Step variety: Difference between revisions

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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. This use of term ''arity'', while generalized from the use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, is a non-conventional extrapolation from it; standard academic usage prefers "word on ''n'' letters".

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 tuning]]s. 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 words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]".

The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]". Our use of term ''arity'', while generalized from the use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, is a non-conventional extrapolation from it; standard academic usage prefers "word on ''n'' letters".

== Difference from scale rank ==

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-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. This use of term ''arity'', while generalized from the use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, is a non-conventional extrapolation from it; standard academic usage prefers "word on ''n'' letters".
+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 tuning]]s. 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 words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]".
+The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]". Our use of term ''arity'', while generalized from the use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, is a non-conventional extrapolation from it; standard academic usage prefers "word on ''n'' letters".
 == Difference from scale rank ==