Step variety: Difference between revisions

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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 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)~~.

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 pages [[rank-3 scale]] and [[ternary scale theorems]].

== Etymology ==

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where <math>\phi</math> is the Euler totient function and <math>S(n, r)</math> is the Stirling number of the second kind which counts ways to partition an ''n''-element set into ''r'' distinguished parts.

== ~~List of named ternary scales~~ ==

== See also ==

~~The following is a list~~ of ~~(temperament-agnostic) names that have been given to~~ ternary scales~~. We ignore the exact arrangement of scale words here.~~

* [[List of ternary scales]]

~~=== 5 notes ===~~

* [[User:Fitzgerald_Lee/Nicepent|nicepent]] (2L 1M 2s)

[[Category:Scale]]

~~=== 7 notes ===~~

[[Category:Terms]]

* [[nicetone]] ~~(3L 2M 2S)~~

* [[omnidiatonic]] (2L 3M 2S)

* [[~~smicot]] (3L 1M 3S)~~

* [[dualdye]] (5L 1M 1S)

~~=== 8 notes ===~~

* [[porcusmine]] (4L 3M 1S)

* [[pinedye]] (5L 2M 1S)

~~=== 9 notes ===~~

* [[diasem]] (5L 2M 2S)

* [[armozan]] (2L 5M 2S)

* [[semizan]] (2L 3M 4S)

* [[balgram]] (2L 2M 5S)

~~=== 10 notes ===~~

* [[blackbuzz]] (5L 3M 2S)

* [[blackdye]] (5L 2M 3S)

* [[lemmagic]] (3L 3M 4S)

* [[blackville]] (2L 5M 3S)

* [[jarlem]] (2L 4M 4S)

~~=== 11 notes ===~~

* [[diamech]] (5L 2M 4S)

* [[semion]] (5L 3M 3S)

~~=== 12 notes ===~~

* [[diachromedye]] (5L 2M 5S)

* [[User:~~Fitzgerald_Lee/Nicechrome|nicechrome~~]] ~~(3L 4M 5s)~~

~~=== 14 notes ===~~

* [[whitedye]] (5L 2M 7S)

* [[chromatic crossdye]] (5L 7M 2S)

[[Category:Terms~~]][[Category:Scale~~]]

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 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 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).
+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 pages [[rank-3 scale]] and [[ternary scale theorems]].
 == Etymology ==
@@ Line 26: / Line 26: @@
 where <math>\phi</math> is the Euler totient function and <math>S(n, r)</math> is the Stirling number of the second kind which counts ways to partition an ''n''-element set into ''r'' distinguished parts.
-== List of named ternary scales ==
+== See also ==
-The following is a list of (temperament-agnostic) names that have been given to ternary scales. We ignore the exact arrangement of scale words here.
+* [[List of ternary scales]]
-=== 5 notes ===
-* [[User:Fitzgerald_Lee/Nicepent|nicepent]] (2L 1M 2s)
+[[Category:Scale]]
-=== 7 notes ===
+[[Category:Terms]]
-* [[nicetone]] (3L 2M 2S)
-* [[omnidiatonic]] (2L 3M 2S)
-* [[smicot]] (3L 1M 3S)
-* [[dualdye]] (5L 1M 1S)
-=== 8 notes ===
-* [[porcusmine]] (4L 3M 1S)
-* [[pinedye]] (5L 2M 1S)
-=== 9 notes ===
-* [[diasem]] (5L 2M 2S)
-* [[armozan]] (2L 5M 2S)
-* [[semizan]] (2L 3M 4S)
-* [[balgram]] (2L 2M 5S)
-=== 10 notes ===
-* [[blackbuzz]] (5L 3M 2S)
-* [[blackdye]] (5L 2M 3S)
-* [[lemmagic]] (3L 3M 4S)
-* [[blackville]] (2L 5M 3S)
-* [[jarlem]] (2L 4M 4S)
-=== 11 notes ===
-* [[diamech]] (5L 2M 4S)
-* [[semion]] (5L 3M 3S)
-=== 12 notes ===
-* [[diachromedye]] (5L 2M 5S)
-* [[User:Fitzgerald_Lee/Nicechrome|nicechrome]] (3L 4M 5s)
-=== 14 notes ===
-* [[whitedye]] (5L 2M 7S)
-* [[chromatic crossdye]] (5L 7M 2S)
-[[Category:Terms]][[Category:Scale]]