Step variety

An n-ary scale is a scale with exactly n distinct step sizes. The terms unary, binary and ternary are used for n = 1, 2 and 3.

A unary scale is an equal tuning. The class of binary scales consists of all MOS scales and every alteration-by-permutation of a MOS scale. Ternary scales are much more subtle, but one well-studied type of ternary scales is the class of generator-offset scales.

History of the term

The terms binary and ternary are already used in some academic literature in reference to scale words; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".

Difference from scale rank

Certain abstract scale theorists in the xen community have taken to using the n-ary terminology, to respect the subtlety of the notion of a scale's rank. Examples of this subtlety are:

• Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1.
• Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs.

The term n-ary disregards the rank of the group generated by the step sizes, although an n-ary scale is still generically rank-n (the group generated by the n step sizes X_i > 0, i = 1, ..., n, has rank n, not lower, for almost all choices of X_i, in the same sense that almost all real numbers between 0 and 1 are irrational). Most known facts about ternary scales on the wiki can be found on the page rank-3 scale.