Other Parsimonious Tritone Progressions

My previous posts have recognized that the major tritone progression is the only progression between consonant triads that can allow both exclusively smooth voice leading and root doubling. One way to understand why the major tritone progression is capable of this is to observe that there are exactly three pitch-class dyads that can each connect with their tritone pitch-class transpositions with exclusively smooth voice leading: the major third or minor sixth (both voices move by whole step), the perfect fourth or fifth (both voices move by half step), and the tritone (both voices stay put). Any chord that is a disjoint union of these dyads can voice lead smoothly to its tritone partner. The root-doubled major triad is one such chord: it is the combination of a perfect fifth and a major third such that the "bottom note" of each doubles that of the other. (Substitute "top note" for "bottom note" and you have the fifth-doubled minor triad, which can also enjoy a parsimonious tritone progression.)

Doublings set aside, there are 18 different four-note chords that can do this, and 35 six-note chords that can do this, up to transposition and inversion. Of these chords, the ones I find most compositionally suggestive are those that can be disassembled into these "large" pitch-class intervals in multiple ways. For example, the major seventh chord, which can partition into two perfect fifths or two major thirds, can voice lead to its tritone partner in at least two different ways. Next month, I'll generalize this (and, in so doing, explain the "at least" in the previous sentence) and use this generalization to make an observation about Schoenberg's Ode to Napoleon.