Why Are There Twenty-Nine Tetrachords? A Tutorial on Combinatorics and Enumeration in Music Theory

"Can you do addition?" the White Queen asked."How much is one and one and one and one and oneand one and one and one and one and one?""I don't know," said Alice. "I lost count.""She can't do addition," the Red Queen interrupted.-Lewis Carroll, Through the Looking Glass[1] Table 1 gives the number of pitch-class sets and the number of Tn/TnI set classes of each cardinality from 0 through 12. The number of different pc-sets of cardinality k, appearing in the central row of the table, may be calculated by a well-known formula (to be reviewed below) for "the number of combinations of 12 things k at a time." The set-class counts in the bottom row are familiar to music theorists not by means of a formula, but through the widespread use of tables enumerating all the set classes. Such tables appear, for example, in Forte 1973, Rahn 1980, Morris 1987 and 1991, and Straus 2005, among various other texts.Table 1. Enumeration of mod-12 pitch-classsets and Tn/TnI set classes[2] The formula giving the number of set classes-if indeed there is such a formula at all-is not an obvious one. On reflection, some aspects of these numbers may even seem somewhat counterintuitive. We might expect that when we count something governed by a high degree of symmetry, such as the number of different types of four-element subsets of a twelve-element set, we should end up with a "round" number-probably a number with many divisors, such as 24, 30, or 36. The fact that the number of tetrachord types turns out to be 29-a number not only odd but prime-challenges this intuition and suggests that the enumeration question may be deeper or more difficult than we imagined.[3] Bernard 1997 and Nolan 2003 show that pitch-class sets and set classes have been effectively enumerated in various ways dating back at least to the late nineteenth century. In general, though, the music scholars who have carried out this exercise have not pursued mathematical explanations for the numbers appearing in the bottom row of Table 1. That is, they have not attempted to calculate the number of set classes except by exhaustively listing all of the set classes-a process akin to the White Queen's way of "doing addition" in the quotation above.[4] Beyond simple curiosity, a further motivation for exploring the mathematics behind set-class enumeration is its potential for generalization. If we develop a formula that explains why there are 29 tetrachords, that formula will likely be applicable in a variety of other situations: other cardinalities of set (besides 4), other cardinalities of the underlying scale (besides 12), other definitions of "set class" (besides the usual one based on Tn/TnI equivalence), and perhaps enumeration problems of entirely different sorts, involving twelve-tone rows or rhythms, for example. Listing and counting all the mod-12 set classes is feasible but already somewhat unwieldy, as the multi-page lengths of the tabulations cited above reveal; it is not hard to devise similar problems for which the numbers involved are much larger and the exhaustive approach becomes impractical.[5] This article is organized in two parts. Part I, which might be subtitled "What every music theorist should know about combinatorics," offers a primer on standard techniques in classical combinatorial theory (including the "combinations" formula mentioned above) and offers some simple applications to music theory (including the enumeration of pitch-class sets) and historical notes (including a brief overview of the ars combinatoria that found its way into eighteenth-century musical discourse). The computational methods of Part I are familiar to every mathematician and will be familiar as well to many readers of this article. I believe it is useful to include this crash course here because, even though the mathematics involved is elementary, it has never become a part of the standard music theorist's toolbox and has seldom if ever been given a unified presentation oriented toward musical applications. …