Generator form manipulation - Revision history

Dummy index: add ja interwiki

2025-07-28T16:12:05Z

add ja interwiki

← Older revision		Revision as of 16:12, 28 July 2025
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			{{Interwiki
			\| en = Generator form manipulation
			\| ja = ジェネレーター読み替え操作
			}}
	A [[canonical_form\|canonical mapping form]] is an important standard to have as a community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]] (to ultimately generate the same tempered intervals). Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form\|positive generator form]], [[Normal_lists#Equave-reduced_generator_form\|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form\|minimal-generator form]].		A [[canonical_form\|canonical mapping form]] is an important standard to have as a community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]] (to ultimately generate the same tempered intervals). Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form\|positive generator form]], [[Normal_lists#Equave-reduced_generator_form\|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form\|minimal-generator form]].

Cmloegcmluin: /* Easy instructions table to achieve mingen form for rank-2 mapping */

2024-06-16T16:30:58Z

Easy instructions table to achieve mingen form for rank-2 mapping

← Older revision		Revision as of 16:30, 16 June 2024
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	\|1201.4		\|1201.4
	\|−1898.4		\|−1898.4
	\|g < −p		\|<math>g < −p</math>
	\|g + p		\|<math>g + p</math>
	\|𝒎₁ − 2𝒎₂		\|<math>𝒎₁ − 2𝒎₂</math>
	\|		\|
	\|yes		\|yes
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	\|1201.4		\|1201.4
	\|−697.049		\|−697.049
	\|−p <= g < −p/2		\|<math>−p <= g < −p/2</math>
	\|p + g		\|<math>p + g</math>
	\|𝒎₁ − 𝒎₂		\|<math>𝒎₁ − 𝒎₂</math>
	\|		\|
	\|no, you're done		\|no, you're done
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	\|1201.4		\|1201.4
	\|−504.348		\|−504.348
	\|−p/2 <= g < 0		\|<math>−p/2 <= g < 0</math>
	\|−g		\|<math>−g</math>
	\|		\|
	\|−𝒎₂		\|<math>−𝒎₂</math>
	\|no, you're done		\|no, you're done
	\|-		\|-
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	\|1201.4		\|1201.4
	\|504.4		\|504.4
	\|0 <= g <= p/2		\|<math>0 <= g <= p/2</math>
	\|g		\|<math>g</math>
	\|		\|
	\|		\|
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	\|1201.4		\|1201.4
	\|697.049		\|697.049
	\|p/2 < g <= p		\|<math>p/2 < g <= p</math>
	\|p - g		\|<math>p - g</math>
	\|𝒎₁ + 𝒎₂		\|<math>𝒎₁ + 𝒎₂</math>
	\|−𝒎₂		\|<math>−𝒎₂</math>
	\|no, you're done		\|no, you're done
	\|-		\|-
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	\|1201.4		\|1201.4
	\|1898.4		\|1898.4
	\|p < g		\|<math>p < g</math>
	\|g - p		\|<math>g - p</math>
	\|𝒎₁ + 2𝒎₂		\|<math>𝒎₁ + 2𝒎₂</math>
	\|−𝒎₂		\|<math>−𝒎₂</math>
	\|yes		\|yes
	\|}		\|}

Cmloegcmluin: /* Tuning strategy */

2024-06-16T16:28:36Z

Tuning strategy

← Older revision		Revision as of 16:28, 16 June 2024
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	Using these two tricks, you do not have to worry about enfactoring the mapping, i.e. introducing a common factor in one of the maps. This is because neither of these tricks ever involve replacing a map with a multiple of that map; we always replace a map with a combination of at least one each of two different maps, as in trick 1, or with the map negated, as in trick 2.		Using these two tricks, you do not have to worry about enfactoring the mapping, i.e. introducing a common factor in one of the maps. This is because neither of these tricks ever involve replacing a map with a multiple of that map; we always replace a map with a combination of at least one each of two different maps, as in trick 1, or with the map negated, as in trick 2.

	== Tuning ~~strategy~~ ==		== Tuning ==

	In order to ~~define~~ the size of the generators, you need to ~~specify~~ a tuning ~~strategy. Though if you have two generators that are close enough that their size ranking depends on the tuning, then you probably have other problems~~. ~~In any case, the~~ tuning ~~strategy that~~ we'~~ll be~~ using here is [[minimax-ES]], because ~~it's~~ decent enough and easy to compute.<ref>Note from Douglas Blumeyer: though if I had written this article today having done a lot of tuning theory and built a library for optimizing tunings, I would have gone with TILT minimax-U instead.</ref>		In order to determine the relative size of the generators, you need to have chosen a tuning for each them. The tuning we're using here comes from the [[minimax-ES]] tuning scheme, because this scheme is decent enough and easy to compute.<ref>Note from Douglas Blumeyer: though if I had written this article today having done a lot of tuning theory and built a library for optimizing tunings, I would have gone with TILT minimax-U instead.</ref>

	== Easy instructions table to achieve mingen form for rank-2 mapping ==		== Easy instructions table to achieve mingen form for rank-2 mapping ==

Cmloegcmluin: /* Second trick: negating a generator */

2024-06-16T16:24:44Z

Second trick: negating a generator

← Older revision		Revision as of 16:24, 16 June 2024
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	So at this point, the generators are an octave and a negative perfect fifth. It's time for our second step.		So at this point, the generators are an octave and a negative perfect fifth. It's time for our second step.

	The second step of achieving a generator with the size of a perfect fourth should be familiar: we need to increase our negative perfect fifth to a perfect fourth, and so we need to add one octave, and therefore we need to change <math>𝒎_1</math> to be <math>𝒎_1 - ~~v_2~~</math>, so we end up with:		The second step of achieving a generator with the size of a perfect fourth should be familiar: we need to increase our negative perfect fifth to a perfect fourth, and so we need to add one octave, and therefore we need to change <math>𝒎_1</math> to be <math>𝒎_1 - 𝒎_2</math>, so we end up with:

	<math>		<math>
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	</math>		</math>

	And we're done, having found the mapping {{rket\|{{map\|1 2 4}} {{map\|0 -1 -4}}}}.		And we're done, having found the mapping {{rket\|{{map\|1 2 4}} {{map\|0 -1 -4}}}}.

	=== Beyond rank-2 ===		=== Beyond rank-2 ===

Cmloegcmluin: update templates, fix typo, clarifying footnotes

2022-12-07T08:16:48Z

update templates, fix typo, clarifying footnotes

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Cmloegcmluin: Cmloegcmluin moved page Generator size manipulation to Generator form manipulation: "size" is misleading, suggesting exact sizes in cents, when this is really about a more abstract concept of relative size and size category

2022-12-07T08:15:38Z

Cmloegcmluin moved page Generator size manipulation to Generator form manipulation: "size" is misleading, suggesting exact sizes in cents, when this is really about a more abstract concept of relative size and size category

← Older revision	Revision as of 08:15, 7 December 2022
(No difference)

Cmloegcmluin: prefer m for map to r for row, for better consistency with mathematical conventions of using the lowercase version of the same letter for a row/col of a matrix

2022-11-01T18:00:53Z

prefer m for map to r for row, for better consistency with mathematical conventions of using the lowercase version of the same letter for a row/col of a matrix

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Cmloegcmluin: standardizing in our articles to bold italic for maps

2022-10-25T19:01:16Z

standardizing in our articles to bold italic for maps

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Cmloegcmluin: the name "Frobenius" should not be used; T2 is a better name for this tuning

2022-03-14T03:44:45Z

the name "Frobenius" should not be used; T2 is a better name for this tuning

← Older revision		Revision as of 03:44, 14 March 2022
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	=== Wolfram Language implementation of mingen form instructions above ===		=== Wolfram Language implementation of mingen form instructions above ===

	The below code essentially works through the input matrix M two rows at a time, beginning with the first two rows. Each pair of rows is "fixed" so that the second row is less than half of the first row. The same set of changes and potential recursions as described in the table in the previous section is used for each pair of rows. When the sizes of the generators is computed, ~~a weighted Frobenius~~ tuning is used for its computational frugality and reasonableness.		The below code essentially works through the input matrix M two rows at a time, beginning with the first two rows. Each pair of rows is "fixed" so that the second row is less than half of the first row. The same set of changes and potential recursions as described in the table in the previous section is used for each pair of rows. When the sizes of the generators is computed, T2 tuning is used for its computational frugality and reasonableness.

	<nowiki>		<nowiki>

Cmloegcmluin: use conventional mathematical formatting of variables

2022-02-28T20:06:07Z

use conventional mathematical formatting of variables

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