Square Dances with Cubes

My earlier paper on hexatonic systems (Cohn 1996) observed, in passing, a rough correlation between the position of two triads in the hyperhexatonic system (see Figure 3, page 175) and the efficiency of the voiceleading between them. The current paper refines the correlation by replacing voice-leading efficiency with a closely related construct, uses this new construct to illuminate brief passages from music of Schubert, Liszt, and Brahms, and explores its relationship to several other neo-Riemannian configurations of triadic space. Voice-leading efficiency, informally sketched in my earlier paper, is here defined as a function that acts on a pair of triads (X = {X1,X2,X3}, Y= Y1,Y2,Y3}) whose pitch classes are 1-to-1 paired {Xn,yn}. Each of the three pairs constitutes a "voice." We stipulate further that the pcs are paired such that the sum of the distances travelled by the three voices is as small as possible, i.e., the "voice-leading" involves the "principle of least motion." There follow two preliminary definitions, both familiar. Arithmetic is modulo 12 here and throughout this paper, unless otherwise stipulated.