The legacy of John Clough in mathematical music theory

This special issue of Journal of Mathematics and Music is dedicated to the memory of John Clough (1930–2003). The three authors herein were not among his students, but Clough's work helped create a disciplinary space for theirs; moreover, direct connections linking his work to theirs may be drawn, as discussed below.

It is certain that Clough would have rejoiced at the founding of the Society for Mathematics and Computation in Music and of the affiliated Journal. Clough had a vision for a subfield of music theory that not only drew upon the subject matter of mathematics, but also modeled its working arrangements on those of mathematics and the sciences generally. He fostered the collaborative work that is more characteristic of the sciences than the humanities, on several fronts. He often co-authored articles, both with senior colleagues and with his graduate students, with fellow music theorists and with colleagues in other disciplines. For example, his article 1 on ancient Indian scales, written together with three co-authors: mathematician Jack Douthett, Indian music specialist N. Ramanathan, and music theorist Lewis Rowell. Indeed, Clough is that rara avis, a music theorist with a finite Erdós number (through his work with Douthett). By virtue of his position as Slee Professor of Music Theory at the University at Buffalo—at that time the only endowed chair in music theory in North America—Clough sponsored three quadrennial Buffalo conferences that were highly significant in the development of mathematical music theory, especially transformational theory and analysis. The 1993 Buffalo conference marked the beginning of neo-Riemannian theory and analysis, stimulated by Richard Cohn's work; the 1997 conference yielded a special issue of Journal of Music Theory (1998) that was emblematic of the strength of the field of transformational theory; by the 2001 conference, the maturity that the field had achieved was indicated by the disparate nature of the contributions and by invited critical responses from Fred Lerdahl, Charles Smith, and Daniel Harrison. The conduct of these conferences was reflective of Clough's quiet generosity, thoughtfulness, and openness. He made a point of including graduate students and junior scholars among the participants, just as he included them among his co-authors. He encouraged interdisciplinary engagement: Douthett from mathematics and Carol Krumhansl from psychology of music were involved in each conference, and he was open to the contributions of those outside of traditional academic circles. He supported international cooperation, insofar as he could within the framework of the Buffalo conferences and at the 1994 Bucharest conference, and also by extending invitations to such scholars as Guerino Mazzola and Eytan Agmon to lecture at the University at Buffalo.

Clough's training was in music—performance, composition, and theory. He has the distinction of having written the first article in the first issue in Journal of Music Theory, 2, and he was among the founders of the Society for Music Theory. He served a term as editor of the Society's journal, Music Theory Spectrum, and he attended every conference of the Society from the time of its founding until his final illness prevented him from attending the 2002 conference. He was largely self-taught in mathematics; working in a pre-Google time, he missed some contributions from the mathematical side that would have undoubtedly informed his work—as did most of us working in this area, it should be said. In particular the work of Yves Hellegouarch 3–5 should be noted because it is not only relevant to, but also in its first appearances preceded, much of the work under discussion here. Such failures to connect should become far less frequent in a community that now has an international Society in place.

The following discussion will trace just some of the lines of development of Clough's work, those that lead toward the contributions in this second issue of Journal of Mathematics and Music. (See also ‘Remarks on Scale Theory’ in Thomas Noll's article, and parts of Norman Carey's summary.) It will have to slight, among other things, Clough's early work on computer music and computers in music 6, and his contributions to pitch-class set theory (e.g., 7 8). The ramifications of his work with Douthett on maximally even sets 9 are manifold: the application to a one-dimensional Ising model within physics 10 and the application to similarity relations within pitch-class set theory 11, are just two of the more remarkable ones. These too will be given short shrift here—Douthett and Krantz have undertaken such a review of the implications of maximally even sets in a forthcoming article 12 in Clough's honor.

1 Generic and specific interval measures, Myhill's property

One of Clough's most fruitful conceptual moves was to take seriously the dichotomy in traditional musical nomenclature between generic and specific intervals: the former construed with the usual 7-note diatonic scale as referential universe, the latter construed according to the usual 12-note chromatic scale. He understood the generic perspective as modeled after the specific perspective, and just as traditional musical set theory employs arithmetic modulo 12 to treat relations of pitch classes under octave equivalence, in 13 and 14 he employed arithmetic modulo 7 within a diatonic context. Just as enharmonic equivalence (e.g., F-sharp≡G-flat) is assumed in mod 12 applications, diatonic equivalence, which erases the distinctions among step intervals of different sizes (as in the usual terminology, ‘seconds’, ‘thirds’, etc.), is assumed in mod 7 applications, and similarly for the generalized situation, in which a diatonic set of cardinality d is embedded in a chromatic universe of cardinality c: diatonic set D={c _j |0 ≤ c _j <c, j=0,1, … d − 1}. In Clough's later work with Gerald Myerson and with Douthett 9 15 16, he quantified the generic/specific distinction through diatonic length vs. chromatic length, i.e., differences mod d and mod c. In Carey's and my work 17, where we do not assume that our sets are embedded within chromatic universes, we refer to span and size, where the latter is assigned a real number. It is worth noting that despite the fact that Clough only ever studied embedded sets, he insisted that this did not imply a commitment to equal temperament or to any particular tuning system. In reading Julian Hook's article in the present issue, one should similarly avoid acoustical assumptions.

Clough's Einfall truly bore fruit when he brought the two perspectives together in 15 16, enlisting the number theorist Myerson as co-author, at the suggestion of the logician John Myhill. Clough and Myerson extended the double description of diatonic intervals to ordered diatonic subsets, or lines. For example, the generic line class of three-note segments of the diatonic scale might be described as ‘a step, followed by a step’. This generic class includes the specific classes ‘whole step, whole step’, ‘whole step, half step’, and ‘half step, whole step’. They observed that in the usual diatonic, generic intervals that are non-zero mod 7 come in two specific varieties. Moreover, they observed that generic line classes consisting of k distinct diatonic pitch classes for k≤7 come in k specific varieties. Myhill conjectured the equivalence of the 2-to-1 and k-to-1 conditions. Clough and Myerson therefore named the 2-to-1 non-zero generic-to-specific property Myhill's Property (MP), and named the second property Cardinality equals variety for lines (CV). They proved, among other things, the equivalence of MP and CV for subsets D of cardinality d in chromatic universes U _c of cardinality c.

In the course of this proof, Clough and Myerson introduced the auxiliary construction of a semireduced set: if a subset D has MP, D becomes semireduced by the deletion of extraneous elements of the chromatic set so that the chromatic lengths of the two step-interval sizes in D form a set of consecutive positive integers. The point is that for D semireduced, c and d are coprime. Thus, there exist integers c′ and d′ such that

1

Clough and Myerson then demonstrated the existence in D of a generalized circle of fifths: d−1 intervals, each of diatonic length c′ and of chromatic length d′, with the remaining interval of diatonic length c′ and chromatic length d′ − 1.

2 Subsequent developments

The relation (1), implicit in Clough's article with Myerson, may suggest to a mathematician several contexts, all of which come into play in subsequent developments in mathematical music theory: continued fractions 5 17–19, also Carey's article in this special issue; Farey series 20; the Stern-Brocot tree 21 22; and SL(2,ℤ) 23, also Noll's article in this issue.

The generator of constant span, essentially equivalent to the generalized circle of fifths, was the starting point for the study of well-formed scales 18, significant for the articles in this issue by Carey and Noll. For θ real, 0 < θ<1, and an integer N>1, consider S={nθ–⌊nθ⌋|0 ≤ n<N}. Then S={s ₀, s ₁, … s _N−1}, where 0 = s ₀< …  < s _N−1, is well-formed if and only if there exists a unit u mod N such that µ:ℤ _N →ℤ _N maps z to uz mod N where s_µ(z)=zθ–⌊zθ⌋. S is non-degenerate well-formed if it is well-formed and its step intervals (differences s _i + 1–s _i mod 1) come in two sizes. A generalized Myhill's property may be defined in a natural way for S, and we showed in 17 that S has generalized MP if and only if S is non-degenerate well-formed.

In the definition above, it is clear that u plays the role of the span or diatonic length of the generator θ, equivalent to Clough and Myerson's c′, which they define as the negative of the multiplicative inverse of c mod d. In this context, no embedding in a chromatic universe of cardinality c is assumed, but since (u,N) = 1, u has a multiplicative inverse mod N that has other meanings (keeping in mind that here N replaces d). Let g be the least non-negative integer that satisfies ug≡1 mod N. From the definition, we see that s ₁=s _µ(g), that is, s _ug is the size of one of the step intervals; moreover, g determines µ⁻¹. It follows that g is the winding number discussed by Vittorio Cafagna and Domenico Vicinanza 22 23. It also follows that g and N−g are the multiplicities of the two step-interval sizes of S 17.

S is well-formed if and only if N is the denominator of a (semi-)convergent (i.e., full or intermediate convergent) in the continued fraction expansion of θ, a result that is a corollary of the Three Gap Theorem (see 18 and V. Sós, 24). If M/N is such a (semi-) convergent, and the kth convergent a _k/b _k is the full convergent immediately preceding M/N in the continued fraction of θ, then from basic continued fraction theory we have Mb _k −Na _k =(−1) ^k . M is the span of the generating interval of size θ (i.e., M=u). Depending on the parity of k, either b _k or N−b _k plays the role of g, i.e., is the multiplicative inverse of M mod N and is the multiplicity of a step interval. Similarly, N − M is the span of a generating interval whose size is 1 − θ, with the multiplicity of the other step interval the multiplicative inverse mod N of N − M, N − b _k or b _k , depending on the parity of k.

The generator θ thus determines a hierarchy of well-formed scales, according to its continued fraction expansion, as discussed in Carey's article. Noll applies a transformational approach to navigate this hierarchy, from multiple perspectives. The values determined by the continued fraction, with their music-theoretical meanings, play fundamental roles in his transformations, and in their matrix representations. As he acknowledges, the general basis for his approach lies in the work of Mazzola, and in that of the late David Lewin, a highly significant figure in our field whom we also lost, along with Clough, in 2003. Clough played a role in furthering Lewinian transformational theory: through his important review 25 of Lewin's 1987 book 26; through his organization of the Buffalo conferences, which Lewin participated in; and in his own work, notably 27. Posthumous articles on transformational theory by Lewin and by Clough will appear in 28, a forthcoming book in honor of Clough, with contributions from some of his former students and colleagues closely associated with him.

Hook's article (whose companion article on signature transformations is to appear in 28) stems both from the Lewinian transformational tradition and from the Clough tradition. Hook treats the structure of modern Western musical notation and its implications, departing from the assumption of diatonic equivalence that staff notation suggested to Clough. His assumption of diatonic equivalence is thoroughgoing: he attributes only a relational meaning to the placement of notes on lines and spaces of the staff, as is conventional in diatonic set theory; moreover, he attributes only a relational meaning to the prime ordering or circle-of-fifths ordering—this ordering is privileged in that it defines the order of sharps and flats in the signatures, but otherwise there are no a priori intervallic assumptions. Hook intertwines the notion of diatonic equivalence with that of enharmonic equivalence, which returns us to the point of departure for Clough's work discussed above. The context here is different, however, and Hook takes it in an original direction by dropping the conventional identification of enharmonic equivalence with mod 12 congruence.

With his discussion of Clough words, Noll makes reference to another instance where a mapping between mathematics and music theory remains to be fully explored. The combinatorial theory of words, which in its modern form extends back to the work of Morse and Hedlund 29 30, has been a very active subfield in the past several decades. The manifold connections between combinatorial word theory and mathematical music theory remained largely unknown, it appears, until very recently. Marc Chemillier and Charlotte Truchet 31 have notably applied word theory to Central African rhythmic patterns. Preliminary steps toward effecting a translation between the subfields were taken in the recent meetings in Berlin, at the Helmholtz Workshop at Humboldt University, May 12–13, and at the first meeting of the Society for Mathematics and Computation in Music, May 18–20. Talks towards that end were given by Thomas Noll, 32, and by Manuel Dominguez, David Clampitt, and Thomas Noll, 33, from the music theory side; and by Jean-Paul Allouche, from the mathematics side 34. The bridging of this divide is just one herald of a new era in mathematical-musical scholarship that the Society and Journal of Mathematics and Music have ushered in, one which John Clough would have welcomed with joy.