Generator preimage - Revision history

Sintel: typo again

2025-03-28T00:22:56Z

typo again

← Older revision		Revision as of 00:22, 28 March 2025
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	Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.		Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

	Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is ~~generator~~ by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].		Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generated by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].

	=Technical Definition=		=Technical Definition=

Sintel: typo

2025-03-28T00:20:46Z

typo

← Older revision		Revision as of 00:20, 28 March 2025
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	Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.		Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

	Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generator by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. ~~For~~ [[~~miracle~~]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].		Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generator by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].

	=Technical Definition=		=Technical Definition=

Sintel: phrasing

2025-03-28T00:20:31Z

phrasing

← Older revision		Revision as of 00:20, 28 March 2025
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	Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.		Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

	Generator preimages are commonly used to describe temperaments. For example, ~~the preimage generators of~~ [[meantone]] ~~are~~ [[2/1]] and [[3/2]]~~, and those of~~ [[miracle]] ~~are~~ [[2/1]] and [[16/15]].		Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generator by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. For [[miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].

	=Technical Definition=		=Technical Definition=

Sintel: add some basic examples to intro

2025-03-28T00:14:40Z

add some basic examples to intro

← Older revision		Revision as of 00:14, 28 March 2025
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	Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.		Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

			Generator preimages are commonly used to describe temperaments. For example, the preimage generators of [[meantone]] are [[2/1]] and [[3/2]], and those of [[miracle]] are [[2/1]] and [[16/15]].

	=Technical Definition=		=Technical Definition=

Sintel: Sintel moved page Transversal generators to Generator preimage: Use more common term over transversal

2025-03-27T23:26:09Z

Sintel moved page Transversal generators to Generator preimage: Use more common term over transversal

← Older revision	Revision as of 23:26, 27 March 2025
(No difference)

Cmloegcmluin: I agree it's no good that this page doesn't link to Detempering, but isn't it better to give the explanatory link in the opening, since it's half of the name for this concept?

2025-03-26T19:28:23Z

I agree it's no good that this page doesn't link to Detempering, but isn't it better to give the explanatory link in the opening, since it's half of the name for this concept?

← Older revision		Revision as of 19:28, 26 March 2025
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	Also known as a '''generator preimage transversal''' or a '''generator detempering'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.		Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

	=Technical Definition=		=Technical Definition=
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	getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)		getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
	</syntaxhighlight>}}		</syntaxhighlight>}}

	~~== See also ==~~
	* [[Detempering]]

	[[Category:generator]]		[[Category:generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:todo:reduce_mathslang]]		[[Category:todo:reduce_mathslang]]

Sintel: cleanup template text

2025-03-26T19:24:43Z

cleanup template text

← Older revision		Revision as of 19:24, 26 March 2025
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	=Finding the generator preimage transversal=		=Finding the generator preimage transversal=
	{{todo\|cleanup}}		{{todo\|cleanup\|inline=1\|text=Add simpler algorithm}}

	Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel\|Sintel]] in December 2021, which uses the [[Smith normal form]].		Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel\|Sintel]] in December 2021, which uses the [[Smith normal form]].

Sintel: todo cleanup, see also

2025-03-26T19:23:24Z

todo cleanup, see also

← Older revision		Revision as of 19:23, 26 March 2025
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	=Finding the generator preimage transversal=		=Finding the generator preimage transversal=
			{{todo\|cleanup}}

	Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel\|Sintel]] in December 2021, which uses the [[Smith normal form]].		Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel\|Sintel]] in December 2021, which uses the [[Smith normal form]].
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	getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)		getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
	</syntaxhighlight>}}		</syntaxhighlight>}}

			== See also ==
			* [[Detempering]]

	[[Category:generator]]		[[Category:generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:todo:reduce_mathslang]]		[[Category:todo:reduce_mathslang]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:48:32Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:48, 5 March 2023
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	[[Category:generator]]		[[Category:generator]]
	~~[[Category:theory]]~~
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:todo:reduce_mathslang]]		[[Category:todo:reduce_mathslang]]

Cmloegcmluin: /* Method using the Smith Normal Form */ add link

2023-02-21T18:18:05Z

Method using the Smith Normal Form: add link

← Older revision		Revision as of 18:18, 21 February 2023
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	and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a prime-count vector represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.		and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.

	Essentially we need a way to do:		Essentially we need a way to do: