Step variety: Difference between revisions
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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. | 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. | |||
== Difference from scale rank == | == Difference from scale rank == | ||
To respect the subtlety of the notion of a scale's [[rank]], certain abstract scale theorists in the xen community have taken to using this ''n-ary'' terminology. Examples of this subtlety are: | To respect the subtlety of the notion of a scale's [[rank]], certain abstract scale theorists in the xen community have taken to using this ''n-ary'' terminology. Examples of this subtlety are: | ||
Revision as of 21:09, 7 June 2023
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.
Difference from scale rank
To respect the subtlety of the notion of a scale's rank, certain abstract scale theorists in the xen community have taken to using this n-ary terminology. 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 Xi > 0, i = 1, ..., n, has rank n, not lower, for almost all choices of Xi, 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.