On a Logical Principle Useful in Music Theory

ON A LOGICAL PRINCIPLE USEFUL IN MUSIC THEORY ROY WHELDEN Our life is frittered away by detail…Simplify, simplify. —H. D. Thoreau in Walden N 1860 THE BRITISH LOGICIAN Augustus De Morgan published a challenging puzzle in a small gazette. The solution to the puzzle, which four years later he was referring to as Theorem K, proves itself to be of interest to music theorists.1 (A few logicians still call it Theorem K. Most logicians, if they know it at all, call it Schroeder’s theorem.) Although De Morgan’s Theorem K holds in very general systems of relations, musicians may be most interested in the theorem’s applications within group theory, since many common musical structures— the classical twelve-tone operations of the Viennese school, Babbitt’s duration series, the operations on Forte’s pitch-class sets, the cubic symmetries used by Xenakis, the operations on pitch-class sets from expanded microtonal systems—can be studied as groups. Within the context of group theory, then, here is a brief introduction to the notation used. (As an example group to hold in mind, used here because of its connection to twelve-tone equal temperament, take the group of integers 0, 1, 2, . . ., 11 with addition modulo 12. I 190 Perspectives of New Music As is customary, we’ll let 0, 1, 2, . . ., 11 represent the equal-tempered pitch classes C, C#, D, . . ., B.) Letting x and y be arbitrary elements of some group, simple concatenation xy will denote the group operation. (Thus, in our example, xy denotes addition x + y mod 12.) Let S, T, and U represent arbitrary subsets of G. (These subsets can be thought of as sonorities or as chords or, more accurately, as pitchclass sets.) • Then S-1 will represent the set of inverses of every element in S. (In our example, the inverse of an element x is 12 − x.) • The complement of S, that is, set of group elements not in S, will be denoted by S′. • ST will denote all possible concatenations of elements taken from S and T, in that order. • Finally, we use the sign ⊂ as the inclusion relation “is contained in.” De Morgan’s Theorem K says that the following statements are equivalent : ST ⊂ U, S-1U′ ⊂ T′, and U′T-1 ⊂ S′. De Morgan suggested a handy mnemonic for this: “Given an inclusion of the form ST ⊂ U, an equivalent form can be obtained by inverting one of the ‘multipliers’ S or T, then interchanging the complements of the other two members.” Our application of Theorem K will often require the use of some well known logical and group laws, including the law of double negation S′′ = S, the law of double inverse (S-1)-1 = S, and the law of contraposition, which says that S ⊂ T and T′ ⊂ S′ are equivalent. Before showing the proof of Theorem K, let’s apply the theorem to the subject of combinatoriality, the well known concept introduced by Milton Babbitt in 1955.2 I hasten to add that combinatoriality has been well understood since its introduction. The composer Donald Martino in particular offered insightful discussions of combinatoriality.3 My purpose here is to show how De Morgan’s powerful Theorem K provides a direct, simple, and easily programable approach to combinatoriality , generalizing it as well as several related musical concepts. 194 Perspectives of New Music the Common Tone theorem. For example, let S be the (just discussed ) sonority 6-32. We know that S is prime combinatorial, using traditional tools, from the fact that its interval vector (143250) has a rightmost entry of 0. It’s the presence of the 0 entry which informs us immediately (or at least after computing or looking up in a table the sonority’s interval vector) that S has prime combinatoriality. The reasoning is as follows: the interval vector (143250) of S tells us that the interval class of the tritone (6) is lacking. The Common Tone theorem assures us that the number of tritones to be found in S is the same as the size of the intersection of {6}S and S. But since the number of tritones in S is 0, the...