Last month’s method of partitioning into dyads falls short as a way to enumerate the different ways a chord can voice lead smoothly to its tritone partner. For one example, the major seventh chord divides into “large” dyads in two ways, but it can voice lead to its tritone partner in four different ways. A better, more generalized way to enumerate the number of possibilities is to use permutations. A permutation of a set can be understood as its partition into one or more cyclic orderings. For example, (spring, summer, fall, winter) is a permutation of the seasons, which is equivalent to (winter, spring, summer, fall) but different from (spring, fall, summer, winter) and from (spring, fall)(summer, winter). Y is “followed by” X in a permutation if (…XY…) or (Y…X).
Many permutations—21, to be exact—can be invoked on a chord like a CM7 {C, E, G, B}, but present purposes favor only those permutations in which some registral realization of every note is followed by another note that can be realized 4, 5, 6, 7, or 8 semitones higher. There are four such permutations of {C, E, G, B}: not only (CE)(GB) and (CG)(EB), which reproduce the dyadic partition from last month, but also (CEBG) and (CGBE). These four possible permutations match one-to-one with the four possible smooth voice leadings to CM7’s tritone partner.
Caught between a stated commitment to bring up Ode to Napoleon during this entry and an unstated commitment to keep these entries relatively short, I have decided to honor the latter and delay meeting the former until next month.
