Step variety

An n-ary scale is a scale with exactly n distinct step sizes; a scale's arity is the number of distinct step sizes it has. Unary, binary and ternary scales are scales with exactly 1, 2 and 3 step sizes, respectively.

Unary scales are equal tunings. The class of binary scales consists of all MOS scales and every alteration-by-permutation of a MOS scale, but do not include altered MOS scales such as the harmonic minor scale, msmmsLs, which gain additional step sizes from the alteration. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of generator-offset scales. Most known facts about ternary scales on the wiki can be found on the page rank-3 scale (which is mostly about specifically ternary scales).

History of the term

The terms binary and ternary are already used in some academic literature in reference to scales; 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 in view of 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).

List of named ternary scales

The following is a list of (temperament-agnostic) names that have been given to ternary scales.

7 notes

• nicetone (3L 2M 2S)
• omnidiatonic (2L 3M 2S)
• smicot (3L 1M 3S)
• dualdye (5L 1M 1S)

8 notes

• porcusmine (4L 3M 1S)
• pinedye (5L 2M 1S)

9 notes

• diasem (5L 2M 2S)
• armozan (2L 5M 2S)
• balgram (2L 2M 5S)

10 notes

• blackdye (5L 2M 3S)
• blackville (2L 5M 3S)
• blackbuzz (5L 3M 2S)

11 notes

• diamech (5L 2M 4S)
• semion (5L 3M 3S)

12 notes

• diachromedye (5L 2M 5S)

14 notes

• whitedye (5L 2M 7S)