General Equal-Tempered Harmony: Parts 2 and 3
Ian Quinn · Perspectives of New Music · 2007
General Equal-Tempered HARMONY: Parts 2 and 3 L__T7tl Ian Quinn Part 2: Generic Prototypes, Maximally Even Sets, and the Intervallic Half-Truth Now that we have found common cause among all of the various approaches to chord quality we have surveyed, and seen that they are rather tightly interrelated to one another, the complexity of the theoreti cal task ahead should be clear. It seems that three abstract concepts?pro totypes, intrageneric affinities,and intergeneric affinities?can provide the infrastructureof a theory of chord quality that unifies the distance- and taxonomy-oriented modes of theorizing; our task is to make these con cepts concrete in the domain of chords and species. Rather than starting from the bottom up, proceeding from any one of the approaches covered in Part 1, we will take a top-down approach, wip General Equal-Tempered Harmony 5 ing the slate clean and taking as our point of departure a theoretical framework from scale theory (Clough and Douthett 1991), the original intent ofwhich was to generalize certain aspects of the diatonic scale. (In the broadest sense, that isour intent aswell: the diatonic scale, after all, is a generic prototype.) As we generalize Clough and Douthett's work out ward into a theory of generic prototypes, we will find ourselves subsum ing certain key notions from each of the approaches that have been covered so far, fitting them into the emerging unified framework. At times thework will, of necessity, be highly technical. The business of Part 3 will be to develop an approach to affinities that interfaces with this theory of prototypes; our treatment of that approach will retroactively clarifyand simplify much of the technical language that comes up below. ?2.1. MAXIMALLY EVEN SUBGENERA Although it is assumed that the reader is familiar with Clough and Douthett's remarkable paper on maximally even chords and chord spe cies, we will begin by summarizing and recasting some of their results, periodically adding some new terms concepts into the mix. An asterisk (*) denotes a term or usage original to this work. Page and theorem numbers in this section refer toClough and Douthett's paper unless oth erwise specified. Following Clough and Douthett's usage, the variable c will stand for the number of pes in a given pitch class universe; the vari able dwill generally stand for the cardinality of a chord or species. A maximally even (ME) chord isone "whose elements are distributed as evenly as possible around the chromatic circle" (96). There is at least one ME chord of every cardinality in every chromatic universe (Theorem 1.2, 102). Any transposition of aME chord is also ME, and conversely, allME chords of a given cardinality in a given chromatic universe are related by transposition (Theorems 1.6 and 1.7, 108). Furthermore, all ME chords are inversionally invariant (Theorem 1.8, 109). The foregoing may be summarized by observing that allME chords of the same cardinality in the same chromatic universe belong to the same inversionally invariant species, and, conversely, that all members of that species areME. Let M(c^ d) stand for the unique ME species of cardinal ityd in a universe of cpes. A ME chord is a singleton ifd = 1, an anti singleton * ifd = c 1 , and trivial ifd = 0 or d = c. Singletons and antisingletons (and various equivalence classes of them) are semitrivial* (but not trivial). We recall that the abstract complement of a ME species is also ME (Theorem 3.3, 150; Corollary 3.2, 151). The chords in aME species, 6 PerspectivesofNew Music together with their complements, constitute a ME subgenus*. Let M(?, d) stand for the ME subgenus includingM(c, d) and itscomplement. The number of ME subgenera in a ?-pc universe is equal to c/2 rounded down to the nearest integer. (This is the same as the number of interval classes in the universe.) As a preview of what is to come, our theory of chord quality will entail making a one-to-one correspondence between nontrivial ME subgenera and qualitative genera; a sense of this can be had by inspecting Example 2 (inwhich ME species are indicated with thicker circles) and considering the relationship between...