Step variety: Difference between revisions
No edit summary |
|||
| Line 8: | Line 8: | ||
* 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. | * 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. | * 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<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | 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<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | ||
[[Category:Terms]] | [[Category:Terms]] | ||
Revision as of 21:16, 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 less well-known than binary ones, 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 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).