Last month I promised to bring this multi-post thread to Schoenberg: here goes. The six-note collection of notes that can voice lead smoothly to its tritone partner in the most ways—36, to be exact—is some transposition of {C, C#, D, F#, G, G#}. Tied for second place at 20 ways are the whole-tone scale—some transposition of {C, D, E, F#, G#, A#}—and the hexatonic scale—some transposition of {C, Db (or C#), E, F, G# (or Ab), A}. The 20 permutations that correspond to the 20 possible voice leadings of the hexatonic scale are given below, organized into rows by the kind of chord in each permutation partition (or "orbit"):
Three perfect fifths: (CF)(C#G#)(EA)
Two major thirds, one perfect fifth: (CE)(C#G#)(FA) || (CF)(C#A)(EG#) || (CG#)(C#F)(EA)
Two augmented triads: (CEG#)(DbFA) || (CEG#)(DbAF) || (CG#E)(DbFA) || (CG#E)(DbAF)
One major-7th chord, one perfect fifth: (CEAF)(DbAb) || (CF)(C#G#EA) || (CF)(C#AEG#) || (CFAE)(DbAb) || (CFDbAb)(EA) || (CAbDbF)(EA)
One hexatonic collection: (CEG#C#AF) || (CEAFDbAb) || (CFC#AEG#) || (CFAC#G#E) || (CG#EAC#F) || (CG#C#FAE)
However, of these three kinds of collections, only the hexatonic scale and its tritone transposition have different notes. In fact, they are complements, in that they share no notes, and, together, they contain all twelve tones. Moreover, of all the kinds of six-note collections, only the hexatonic scale can voice lead smoothly to its complement in the most possible ways of 20—the next highest number of ways is 10. This might make the hexatonic collection attractive to a twelve-tone composer who is interested in a variety of smooth voice leadings.
Schoenberg’s twelve-tone work Ode to Napoleon frequently juxtaposes complementary hexatonic-scale collections, and, at times in the work, each note in one hexatonic collection is no more than a pitch-class whole step away—in register, in a single instrument, or both—from a note in an adjacent complementary hexatonic collection. It remains to be shown whether or not Schoenberg uses, or at least implies, all twenty possible voice leadings, but he does so for at least one voice leading in each of the five rows above, in mm. 23 (strings), 21 (piano), 21 (strings), 219 (piano), and 187 (strings), respectively.
