Maximal evenness

In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are either one or two consecutive (adjacent) specific intervals, in other words a scale which is "spread out as much as possible." This property was first described by music theorist John Clough and mathematician Jack Douthett in "Maximally Even Sets" (1991). (Johnson 2003, p.27, 150)

Myhill's property, that there are two specific intervals for every generic interval, follows from maximally evenness, and both properties are true of the diatonic collection, for instance adjacent notes are separated only by half-tones and whole-tones (1 and 2). The whole-tone scale is also maximally even, for instance adjacent notes are separated only by whole-tones.

Second-order maximal evenness is maximal evenness of subcollections with regards to a larger collection which is maximally even. Diatonic triads and seventh chords possess second-order maximal evenness, being maximally even in regard to the maximally even diatonic scale but are not maximally even with regard to the chromatic scale. (ibid, p.115) This nested quality resembles Fred Lerdahl's "reductional format" for pitch space from the bottom up: