Convex scale - Revision history

Hkm at 20:39, 25 June 2025

2025-06-25T20:39:39Z

← Older revision		Revision as of 20:39, 25 June 2025
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	{{Todo}}[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]		{{Todo\|inline=1\|expand\|comment=explain musical application --[[User:Hkm\|hkm]] ([[User talk:Hkm\|talk]]) 20:39, 25 June 2025 (UTC)}}[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]

	In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

Hkm at 20:39, 25 June 2025

2025-06-25T20:39:14Z

← Older revision		Revision as of 20:39, 25 June 2025
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	[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]		{{Todo}}[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]

	In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

Hkm: too much formality, man. you're making it hard to read

2025-06-25T20:37:51Z

too much formality, man. you're making it hard to read

← Older revision		Revision as of 20:37, 25 June 2025
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	[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]		[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]

	In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

	~~{{todo\|clarify\|comment=Is there maybe somebody who can explain this~~ in ~~plain English?\|increase applicability\|inline=1}}~~		A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. Alternatively, a convex set in a lattice is a set where any weighted average of elements (where no element has negative weight) is within the set if it is on the lattice.

	A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.		The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. (Every scale has a unique convex hull.) See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

	The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

	~~==Formal definition==~~
	The following definitions make sense in the context of any Z-[https://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [https://en.wikipedia.org/wiki/Abelian_group abelian group].

	~~===Convex combination===~~
	A '''convex combination''' of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that

	~~<math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>~~

	Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [https://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by

	~~<math>$c = c_1 + c_2 + \dots + c_k$</math>~~

	~~we obtain~~

	~~<math>$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</math>~~

	~~where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.~~

	~~===Convex set===~~
	A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.

	==Examples==		==Examples==
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	* Every [[Fokker block]] is convex.		* Every [[Fokker block]] is convex.
	* Every untempered [[tonality diamond]] is convex.		* Every untempered [[tonality diamond]] is convex.
	* [[Gallery of Z-polygon transversals]]

	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Todo:clarify]]		[[Category:Todo:clarify]]

BudjarnLambeth: I noticed the article had a todo:clarify tag in the article, but it wasn't in the todo:clarify category. So I added it to the todo:clarify category.

2023-04-18T07:38:43Z

I noticed the article had a todo:clarify tag in the article, but it wasn't in the todo:clarify category. So I added it to the todo:clarify category.

← Older revision		Revision as of 07:38, 18 April 2023
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	{{todo\|clarify\|comment=Is there maybe somebody who can explain this in plain English?\|increase applicability\|inline=1}}		{{todo\|clarify\|comment=Is there maybe somebody who can explain this in plain English?\|increase applicability\|inline=1}}

	A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [~~http~~://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.		A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.

	The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.		The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

	==Formal definition==		==Formal definition==
	The following definitions make sense in the context of any Z-[~~http~~://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [~~http~~://en.wikipedia.org/wiki/Abelian_group abelian group].		The following definitions make sense in the context of any Z-[https://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [https://en.wikipedia.org/wiki/Abelian_group abelian group].

	===Convex combination===		===Convex combination===
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	<math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>		<math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>

	Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [~~http~~://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by		Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [https://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by

	<math>$c = c_1 + c_2 + \dots + c_k$</math>		<math>$c = c_1 + c_2 + \dots + c_k$</math>
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Math]]		[[Category:Math]]
			[[Category:Todo:clarify]]

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:51Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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	* [[Gallery of Z-polygon transversals]]		* [[Gallery of Z-polygon transversals]]

	[[Category:Scale ~~theory~~]]		[[Category:Scale]]
	[[Category:Math]]		[[Category:Math]]

Fredg999: Categories

2021-08-17T22:27:53Z

Fredg999: Categories

2021-08-15T17:51:20Z

Fredg999: Added an illustration

2021-08-15T17:50:43Z

Added an illustration

← Older revision		Revision as of 17:50, 15 August 2021
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			[[File:Lattice Marvel Convex12.png\|400px\|thumb\|A convex set of 12 tones from the marvel lattice.]]

	In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

Xenwolf: is there maybe somebody who can explain this in plain English?

2021-01-01T23:38:45Z

is there maybe somebody who can explain this in plain English?

← Older revision		Revision as of 23:38, 1 January 2021
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	In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

			{{todo\|clarify\|comment=Is there maybe somebody who can explain this in plain English?\|increase applicability\|inline=1}}

	A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.		A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.

Xenwolf: some cleanup, cats

2021-01-01T23:30:55Z

some cleanup, cats

← Older revision		Revision as of 23:30, 1 January 2021
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	In a [[~~Regular_Temperaments\|~~regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[~~Just_intonation~~\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.		In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation\|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

	A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.		A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.

	The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[~~Gallery_of_Z-polygon_transversals\|~~Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.		The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

	==Formal definition==		==Formal definition==
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	==Examples==		==Examples==
	~~<ul><li>~~Every [[MOSScales\|MOS]] is convex.~~</li><li>~~In fact, every [[~~distributionally_even\|~~distributionally even]] scale is convex.~~</li><li>~~Every [[~~Fokker_blocks\|~~Fokker block]] is convex.~~</li><li>~~Every untempered [[~~Tonality_diamond\|~~tonality diamond]] is convex.~~</li><li>~~[[~~Gallery_of_Z-polygon_transversals\|~~Gallery of Z-polygon transversals]]~~</li></ul> [[Category:math]]~~
	[[Category:~~scale~~]]		* Every [[MOSScales\|MOS]] is convex.
	[[Category:theory]]		*In fact, every [[distributionally even]] scale is convex.
			* Every [[Fokker block]] is convex.
			* Every untempered [[tonality diamond]] is convex.
			* [[Gallery of Z-polygon transversals]]

			[[Category:Math]]
			[[Category:Scale theory]]

← Older revision		Revision as of 22:27, 17 August 2021
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	* [[Gallery of Z-polygon transversals]]		* [[Gallery of Z-polygon transversals]]

	[[Category:Scale]]		[[Category:Scale theory]]
	[[Category:Math]]		[[Category:Math]]

← Older revision		Revision as of 17:51, 15 August 2021
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	* [[Gallery of Z-polygon transversals]]		* [[Gallery of Z-polygon transversals]]

			[[Category:Scale]]
	[[Category:Math]]		[[Category:Math]]
	~~[[Category:Scale theory]]~~