Neo-Riemannian Transformations and Wyschnegradsky's DC-scale

This chapter starts by looking at Richard Cohn's generalized parsimonious trichord and the generalized neo-Riemannian transformations that derive from it. Cohn himself includes some quarter-tone examples in his article "Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations" (JMT 1997) so it was natural to cite Cohn's work in this dissertation. There was one problem, though, and that is that the quarter-tone version of Cohn's generalized trichord does not appear to have much significance in the music I have studied. Indeed, quarter-tone composers seem to be more interested in tetrachords than trichords. I wanted to find a chord from an actual composer's music that I could subject to various transformations.

It turns out that if we assume that Wyschnegradsky's diatonicized chromatic scale is analogous to the major scale, and that the tonic tetrachord I found in Chapter 5 is analogous to the major triad, then we can come up with quarter-tone equivalents for the canonic neo-Riemannian transformations, P, L, and R, and that these transformations can even participate in transformation cycles such as those found in Cohn.