Maximum variety - Revision history

FloraC: Changed redirect target from Interval variety to Interval variety#Maximum variety

2025-07-07T13:31:15Z

Changed redirect target from Interval variety to Interval variety#Maximum variety

← Older revision		Revision as of 13:31, 7 July 2025
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	#redirect [[Interval variety]]		#redirect [[Interval variety #Maximum variety]]

Inthar: Redirected page to Interval variety

2024-03-21T03:07:49Z

Redirected page to Interval variety

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Inthar: /* Max-variety-3 scales */

2024-02-15T17:38:01Z

Max-variety-3 scales

← Older revision		Revision as of 17:38, 15 February 2024
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	The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#Definition\|as well as in other equivalent ways]]: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) is the generalization of the MV2 characterization of the MOS property to [[ternary scale]]s. Other characterizations of the MOS property, such as [[distributional evenness]] and [[generator\|having a generator]], do not generalize to properties that are equivalent to MV3 in higher [[arity\|arities]].		The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#Definition\|as well as in other equivalent ways]]: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) is the generalization of the MV2 characterization of the MOS property to [[ternary scale]]s. Other characterizations of the MOS property, such as [[distributional evenness]] and [[generator\|having a generator]], do not generalize to properties that are equivalent to MV3 in higher [[arity\|arities]].

	When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''~~unconditionally~~ MV3'' scale patterns ~~or ''abstractly MV3'' ones~~, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.		When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''abstractly MV3'' scale patterns, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.
	There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].		There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].

Inthar: /* Max-variety-3 scales */

2024-02-09T15:50:08Z

Max-variety-3 scales

← Older revision		Revision as of 15:50, 9 February 2024
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	When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.		When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.
	There is a theorem classifying MV3 scales; see [[Ternary scale theorems]].		There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].

	=== Generating MV3 scales ===		=== Generating MV3 scales ===

Inthar: /* Classification of MV3 scales */ Linked to the relevant article, so I won't have to maintain a separate statement anymore.

2024-02-09T15:45:14Z

Classification of MV3 scales: Linked to the relevant article, so I won't have to maintain a separate statement anymore.

← Older revision		Revision as of 15:45, 9 February 2024
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	When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.		When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.
	~~=== Classification of~~ MV3 scales ~~===~~		There is a theorem classifying MV3 scales; see [[Ternary scale theorems]].
	~~{{main\|~~Ternary scale theorems}}
	~~# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) [[generator-offset~~]] ~~with odd length, (4) constructed from 2aX 2bY with a odd and gcd(a, b) = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:~~
	~~## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.~~
	~~## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''.~~
	~~## Replace every other X with Y in ''w''.~~
	~~# Single-period MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the n/2-step (n = scale length) which is variety 2.~~
	~~# Single-period MV3 scales not of type (1) or (5) are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size~~.

	=== Generating MV3 scales ===		=== Generating MV3 scales ===

Inthar: /* Conditional */

2023-12-25T00:34:29Z

Conditional

← Older revision		Revision as of 00:34, 25 December 2023
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	So this scale is Conditionally MV3 (MM=Ls).		So this scale is Conditionally MV3 (MM=Ls).

	[[Category:Scale]]		[[Category:Scale]][[Category:Terms]]
	[[Category:Terms]]
	~~[[Category:Combinatorics on words~~]]

Inthar: /* Mathematical overview */ perio

2023-12-25T00:28:57Z

Mathematical overview: perio

← Older revision		Revision as of 00:28, 25 December 2023
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	Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.		Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
	== Mathematical overview ==		== Mathematical overview ==
	A standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρ<sup>ab</sup> : '''N''' -> '''N''' where ρ<sup>ab</sup>(''n'') is the number of distinct "sizes" that length-''n'' subwords can have in a word		A standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρ<sup>ab</sup> : '''N''' -> '''N''' where ρ<sup>ab</sup>(''n'') is the number of distinct "sizes" that length-''n'' subwords can have in a word.
	* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.		* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.

Inthar: /* Mathematical overview */ abelian complexity is a function rather than a maximum of that function

2023-12-25T00:28:37Z

Mathematical overview: abelian complexity is a function rather than a maximum of that function

← Older revision		Revision as of 00:28, 25 December 2023
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	Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.		Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
	== Mathematical overview ==		== Mathematical overview ==
	A standard academic counterpart to ~~this~~ xen term is the ''abelian complexity of a [[word]]''.		A standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρ<sup>ab</sup> : '''N''' -> '''N''' where ρ<sup>ab</sup>(''n'') is the number of distinct "sizes" that length-''n'' subwords can have in a word
	* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.		* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.

Inthar: /* Mathematical overview */

2023-12-25T00:21:32Z

Mathematical overview

← Older revision		Revision as of 00:21, 25 December 2023
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	Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.		Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
	== Mathematical overview ==		== Mathematical overview ==
	A standard academic counterpart to this xen term is the ~~'''~~''abelian complexity~~'''~~ of a [[word]]''.		A standard academic counterpart to this xen term is the ''abelian complexity of a [[word]]''.
	* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.		* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.

	==Max-variety-3 scales==		==Max-variety-3 scales==
	The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#Definition\|as well as in other equivalent ways]]: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) is the generalization of the MV2 characterization of the MOS property to [[ternary scale]]s. Other characterizations of the MOS property, such as [[distributional evenness]] and [[generator\|having a generator]], do not generalize to properties that are equivalent to MV3 in higher [[arity\|arities]].		The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#Definition\|as well as in other equivalent ways]]: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) is the generalization of the MV2 characterization of the MOS property to [[ternary scale]]s. Other characterizations of the MOS property, such as [[distributional evenness]] and [[generator\|having a generator]], do not generalize to properties that are equivalent to MV3 in higher [[arity\|arities]].

Inthar at 00:20, 25 December 2023

2023-12-25T00:20:25Z

← Older revision		Revision as of 00:20, 25 December 2023
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	Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.		Any scale with all equal steps (such as an [[EDO\|EDO]]) has maximum variety 1. All [[MOSScales\|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series\|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
	== Mathematical ~~properties~~ ==		== Mathematical overview ==
			A standard academic counterpart to this xen term is the '''''abelian complexity''' of a [[word]]''.
	* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.		* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.
	==Max-variety-3 scales==		==Max-variety-3 scales==