Diregular scale

A diregular scale is a type of ternary scale with an even number of notes. A diregular scale consists of two identical generator chains, where all generators are identical and subtend the same step class. The two chains are offset by an interval that subtends k steps in a 2k-note diregular scale. Another characterization of diregular scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One diregular scale is the achiral variant of diachrome.

In terms of guide frames and interleaved scales, in diregular scales the interleaving offset is generated by the guided generator sequence GS(g), and the 2-note strand scale [0, len(scale)/2-step] is the offset for the guide frame. The other type of generator-offset scale is represented by scales including bipentatonic scales (such as blackdye), where the strand is generated by GS(g) and the interleaving offset is the offset.

Diregular scales are MV3 (but not SV3), and by the MV3 classification theorem a balanced single-period MV3 scale that has an even number of notes is always diregular and has step signature aXaYbZ where a is odd and b is even.

Terminology

The term diregular has been coined by akselai and Inthar.