Step variety - Revision history

Fredg999: Split List of ternary scales into its own page

2025-02-26T04:50:15Z

Split List of ternary scales into its own page

@@ 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 ===
-* [[nicepent]] (2L1m2s)
+[[Category:Scale]]
-=== 7 notes ===
+[[Category:Terms]]

Inthar at 16:31, 4 January 2025

2025-01-04T16:31:37Z

← Older revision		Revision as of 16:31, 4 January 2025
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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.		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 ==		== Etymology ==

Inthar: /* List of named ternary scales */

2024-12-30T14:08:37Z

List of named ternary scales

← Older revision		Revision as of 14:08, 30 December 2024
Line 29:		Line 29:
	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.		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.
	=== 5 notes ===		=== 5 notes ===
	* [[nicepent]] (~~2L 1M 2s~~)		* [[nicepent]] (2L1m2s)
	=== 7 notes ===		=== 7 notes ===
	* [[nicetone]] (~~3L 2M 2S~~)		* [[nicetone]] (3L2m2s)
	* [[omnidiatonic]] (~~2L 3M 2S~~)		* [[omnidiatonic]] (2L3m2s)
	* [[smicot]] (~~3L 1M 3S~~)		* [[smicot]] (3L1m3s)
	* [[dualdye]] (~~5L 1M 1S~~)		* [[dualdye]] (5L1m1s)
	=== 8 notes ===		=== 8 notes ===
	* [[porcusmine]] (~~4L 3M 1S~~)		* [[porcusmine]] (4L3m1s)
	* [[pinedye]] (~~5L 2M 1S~~)		* [[pinedye]] (5L2m1s)
	=== 9 notes ===		=== 9 notes ===
	* [[diasem]] (~~5L 2M 2S~~)		* [[diasem]] (5L2m2s)
	* [[armozan]] (~~2L 5M 2S~~)		* [[armozan]] (2L5m2s)
	* [[semizan]] (~~2L 3M 4S~~)		* [[semizan]] (2L3m4s)
	* [[balgram]] (~~2L 2M 5S~~)		* [[balgram]] (2L2m5s)
	=== 10 notes ===		=== 10 notes ===
	* [[blackbuzz]] (~~5L 3M 2S~~)		* [[blackbuzz]] (5L3m2s)
	* [[blackdye]] (~~5L 2M 3S~~)		* [[blackdye]] (5L2m3s)
	* [[lemmagic]] (~~3L 3M 4S~~)		* [[lemmagic]] (3L3m4s)
	* [[blackville]] (~~2L 5M 3S~~)		* [[blackville]] (2L5m3s)
	* [[jarlem]] (~~2L 4M 4S~~)		* [[jarlem]] (2L4m4s)
	=== 11 notes ===		=== 11 notes ===
	* [[diamech]] (~~5L 2M 4S~~)		* [[diamech]] (5L2m4s)
	* [[semion]] (~~5L 3M 3S~~)		* [[semion]] (5L3m3s)
	=== 12 notes ===		=== 12 notes ===
	* [[diachrome]] (~~5L 2M 5S~~)		* [[diachrome]] (5L2m5s)
	* [[nicechrome]] (~~3L 4M 5s~~)		* [[nicechrome]] (3L4m5s)

	=== 14 notes ===		=== 14 notes ===
	* [[whitedye]] (~~5L 2M 7S~~)		* [[whitedye]] (5L2m7s)
	* [[chromatic crossdye]] (~~5L 7M 2S~~)		* [[chromatic crossdye]] (5L7m2s)
	=== 16 notes ===		=== 16 notes ===
	* [[penslen]] (~~5L 5m 6s~~)		* [[penslen]] (5L5m6s)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

Inthar: /* 14 notes */

2024-12-30T14:07:24Z

14 notes

← Older revision		Revision as of 14:07, 30 December 2024
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	* [[whitedye]] (5L 2M 7S)		* [[whitedye]] (5L 2M 7S)
	* [[chromatic crossdye]] (5L 7M 2S)		* [[chromatic crossdye]] (5L 7M 2S)
			=== 16 notes ===
			* [[penslen]] (5L 5m 6s)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

Inthar: /* 12 notes */

2024-12-30T14:06:54Z

12 notes

← Older revision		Revision as of 14:06, 30 December 2024
Line 53:		Line 53:
	* [[semion]] (5L 3M 3S)		* [[semion]] (5L 3M 3S)
	=== 12 notes ===		=== 12 notes ===
	* [[~~diachromedye~~]] (5L 2M 5S)		* [[diachrome]] (5L 2M 5S)
	* [[nicechrome]] (3L 4M 5s)		* [[nicechrome]] (3L 4M 5s)

	=== 14 notes ===		=== 14 notes ===
	* [[whitedye]] (5L 2M 7S)		* [[whitedye]] (5L 2M 7S)
	* [[chromatic crossdye]] (5L 7M 2S)		* [[chromatic crossdye]] (5L 7M 2S)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

Fitzgerald Lee: /* List of named ternary scales */

2024-12-30T12:13:32Z

List of named ternary scales

← Older revision		Revision as of 12:13, 30 December 2024
Line 29:		Line 29:
	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.		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.
	=== 5 notes ===		=== 5 notes ===
	* [[~~User:Fitzgerald_Lee/Nicepent\|~~nicepent]] (2L 1M 2s)		* [[nicepent]] (2L 1M 2s)
	=== 7 notes ===		=== 7 notes ===
	* [[nicetone]] (3L 2M 2S)		* [[nicetone]] (3L 2M 2S)
Line 54:		Line 54:
	=== 12 notes ===		=== 12 notes ===
	* [[diachromedye]] (5L 2M 5S)		* [[diachromedye]] (5L 2M 5S)
	* [[~~User:Fitzgerald_Lee/Nicechrome\|~~nicechrome]] (3L 4M 5s)		* [[nicechrome]] (3L 4M 5s)
	=== 14 notes ===		=== 14 notes ===
	* [[whitedye]] (5L 2M 7S)		* [[whitedye]] (5L 2M 7S)
	* [[chromatic crossdye]] (5L 7M 2S)		* [[chromatic crossdye]] (5L 7M 2S)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

Fitzgerald Lee: /* List of named ternary scales */

2024-12-23T15:31:10Z

List of named ternary scales

← Older revision		Revision as of 15:31, 23 December 2024
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	== List of named ternary scales ==		== List of named ternary scales ==
	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.		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.
			=== 5 notes ===
			* [[User:Fitzgerald_Lee/Nicepent\|nicepent]] (2L 1M 2s)
	=== 7 notes ===		=== 7 notes ===
	* [[nicetone]] (3L 2M 2S)		* [[nicetone]] (3L 2M 2S)
Line 33:		Line 35:
	* [[smicot]] (3L 1M 3S)		* [[smicot]] (3L 1M 3S)
	* [[dualdye]] (5L 1M 1S)		* [[dualdye]] (5L 1M 1S)

	=== 8 notes ===		=== 8 notes ===
	* [[porcusmine]] (4L 3M 1S)		* [[porcusmine]] (4L 3M 1S)
	* [[pinedye]] (5L 2M 1S)		* [[pinedye]] (5L 2M 1S)

	=== 9 notes ===		=== 9 notes ===
	* [[diasem]] (5L 2M 2S)		* [[diasem]] (5L 2M 2S)
Line 43:		Line 43:
	* [[semizan]] (2L 3M 4S)		* [[semizan]] (2L 3M 4S)
	* [[balgram]] (2L 2M 5S)		* [[balgram]] (2L 2M 5S)

	=== 10 notes ===		=== 10 notes ===
	* [[blackbuzz]] (5L 3M 2S)		* [[blackbuzz]] (5L 3M 2S)
Line 50:		Line 49:
	* [[blackville]] (2L 5M 3S)		* [[blackville]] (2L 5M 3S)
	* [[jarlem]] (2L 4M 4S)		* [[jarlem]] (2L 4M 4S)

	=== 11 notes ===		=== 11 notes ===
	* [[diamech]] (5L 2M 4S)		* [[diamech]] (5L 2M 4S)
Line 56:		Line 54:
	=== 12 notes ===		=== 12 notes ===
	* [[diachromedye]] (5L 2M 5S)		* [[diachromedye]] (5L 2M 5S)
			* [[User:Fitzgerald_Lee/Nicechrome\|nicechrome]] (3L 4M 5s)
	=== 14 notes ===		=== 14 notes ===
	* [[whitedye]] (5L 2M 7S)		* [[whitedye]] (5L 2M 7S)
	* [[chromatic crossdye]] (5L 7M 2S)		* [[chromatic crossdye]] (5L 7M 2S)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

FloraC: +redirect notice

2024-12-22T13:08:47Z

+redirect notice

← Older revision		Revision as of 13:08, 22 December 2024
Line 1:		Line 1:
			{{Redirect\|Ternary\|the temperament\|Ternary (temperament)}}

	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.		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.

CellularAutomaton: /* List of named ternary scales */ added some names of mine

2024-11-03T00:44:16Z

List of named ternary scales: added some names of mine

← Older revision		Revision as of 00:44, 3 November 2024
Line 39:		Line 39:
	* [[diasem]] (5L 2M 2S)		* [[diasem]] (5L 2M 2S)
	* [[armozan]] (2L 5M 2S)		* [[armozan]] (2L 5M 2S)
			* [[semizan]] (2L 3M 4S)
	* [[balgram]] (2L 2M 5S)		* [[balgram]] (2L 2M 5S)

	=== 10 notes ===		=== 10 notes ===
			* [[blackbuzz]] (5L 3M 2S)
	* [[blackdye]] (5L 2M 3S)		* [[blackdye]] (5L 2M 3S)
			* [[lemmagic]] (3L 3M 4S)
	* [[blackville]] (2L 5M 3S)		* [[blackville]] (2L 5M 3S)
	* [[~~blackbuzz~~]] (~~5L 3M 2S~~)		* [[jarlem]] (2L 4M 4S)

	=== 11 notes ===		=== 11 notes ===
Line 53:		Line 56:
	=== 14 notes ===		=== 14 notes ===
	* [[whitedye]] (5L 2M 7S)		* [[whitedye]] (5L 2M 7S)
			* [[chromatic crossdye]] (5L 7M 2S)
	[[Category:Terms]][[Category:Scale]]		[[Category:Terms]][[Category:Scale]]

Fredg999: Misc. edits in lead section and etymology section

2024-08-23T03:32:04Z

Misc. edits in lead section and etymology section

← Older revision		Revision as of 03:32, 23 August 2024
Line 1:		Line 1:
	~~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 ==		== Difference from scale rank ==