The T-Hex Constellation

Abstract

An interval string in Z12 is an ordered n-tuple of integers s = whose elements sum to 12; for example, a = and P = (we will generally use Greek lower-case letters for specific interval strings).' Given a pitch class p and a string s, a pitch-class set (chord, Klang, sonority, scale) in the usual chromatic universe is fully described by naming the ordered pair (p,s) = {Pk I Pk is the sum of p and the first k elements of s mod 12 (k = 0,1,...,n-1)}. Thus (4,a) = {4,7,11 }, an E minor triad, and (9,P) = { 9,1,4,7 }, a dominant seventh chord on A. If we are given a set X we will read $(X) as the interval string associated with X. Thus if X = {0,4,7 }, $(X) = . We will write ps for (p,s) where the context is clear, so in the current example, (4,a) may be written 4a.2 Following Morris's (1987) convention, for any given set A = ps, we define the interval-class (ic) vector as the ordered 7-tuple V(A) = [uou1...u6] where ui = #ici(A) is the multiplicity of ici in A. Thus for aX = , if A = 4a (our E-minor triad example from above), V(A) = [3001110]. We will now count the ics formed between two sets. For any two sets A = ps and B = qt, the vector spanning A and B is the ordered 7tuple V(A,B) = [vovI...v6] where vj = #icj(A,B) is the multiplicity of icj