Even-regular MV3 scale: Difference between revisions

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 [[interval class|step class]]. The two chains are offset by an interval that subtends ''k'' steps in a 2''k''-note diregular scale.

== Notable diregular scales ==

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 frame]]s and interleaved scales, in diregular scales the [[interleaved scale|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 [[ternary scale theorems|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.

Diregular scales always satisfy all 3 of the [[monotone-MOS scale|monotone-MOS]] conditions.

The term ''diregular'' has been coined by akselai and Inthar.