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An Idea of Order 1- + Stephen Dembski Thirty years ago, I began working through Meta-Variations; five years later, I had an idea. I have always thought, but recentlywith increasing conviction, that I could not have had that idea in the general form that it came tome without the singularly stimulating and clarifying benefits ofmy encounter with Ben's work. While several specific aspects of the idea have close relations with details of the argument ofMeta Variations^ I have to admit to a further indebtedness to Ben, one at another level altogether: that is, I'm not sure that even my notion of hav ing an idea in or about music isn't derived from his work. So, delighted to have heard the call for submissions about and around thatwork, I'd like to dedicate to Ben, in celebration of this anniversary of his birth, an initialverbal introduction tomy quarter-century-old idea. Roughly, I'll describe a framework for thinking about pitch materials? materials that can be considered analogous to the chords and scales of the 404 PerspectivesofNew Music tonal system, but are restricted neither to the pitch-intervallic constructs of the tonal system, nor to those of diatonic constructs more broadly understood. But like diatonic music, music informed by it need not depend generally on the exhaustion of the twelve-tone aggregate?nei ther to provide a context within which to determine the relational iden tity of any particular pitch, nor to provide a conventional "unit of harmonic progression." Nevertheless, one of the unusual aspects of this framework is that it is inclusive of both tonal and twelve-tone materials, which are conventionally understood to be associated with two distinct systems, discrete and discontinuous. What brought about its develop ment was the urge to manage?conceptually, and purely for composi tional purposes?that vast array of pitch materials within the twelve pitch-class universe that are neither twelve-tone nor tonal, but thatmight be understood to have relations, by analogy, to both: that is, sets of fewer than twelve pitch classes which, nevertheless, hold order associations as an essential aspect of their relational identity. In itsmost concise form, the idea is a generalization of a rational reconstruction of the tonal system. I'll begin by outlining that recon struction, and then take up the generalization and a few of its implica tions, with reference to some brief passages ofmusic. I'll conclude with some observations about the idea, including what I find useful about it, and how ithas made a difference tome. The Diatonic Model 1. Take the unique1 pair of cyclic orderings of the twelve-pitch-class uni verse inwhich each ordering has a single pitch-interval of order adja cency.2 These orderings are of course those known as the "chromatic scale" (in its cyclic, pitch-class form) and the "circle of fifths"; the pitch intervals of order adjacency will be of complementary-interval classes3 1 and 5, respectively. e 0 17 0 5 (t 10 , e 11 ) t 2 2 t 9 3 9 3 8 4 4 8 -tofo single-interval orderings of the 7 65 e 6 1 twelve-pc universe If we wish, we can represent these as sets of pitch-class/order-class pairs (e.g., for the second, the "circle of fifths": {(0,0) (5,1) (t,2) (3,3) (8,4) (1,5) (6,6) (e,7) (4,8) (9,9) (2,t) (7,e)}).4 An IdeaofOrder 405 2. Take a seven-pitch-class segment5 of the second of these twelve-pitch class orderings, the one that isnot congruent with the registral ordering but rather has a pitch interval between order-adjacent pitch classes of complementary-interval class 5. 7 0 2 9 4 -a seven-pc segment of the "circle of fifths" (a G-major e 6 diatonic collection) 3. Associate each pitch class of the segment with the order position that it holds in that other twelve-pitch-class ordering, the one that is congruent with the registral ordering?that is,with a pitch interval of complemen tary-interval class 1 between order-adjacent pitch classes. e 0 2 9 4 -a G-^najor diatonic collection...
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