A diatonic chord with unusual voice-leading capabilities

Abstract

Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and Clampitt 1989, Carey 1998), applies the tools of standard set theory of 12-tone ET to the heptatonic set of seven tones of the diatonic scale. The two universes differ from each other in a number of ways other than simple cardinality. Although 12 is nearly twice as big as 7, the fact that 7 is prime and 12 composite contributes to a number of subtle differences between them. Every positive integer less than 7 is a unit mod 7, thus every diatonic interval generates the entire set. In the set of 12 tones, there are only four units, 1, 5, 7, and 11. Further, because of tuning, the geographies, if you will, of the sets also differ. In the equal-tempered 12-tone landscape, every place looks like everyplace else. In the diatonic scale, each generic span is inhabited by several different specific intervals. Because of this, the terrain is everywhere distinct, contributing to the phenomenon of gravitational asymmetry and of tonality. © 2009 Springer-Verlag.