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~~ '''step variety''' or '''arity''' is the number of distinct step sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively.

The '''step variety''' (or '''arity''') of a [[scale]] is the number of distinct [[step]] sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively. An ''n'''''-ary scale''' is a scale with exactly ''n'' distinct step sizes.

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).

Unary scales are [[equal tuning]]s. The class of binary scales consists of all [[MOS scale]]s and every alteration-by-permutation of an MOS scale, but do not include [[altered MOS scale]]s such as the harmonic minor scale (abstract [[step pattern]]: 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 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

''~~Step~~ variety'', coined by [[~~User:fredg999~~]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].

== Etymology ==

The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to [[word]]s 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 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.

The term ''step variety'', coined by [[Frédéric Gagné]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].

== Difference from scale rank ==

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-An ''n'''''-ary scale''' is a scale with exactly ''n'' distinct step sizes; a scale's '''step variety''' or '''arity''' is the number of distinct step sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively.
+The '''step variety''' (or '''arity''') of a [[scale]] is the number of distinct [[step]] sizes it has. '''Unary''', '''binary''', '''ternary''', and '''quaternary''' scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively. An ''n'''''-ary scale''' is a scale with exactly ''n'' distinct step sizes.
-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).
+Unary scales are [[equal tuning]]s. The class of binary scales consists of all [[MOS scale]]s and every alteration-by-permutation of an MOS scale, but do not include [[altered MOS scale]]s such as the harmonic minor scale (abstract [[step pattern]]: 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 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
-''Step variety'', coined by [[User:fredg999]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].
+== Etymology ==
+The terms ''binary'' and ''ternary'' are already used in some academic literature in reference to [[word]]s 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 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; combinatorics-on-words literature often instead uses "word on ''n'' letters" or "alphabet with ''n'' letters" in the arbitrary-''n'' case.
+The term ''step variety'', coined by [[Frédéric Gagné]], is in analogy with ''[[interval variety]]'' for the number of distinct interval sizes in each [[interval class]].
 == Difference from scale rank ==