale University Press, 1978); Richard Taruskin, Review of The Harmonic Organization of the Rite of Spring by Allen Forte, Current Musicology vol. 28 (1979): 114-29. 24 apparently. Similarly—even more strongly—the knowledge that a diatonic scale provides the structural tones for a particular piece of music, and that the diatonic scale is wellformed, is almost never of analytical moment. The property is so general, while the particularity of a piece is what an analysis usually seeks to address. In the case of the well-formedness of the diatonic scale, however, one can point to the salience of the property for the potency of the set: tonal music, and diatonic music generally, privileges step-wise motion, and the well-formedness of the diatonic set permits steps and leaps to be unambiguously identified, and allows intervals of every generic description to be systematically and coherently organized in relation to the powerful interval of the perfect fifth. A similar justification for the relative ubiquity of the Hungarian minor is more elusive; certainly part of its appeal lies in the way it can be used to support traditional functional harmony. Nonetheless, its fascinating formal properties also suggest an attractive musical potential. All other pairwise well-formed scales are set apart from scales of the singular type by the potential analytical tool and compositional resource that it uniquely lacks: the ability to support P- or Q-cycles. Any sets that participate in a P- or Q-cycle can be organized into a tightly organized system, what Lewin has defined as a Generalized Interval System (GIS).21 Moreover, the sets embraced by a given P- or Q-cycle may be viewed as the elements of a GIS in at least two, if not three, different but natural and reasonable ways. This yields a double or multiple description, in the sense of Gregory Bateson, who indicates its value in binocular vision and other domains.22 The possibility of invoking a particular GIS in the environment of a piece is first of all a warrant that one can say or hear something coherent in those intervallic or transformational terms. This presupposes that the GIS is well grounded in musical reality. In the analyses Chapter 6, 21Lewin, Generalized Musical Intervals and Transformations, passim. In the interest of clarifying what I am borrowing from Lewin: the notion of a GIS is due to Lewin, but in his work on Cohn functions Lewin does not invoke the GIS concept. 22Gregory Bateson, Mind and Nature (New York: Dutton, 1979). 25 the transformational networks acquire a firm basis in the underlying multiple descriptions that these GIS structures afford. The formal theory of Generalized Interval Systems in the context of pairwise well-formed scales will be the subject addressed in Chapter 5, but in the interest of motivating the study of pairwise well-formed scales I will foreshadow that discussion by introducing the basic material and some examples at this point.23 Generalized Interval Systems and Q-cycles A Lewinian Generalized Interval System is defined as a set or space of musical objects, S, together with a group, IVLS, —(the group of intervals, with the binary operation )—and a function int, that maps the set of all ordered pairs (s,t), where s and t are elements of S, into the group IVLS in such a way that (1) for all r, s, and t in S, int(r,s) int(s,t) = int(r,t); (2) for every s in S and every i in IVLS, there is a unique t in S which is the interval i from s, that is, a unique t which satisfies the equation int(s,t)=i.24 The conditions on the int function ensure that its outputs, the elements of IVLS that span between elements of S, act in ways our intuition leads us to expect musical intervals to act. Because of the uniqueness in condition 2, in any GIS we can define a transposition by the interval i as a mapping Ti: S S: s t where t is the unique element that lies the interval i from s. Ti is thus a transformation of S, and Lewin shows it to be a permutation of S, (or as he calls it, an operation on S), that is, a one-to-one function of S onto itself: if s and s' are distinct elements of s, then Ti(s) Ti(s'), and if t is any element of S, there is a (unique) element s such that Ti(s) = t. Later in his book, Lewin takes a transformational point of view, reinterpreting intervals as transposition operations on a set. It turns out that Generalized Interval 23The mathematical machinery employed below is discussed in more detail in the first part of Chapter 5. 24With very slight (non-substantive) modifications, this is the definition Lewin gives in Generalized Musical Intervals and Transformations, 26. 26 Systems are equivalent to simply transitive group actions on a set. A group G of permutations of S (operations on S) acts on a set S in a simply transitive way when, for any pair of elements s and t in S, there is a unique element g of G such g(s)=t. The elements of the group G are exactly the transpositions; the intervals from s to t and s' to t' are the same if and only if g(s)=t and g(s')=t'. 27 Example 1. 1 Schubert, Sonatina op. 137, no. 3 D. 408 (1816), Andante, mm. 32-39 To introduce the notion of multiple description GIS structures in pairwise wellformed scales, I will consider the hexatonic systems studied by Cohn. In “Maximally Smooth Cycles,” Cohn introduces the P-cycles that are supported in the usual 12-note chromatic, one for each odd cardinality less than 12. In particular, he considers the Pcycles of harmonic triads. (Keep in mind that harmonic triads are pairwise well-formed, though trivially so: any three-element set with three distinct step intervals is pairwise well-formed. Pairwise well-formed trichords are special, as we will see, because they can support three distinct P- or Q-cycles.) The 24 major and minor triads are partitioned into four groups of six by the P-cycles, which Cohn calls Hexatonic Systems. Consider the Schubert Sonatina excerpt in example 1.1. In the second half of this slow movement, starting from m. 32, the music moves harmonically through a chain of five P-relations: B major to B minor, to G major for three bars; G minor in m. 37, to Eb major. Since Eb major is the eventual harmonic goal, one can reasonably conclude the chain here, although in the next eight measures, leading into the recapitulation, Schubert adverts to Eb minor, forming a complete P-cycle through a hexatonic system of triads. We will see that the six triads can be considered to be the elements of two mutually reinforcing Generalized Interval Systems. One can consider the cycle to be primary, and measure intervals between triads in terms of the number of pitch classes displaced or turned over. Thus, the interval between B major and G major is the same as the interval 28 between B minor and G minor. This does not seem at all strange to us, because in both cases the triads are related by the same transposition operator, T8. But in this GIS, the interval from B major to G minor is the same as the interval from B minor to Eb major. In both cases, the triads have no common tones; they are thus as far apart as they can be in this hexatonic system. In this interpretation, which Cohn uses in his analyses, the cycle is the foreground figure, against a background that takes for granted that the sets are members of the same class, here the harmonic triad. If we reverse this perspective, and take the existence of the P-cycle as background, another GIS is brought to the foreground. Here the six triads are again the elements of the GIS, and the intervals are the unique transposition and inversion operations that take one triad to another. Here, for example, the interval between the B major and B minor triads is given by I5, which is also the interval between G minor and Eb major, and between G major and Eb minor. Here, what the ear attends to are the triads between which D and Eb (or D#) are exchanged. Similarly, I9 is a measure of transformation for triads in which pitch classes B and Bb are exchanged. Finally, one can consider how the triadic configuration is altered by the quasi-Riemannian transformations, Parallel and Leittonwechsel, or P and L, and combinations of these transformations. Taking the dualist description of triads, wherein the Riemannian root of a minor triad is what otherwise is its fifth, P transforms triads by sending the third of the triad a semitone away from its fifth, towards its (Riemannian) root. In other words, fifth and root exchange roles under P. Thus, P sends B major to B minor by sending pitch class D# to pitch class D, and B minor to B major by sending pitch class D away from its fifth B, to pitch class D#, towards its root F#, which becomes the fifth of B major. L transforms triads by sending the root of the triad a semitone away from the other two constituents, third and fifth. When L and P are combined to form LP, performing P first, B major is sent to G major, while B minor is sent to Eb minor.25 25Henry Klumpenhouwer, in “Some Remarks on the Use of Riemann Transformations,” discusses in depth the relationship between transformations such as P and L and a dualist 29 Under this interpretation, attention is turned to which constituents of the triadic configuration are altered and in what way. The way these GIS structures are related will now be considered in more detail and from a somewhat more abstract point of view. The P-cycle through one of the hexatonic systems, Hex1, is shown below; each element of the cycle is a major or minor triad, designated by a traditional root with plus or minus sign indicating quality: A- A+ C#- C#+ F- F+ (A-) The rationale for the name “hexatonic” is that the six triads of each system embrace the six pitch classes of a set of set class 6-20: {0, 1, 4, 5, 8, 9}; in the case of the cycle above, the six pitch classes are those of the prime form of the set. The justification of the word “system” is that the six triads actually form a Generalized Interval System. The GIS that Cohn chooses as the setting for his analyses of some nineteenth-century music takes the triads of a hexatonic system as the elements of the set S, and takes the cyclic group induced by the P-cycle as the group that acts in a simply transitive way on the set of triads. The measure of intervallic distance is simply the number of pitch classes displaced (or common tones retained), that is, distance along the P-cycle. Thus the six transpositions in this group are: (1) the identity operation, R0, which maps every triad to itself; (2) one move forward along the P-cycle, the transposition R1, which maps A- to A+, A+ to C#-, and so on; (3) two moves along the P-cycle, R2, which takes A- to C#-, A+ to C#+, and so on; R2 has the same effect on the triads as the ordinary transposition T4; (4) three moves along the P-cycle, R3; (5) four moves along the P-cycle, R4, which acts on the triads in the same way that the usual T4 does, and five moves along P-cycle, R5. Because the transpositions combine as elements of the cyclic group mod 6, (to wit: RmRn = RnRm = R(m+n)mod6), it is possible to write R4 as R(-2) and R5 as R(-1), so that the if we class together an index and its complement, the smaller of the values (if there conception of triads. 30 are two), in other words, the mod 6 interval class, reflects the number of tones displaced. For example, the triads transposed by R3 = R(-3) have three tones displaced (zero common tones). This GIS arises in a natural and reasonable way, and has a simple structure; the GIS is commutative, because the binary operation in the associated group is commutative. The only counterintuitive aspect of this group is that the transpositions with odd indices reflect a non-standard notion of interval: the “interval” from the A minor triad to the C# major triad is the same as the “interval” from the A major triad to the F minor triad, for example. The reader can check that the basic notion of an interval is satisfied, however, because the group is simply transitive on the set of triads. The transformational graphs in Cohn’s analyses are based on this GIS.26 There is, however, a pair of complementary Generalized Interval Systems, separate from the foregoing GIS, that this and every other pairwise well-formed P- or Qcyclic set gives rise to in a very natural and reasonable way. In this instance, the two GISs again have as their space S the set of six triads of Hex1. On the one hand, the same subgroup of atonal operations that maps the underlying hexatonic set of pitch classes onto itself (in a simply transitive way) and thus makes the hexatonic set into the space of a GIS, also acts as the group for the six associated triads in a GIS. This is a slight abus de langage, because it presumes to identify the group that acts on the pitch classes, a subgroup of the Tn/In group, with the group that acts on the triads, a subgroup of what I will designate as the Tn/In group (see footnote 26). Of course, in this case the group acting on the triads can be viewed as transforming the triads by operating on their pitchclass constituents. One can see that this identification is a very natural one if one associates pitch classes and triads in the following way (identifying a triad with its third): 26Cohn symbolizes the transpositions of this GIS by Ti; I prefer to reserve this notation for the usual transpositions. I designate groups that are presumed to act on triads with a different font, thus: Ti. The usual font designates groups that act on pitch classes. In later chapters, this somewhat fussy distinction is abandoned. 31 0 1 4 5 8 9 A- A+ C# - C# + F- F+ I will use the permutation notation that decomposes a permutation into disjoint cycles.27 For example, T4 acts on the hexatonic pitch classes by sending 0 to 4, 4 to 8, and 8 to 0 (one cycle, represented by (048)), and by sending 1 to 5, 5 to 9, and 9 to 1 (another cycle, represented by (159)). Thus the operation may be represented as the product of these two cycles: (048)(159). The second GIS on the triads (and the associated one on the hexatonic set) arises from the operations shown below: T0: ( ) T0: ( ) T4: (048)(159) T4: (A- C#- F-)(A+ C#+ F+) T8: (084)(195) T8: (A- F- C#-)(A+ F+ C#+) I1: (01)(49)(58) I1: (A- A+)(C#- F+)(C#+ F-) I5: (05)(14)(89) I5: (A- C#+)(A+ C#-)(F- F+) I9: (09)(18)(45) I9: (A- F+)(A+ F-)(C#- C#+) The group table (for “both” groups) is as follows: ° T0 T4 T8 I1 I5 I9 T0 T0 T4 T8 I1 I5 I9 T4 T4 T8 T0 I9 I1 I5 T8 T8 T0 T4 I5 I9 I1 I1 I1 I5 I9 T0 T4 T8 I5 I5 I9 I1 T8 T0 T4 I9 I9 I1 I5 T4 T8 T0 27By a theorem of elementary group theory, every permutation of N objects can be uniquely expressed as the composition of disjoint cyclic permutations. The empty cycle, denoted by ( ), leaves all objects fixed. See I. N. Herstein, Topics in Algebra (Waltham (Massachusetts), Toronto, London: Ginn & Co., 1964), 66-67. 32 Again, the notion of interval that this GIS presumes may seem counterintuitive, e.g., the interval between C# minor and F major is the same as the interval between A minor and A major, since the transposition between them (in either direction) is I1, in both cases. What the ear attends to in this GIS with triads that lie the interval I1 apart is the exchange between pitch classes C and C#: this is what distinguishes A minor from A major and C# major and F minor, and is one of the pitch classes that move between C# minor and F major. Another aspect of this GIS that perhaps makes it less intuitive than the GIS associated with the cyclic group is that this group is nonabelian, i.e., the group operation is non-commutative, as the group table shows. As we will see, the existence of yet another hexatonic GIS is indissolubly linked to the nonabelian character of the Tn/In subgroup associated with this GIS. Recall that the hexatonic cycle proceeds by the application, in alternation, of the Parallel transform and the Leittonwechsel transform, designated here by P and L.28 These Riemannian operations are contextually defined inversions, in that they provide rules for inverting triads that depend upon the triadic configuration itself for their definitions.29 Thus, P inverts by altering the third of the triad, while L inverts by altering the Riemannian root of a triad, that is, the root of a major triad and the fifth of a minor triad. It is clear that P and L define permutations of the six triads of Hex1, and together they generate a group that acts in a simply transitive way on these six triads. The group {I, P, L, PL, LP, PLP} acts on the set Hex1 as follows: 28There is an interesting hemiolic relationship between the hexatonic cycle as described by the Riemannian transformations and as described by the Tn/In operations, since P and L alternate three times in the course of a cycle, while the Tn/In operations proceed through two cycles of length three. 29For an understanding of contextually defined inversions as a general concept I am indebted to discussions of them that arose in a seminar taught by John Clough, Fall 1996. In particular, Jon Kochavi and Nora Engebretsen, along with Professor Clough, contributed to an emerging notion of contextual operations. 33 I: ( ) P: (A+ A-)(C#+ C#-)(F+ F-) L: (A- F+)(C#- A+)(C#+ F-) PL: (F+ A+ C#+)(F- C#- A-) LP: (F+ C#+ A+)(F- A- C#-) PLP: (A+ F-)(F+ C#-)(C#+ A-) One can check that these permutations form a group: (1) the identity transformation is included; (2) each of the elements has an inverse (left and right inverse), since P and L are their own inverses, as is PLP, and PL and LP are inverses of each other; (3) the operation, composition of permutations, is closed and associative. As Lewin demonstrates, every non-commutative GIS (S, IVLS1, int1) is paired with another GIS, (S, IVLS2, int2), where all the elements of the group IVLS2 commute with all the elements of the group IVLS1. The P/L Hex1 GIS is linked to the Tn/In Hex1 GIS in that the elements of the P/L group commute with the elements of the Tn/In group that acts on Hex1. For example, I9PL(A+) = I9(C#+) = C#- = PLI9(A+) = PL(F-) = C#-. The groups that commute with each other are isomorphic to each other. As we will see, however, there is a natural mapping between the groups that is an anti-isomorphism, i.e., a one-to-one correspondence i between the groups such that i(AB)=i(B)i(A), for all elements A and B.30 An explicit anti-isomorphism between the groups is displayed below: i: I T0, P I1, L I5, PLP I9, PL T4, LP T8. 30As we will see in Chapter 5, if there exists an anti-isomorphism between two groups, the groups are isomorphic, and conversely, given an isomorphism, one can construct an anti-isomorphism. 34 Thus, for example, i(I1I5) = i(T8) = i(I5)i(I1) = LP, and i(I1I9) = i(T4) = i(I9)i(I1) = (PLP)P = (PL)PP= PL. Also, i(I9I5) = i(T4) = i(I5)i(I9) = L(PLP) = (LP)(LP) = PL, and i(I5I9) = i(T8) = i(I9)i(I5) = (PLP)L = (PL)(PL) = LP. In the GIS with Hex1={A+, A-, F+, F-, C#+, C#-} as the set S and the group IVLS1 = {T0, T4, T8, I1, I5, I9}, the group generated by P and L is the group of intervalpreserving operations for the GIS, and vice versa, taking the P/L group as IVLS2. That is, if h1 and h2 are elements of Hex1, and X is an operation in the P/L group acting on Hex1, then int1(h1,h2) = int1(X(h1), X(h2)), and if Y is an operation in the Tn/In group, int2(h1, h2) = int2(Y(h1), Y(h2)). For example, the interval between A+ and F- is PLP; I1(A+) = A-, and I1(F-) = C#+, and the interval between A- and C#+ is also PLP. The P/L group can be defined on any of the hexatonic systems, because its operations are not sensitive to the labeling of the pitch-class constituents of the triads, but act identically on all major triads, on the one hand, and on all minor triads, on the other. The P/L group is also the commuting group and the group of interval-preserving operations for {T0, T4, T8, I1, I5, I9}, when it acts on Hex3 = {B-, B+, Eb-, Eb+, G-, G+}, amd for the other Tn/In subgroup for the hexatonic systems, {T0, T4, T8, I3, I7, I11}, which acts simply transitively on the sets of triads Hex2 = {Bb-, Bb+, D-, D+, F#-, F#+} and Hex4 = {C-, C+, E-, E+, Ab-, Ab+}.31 One could argue, with good arguments on both sides, as to which of the GISs associated with the triads is the most “natural,” the cyclic group or either one of the nonabelian permutation groups. The more important point, I believe, is one of foreground and background. The cyclic group {R0, R1, . . ., R5} has providing it with a very firm background the GIS of six permutations that are identified with the subgroup of Tn/TnI operations that make the hexatonic pitch-class set itself a GIS. At least in part, that is implicit in the set-class consistency condition of the P-cycle, but it is worth 31Cohn, in “Maximally Smooth Cycles,” labels the hexatonic systems according to the cardinal points of the compass. 35 spelling out. What the cyclic group favored by Cohn foregrounds is precisely the existence of the P-cycle (maximally smooth cycle), with the measure of intervallic distance the number of pitch classes displaced between triads. The fact that in a P-cycle (as opposed to a Q-cycle) the moving pitch class moves by a minimal distance plays no formal role in terms of the GIS (it is a very strong formal constraint, though, as we will see) but it is a further musical justification of the whole construction’s analytical validity because of voice-leading norms. On the other hand, if one chooses the Tn/In group as the basis for the hexatonic Generalized Interval System, one foregrounds the power of the Tn/In operations not only to ensure the set-class consistency condition but to form a group; moreover, a group that acts in a simply transitive way on both the hexatonic pitch classes and on the six triads. In this Generalized Interval System, it is the motion of particular pitch classes that is placed in the foreground. For example, I1 is characterized by the exchange of pitch classes C and C#, in the transformations between A-major and A-minor, F-minor and C#major, and F-major and C#-minor triads. What provides this structure with further musical justification is what is pushed to the background, namely, the P-cycle and the cyclic group GIS that it suggests. For that reason, if one were to relabel Cohn’s analyses with the Tn/In operations, one must not lose sight of the fact that these particular Tn/In operations coexist in a tightly organized group of order 6, not just in the big group of order 24. For this reason, I favor keeping in mind all of these perspectives—the cyclic GIS that measures distance along the P- or Q-cycle, the Tn/In GIS that tracks the motion of particular pcs, and the associated commuting-group GIS that is expressed in terms of the contextually defined inversions that alternate to form the P- or Q-cycle—in any analytical application of P- or Q-cyclic sets associated with pairwise well-formed sets. The multiple description does not arise in well-formed sets, as we will see, because there the group induced by the P- or Q-cycle is the same as a cyclic set of ordinary transpositions. 36 The relative impoverishment of a GIS of six triads that is more one-dimensional, associated with the Tn/In group introduced above, is shown by the following example, suggested by Stephen Soderberg in a personal communication. The set of six triads {Bb+, Ab-, D+, C-, F#+, E-} is the space for a GIS where again the group acting on the triads in a simply transitive way is {T0, T4, T8, I1, I5, I9}, ˚. I would not dismiss the musical significance of this GIS—in fact, I will invoke it in the discussion of a passage by Wagner in Chapter 6—but it provides a contrast to the richer hexatonic system. Here, all 12 pitch classes are engaged by the set of triads, so the group of six Tn/In operations does not form a GIS with the set of pitch classes. More significantly, the justification for invoking the cyclic group is weaker here, (although John Clough has pointed out to me that the whole-tone scale ordering is tonally suggestive), therefore a well-behaved commutative GIS does not arise in such a natural way. Finally, the commuting group for this GIS is somewhat more complicated than the group generated by P and L. Here, it is the group obtained by conjugating the P/L group by the relative transform R, that is, the group R(P/L)R-1 = R(P/L)R = {I, LP, PL, RPR, RLR, RPLPR}.32 Having seen that the Tn/In group that acts simply transitively on the hexatonic pitch classes is congruent with the group that does so on the triads, that is, it preserves triads and transforms them according to the P-cycle while operating on their constituent hexatonic pitch classes, one might ask if there is a cyclic group that acts on the pitch classes and preserves triads. It turns out that there is. The cyclic group acting simply transitively on the pitch classes of Hex1 is defined as follows: R0 : ( ) R1: (094185) R2: (048)(159) (= usual T4) R3: (01)(45)(89) 32This example and the concept of conjugation will be discussed further in Chapter 6. 37 R4: (084)(195) (= usual T8) R5: (058149) This group preserves harmonic triads, and when applied to the triads yields the Ri group previously defined, but most set classes as usually understood are not preserved (are expanded), since although semitones and major thirds are preserved, minor thirds and perfect fourths may exchange, that is, they are in the same “interval class” here. For example, in this GIS, the “(014) set class” determined by the transpositions of the GIS is: {014},{015},{458},{459},{089},{189}. The P-cycle of triads provided a first exemplification of the availability of multiple descriptions through commutative and non-commutative GISs in pairwise wellformed scales that support P- or Q-cycles, that is, in all cases except those of the singular Hungarian minor type. In general, there is only one P- or Q-cycle associated with a given pairwise well-formed set class. Pairwise well-formed trichords, however, including the harmonic triad class, support three distinct P- or Q-relations, and these relations pair up in three different ways to form P- or Q-cycles. In the case of the harmonic triads, the three relations are the Riemannian P-relations P and L and the Q-relation R. Applying P and R in alternation produces a Q-cycle of triads of length 8: P R P R P R P R C- C+ A+ F#+ Eb+ A- F#- Eb- (C-) As before, the eight triads form a GIS with the cyclic group, as well as with the Tn/In group acting on the octatonic triads, here {T0, T3, T6, T9, I1, I4, I7, I10}, and with the group generated by R and P. Again, these two nonabelian groups are commuting 38 groups for each other, and each provides the interval-preserving operations for the GIS of the other group. Applying R and L in alternation, on the other hand, exhausts the entire set class, and again, a cyclic GIS can be formed, or non-commutative GISs with either the whole Tn/In group or the R/L group. For pairwise well-formed trichords in general, three distinct P- or Q-cycles arise from the three pairs of contextually defined operations. If the trichord has prime form <abc>, with a<b<c, (a, b, c are necessarily distinct), the three operations are Qa/b, Qb/c, and Qa/c, where Qx/y exchanges intervals x and y. These operations can be readily seen to produce P- or Q-relations. In the usual 12-note universe, there are seven pairwise well-formed trichords, each supporting three P- or Q-cycles. As Table 1.1 confirms, these trichords are the set classes 3-2:{013}, 3-3:{014}, 3-4:{015}, 3-5:{016}, 3-7:{025}, 3-8:{026}, and 3-11:{037}. The harmonic triad, 3-11, is the only trichord that supports a strict P-cycle, the cycle alternating P and L discussed above, so in the remainder of this discussion I will refer to Q-cycles. There are two other hexatonic cycles, for example, one arising from 3-3, with Qa/c followed by Qb/c (both Q-relations involving motion by interval class 5), the other from 3-4, with Qa/b followed by Qb/c (both Q-relations, with motion by interval class 3). These cycles are shown below, embracing the set Hex1. 014 145 458 589 890 901 015 045 459 489 891 801 The first statement of the row of Webern’s Concerto, op. 24, is shown below:33 33Here and elsewhere, I use the pitch notation suggested by the Acoustical Society of America. The number following a note indexes the register of the note. Middle C is C4, 39 Oboe: B5 Bb4 D5 Flute: Eb6 G6 F#5 Trumpet, muted: G#4 E4 F5 Clarinet: C5 C#6 A5 The row, as is well known, comprises four serial transformations of the 3-3:{014} trichord, played by four instruments, with the last notes of the first three trichords overlapping the first notes of the second, third, and fourth trichords. In Chapter 6, the organization of the row and of the row forms in terms of the 3-3 trichord will be discussed. Here only a few observations on the 3-4:{015} trichords embedded in the first statement of the row will be entertained. The first two trichords embrace a hexatonic set, {te2367}. The overlapping 3rd and 4th row elements suggest a chain of four Q-related 3-4:{015} trichords: I5 3 e t 3 t 2 3 7 2 7 6 2 I1 I5 I9 Qa/b Qb/c Q a/b e+3=2 t-3=7 3+3=6 Figure 1.1. Q-related 3-4 trichords in the first hexachord of op. 24 the C an octave above is C5, and so forth. Inflected notes are labelled according to their position on the staff; thus, B# above middle C is B#4, enharmonically C5. 40 The chain ends when the turnover of pitch classes is complete, and the hexachord is completed. Thus, the hexachord is partitioned into two disjoint 3-4 trichords (bold and underlined in the equations above): the first trichord, representing those pitch classes cast off in the course of the chain, and the last, repesenting those picked up in the course of the chain. Were the chain to continue, with a Qb/c relation, here matched with I1 (a manifestation of the hemiolic property), the first note of the row would reappear, violating the row structure. Note that I5 is the “interval” between the disjoint trichords and between the adjacent trichords. There are other aspects of the compositional design that help set in relief the Q-chain of 3-4 trichords. Because the pattern of order numbers for these trichords is symmetrical, and because the arrangement of the row in pitch-space creates symmetries for each of the hexachords, the trichords are organized symmetrically in pitch space. The row is shown below, together with the directed pitch intervals connecting adjacent row elements: e t -13 2 +4 3 +13 7 +4 6 || -13 (-10) 8 4 -4 5 +13 0 -5 1 +13 9 -4 The four 3-4 trichords in the first hexachord thus involve the following symmetrical pattern of pitch intervals: -13 +17 +4 +13 +13 +4 +17 -13 If the pitch space of the first hexachord, Bb4 to G6, is divided into two registral bands of equal width, with each move along the Q-chain the moving pitch class changes register: 41 HLH LHL HLH LHL Because the serial operations on the 3-3:{014} trichords are fundamental, in the second hexachord the Q-chain of 3-4 trichords does not follow chronological order in the outer trichord-pairs (the chronological order is indicated by the unidirectional arrows in figure 1.2 below). The pitch-space symmetries discussed above carry over to the second hexachord. T4 0 1 8 I1 Qb/c I5 0 1 5 T4 0 4 5 I5 Qa/b 9 4 5 I9 Qb/c Figure 1.2. Q-related 3-4 trichords in the second hexachord of op. 24 Table 1.2 displays information about the Q-cycles available for the seven pairwise well-formed trichords in the usual 12-note universe. For each trichord there are three cycles. Each of the trichordal set classes has 24 members, and the lengths of the cycles divide 24 (for reasons to be discussed in Chapter 5). Because the cycles are non-trivial, their lengths are necessarily greater than two. The Forte labels beside the entries for each trichord-cycle indicate the set of pitch classes embraced in the course of the cycle. Thus, 12-1 is the aggregate, and is obviously the set of pitch classes embraced by any cycle that exhausts the trichordal set class; 8-28 is the octatonic; 6-35 is the whole-tone hexachord; 42 6-20 is the hexatonic; 4-25 is the French sixth; and 4-9 is the tetrachord formerly known as Z, {0167}. The length of the cycle is also the number of elements in whichever GIS may be under consideration, as well as the order of the group for that GIS. Table 1.2 Cycle length: 24 12 8 3-2: {013} 1 (12-1) 1 (12-1) 1 (8-28) 3-3: {014} 1 (12-1) 3-4: {015} 2 (12-1) 3-5: {016} 2 (12-1) 3-7: {025} 1 (12-1) 6 4 Trichord: 3-8: {026} 3-11:{037} 1 (8-28) 1 (6-20) 1 (4-9) 1 (12-1) 1 (8-28) 1 (6-35) 1 (12-1) 1 (6-20) 1 (6-35) 1 (8-28) Trichordal P- and Q-cycles 1 (6-20) 1 (4-25) 43 In the mod 7 universe of the diatonic pitch classes, there is only one pairwise well-formed trichord, the all-interval <124> trichord.34 This trichord has some special properties, to be discussed in Chapter 3. Of course, all this discussion of trichords should not cause the reader to forget that all non-singular pairwise well-formed sets also participate in Q-cycles, and the remarks concerning multiple GIS descriptions hold equally for these sets. An example is provided by the “Romanian major” or octatonic-minus-one set, 7-31, which participates in Octatonic Systems, analogous to Cohn’s Hexatonic Systems. Here we have the Q-cycle through the 8 seven-note subsets of an octatonic set, previously introduced. In this case an octatonic set of pitch classes is engaged, and along with a cyclic group of order 8 acting on the 7-31 sets that is induced by the Q-cycle, there is also a nonabelian subgroup of order 8 of the Tn/In group that acts simply transitively on the octatonic set to form a GIS, and an associated Tn/In group that is understood as acting on the 8 forms of 7-31. To conclude this introduction to pairwise well-formed scales, I will sketch how the unification of well-formedness and pairwise well-formedness will be achieved in the formal development of the theory, via lumpiness vs. smoothness. Lumpiness vs. smoothness provides a generative scheme for well-formed and pairwise well-formed sets. Figure 1.3 is a 7X7 matrix of as and bs. If one focuses first on the columns, one observes that each is a maximally lumpy distribution of as and bs, or a partition of those elements into separate connected segments. The clump of as moves cyclically, two rotations downward each time. One can consider this a kind of abacus, effecting multiplication by 2 modulo 7: the leading a takes on successively positions 2, 4, 6, 1, 3, 5, 7 or 0 mod 7. 34The context here is the generic interval measure discussed in John Clough, “Aspects of Diatonic Sets,” Journal of Music Theory vol. 23, no. 1 (1979): 45-61, and applied in Clough and Myerson, “Variety and Multiplicity.” 44 2 4 1mod 7 N=7 m=2 v=4 abba bbb abbba bb babba bb babbba b bbabba b bbabbba bbbabba Figure 1.3. “Lumpiness is orthogonal to smoothness.” Now looking at the rows, and interpreting the as and bs as semitones and whole steps, respectively, one can see that these rows represent the modes of the diatonic scale, in circle of fifths order, from Lydian on the bottom to Locrian at the top. Note that in the rows, of course, the semitones are maximally evenly distributed. Because both the rows and columns are understood cyclicly, this matrix might be superimposed on a torus, identifying top and bottom edges and left and right edges. Now m=2 is the multiplicity of the half step, and 4 is its multiplicative inverse mod 7. Note the interval bracketed under v=4: two half steps separated by two whole steps, the diminished fifth. With this construction, a unique interval of diatonic span or generic length v will always occur on the top row of the array: it will enclose one more a step interval than every other interval of that diatonic length; every other interval of that length will enclose one more b step interval, and one less a interval. We have thus constructed a generating interval of constant diatonic length, and therefore we have a mechanism for constructing a well-formed set. We have illustrated in doing this what 45 will be proven algebraically in Chapter 2, that in a well-formed set the “multiplicity of a step interval is the multiplicative inverse of the diatonic length of a generator,” the fact that provided us with an algorithm for determining the status of a given sequence of step intervals as a well-formed or pairwise well-formed scale. Figure 1.4 shows the general case. Each row represents a mode of the scale, from the bottom, each one beginning v notes above the next. The interval enclosing steps in positions 1 to v contains one more a interval, and one less b interval, than every other interval of length v. mv 1mod N v m 1 2 3 1-v . . . . . . . . 2v 2v+1 . v v+1 . 0 1 . . . . . . . 2v+1 . . v+1 v+2 . . . . . . . . . . . . . . . . . . . . . . v . . 0 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . v Figure 1.4. “The multiplicity of a step interval is the multiplicative inverse of the diatonic length of a generator.” 46 Turning now to figure 1.5, we show how a similar generative scheme produces modes of a pairwise well-formed set of cardinality 7, of the type represented by the diatonic in just intonation, or by the sa-grama, <a b c a c b c>. For the moment we will disregard the array. Consider one row, say the top row. Equivalencing a and c, we have a b a a a b a, in other words something formally of the type of the Dorian mode. Equivalencing b and c, a b b a b b b, thus formally like the Locrian mode. Now equivalencing a and b, a a c a c a c is not a diatonic pattern, but it is well-formed. Thus, the top row represents a mode of the pairwise well-formed set, and each of the rows in the array is a mode of the pairwise well-formed set. Taking into consideration the whole array, equivalencing a and b will produce modes of <a a c a c a c>. In a sense, this is the least problematic case, since equivalencing a and b results in the lumpy vs. smooth array similar to the diatonic case in figure 1.3, but now with m=4 and v=2. The generative scheme works to produce well-formed sets in the cases of the other two equivalences because the matrix can be automorphically transformed into the lumpy vs. smooth diatonic matrix that we saw above, as figure 1.6 suggests. m=4 N=7 v=2 abc acbc bca bcac ac bcabc bca cbca ca bcacb cbc abca ca cbcab Figure 1.5. Generative scheme for a pairwise well-formed set nij=[j+5(i-1)]mod 7 nij=[j+3(i-1)]mod 7 47 row i 1 2 3 4 5 6 7 1234560 6012345 4560123 2345601 0123456 5601234 3456012 X 1234560 4560123 0123456 3456012 6012345 2345601 5601234 Y 1234560 4560123 0123456 3456012 6012345 2345601 5601234 Z Figure 1.6 Transformation of generative scheme for the pairwise well-formed set Matrix X in figure 1.6 reproduces the pairwise well-formed set in figure 1.5, with numbers representing order positions of the step intervals of the scale in the top row, where bold font represents step intervals of type a, italic font represents step intervals of type b, and the unmarked font represents type c step intervals. Matrix Y is an automorphic transformation of matrix X: the entry nij in the ith row and the jth column of matrix X is nij=[j+5(i-1)]mod nij=[j+3(i-1)]mod7. 7, while the comparable entry in matrix Y is In matrix Z, step intervals of types b and c are equivalenced, eliminating italic font, and matrix Z is now in the lumpy vs. smooth formation of the usual diatonic, with N=7, m=2, and v=4. Generalizing and making precise this heuristic example will be the burden of Chapter 4. It is hoped that this introductory chapter has suggested enough of the structural interest and transformational power of pairwise well-formed sets to motivate the formation of a theory for them. To lay the groundwork for such a theory, we turn next to a survey of scale theory.