Step variety: Difference between revisions

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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]". Our use of the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

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 the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words; standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

== Difference from scale rank ==

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 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]". Our use of the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words, standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
+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 the term ''arity'' borrows an {{w|Arity|existing technical term}} and generalizes from this use of ''binary'', ''ternary'', and ''n-ary'' to refer to the number of letters in an alphabet in combinatorics on words; standard academic usage often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
 == Difference from scale rank ==