Equivalence continuum - Revision history

Xenllium: /* List of equivalence continua */

2026-02-08T09:46:07Z

List of equivalence continua

← Older revision		Revision as of 09:46, 8 February 2026
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	** the [[syntonic–ptolemismic equivalence continuum]] ([[calliope]])		** the [[syntonic–ptolemismic equivalence continuum]] ([[calliope]])
	** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])		** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])
			** the [[biyatismic–ptolemismic equivalence continuum]] ([[porkypine]])
	** the [[rastmic–ptolemismic equivalence continuum]] ([[tetracot]])		** the [[rastmic–ptolemismic equivalence continuum]] ([[tetracot]])

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	* [[11-limit]] rank-4 continua include:		* [[11-limit]] rank-4 continua include:
			** the [[jubilismic equivalence continuum]] ([[Jubilismic\|undecimal jubilismic]])
	** the [[keemic equivalence continuum]] ([[Keemic\|undecimal keemic]])		** the [[keemic equivalence continuum]] ([[Keemic\|undecimal keemic]])
	** the [[marvel–keenanismic equivalence continuum]] ([[Marvel\|undecimal marvel]])		** the [[marvel–keenanismic equivalence continuum]] ([[Marvel\|undecimal marvel]])

Xenllium: /* List of equivalence continua */

2026-01-31T00:46:28Z

List of equivalence continua

← Older revision		Revision as of 00:46, 31 January 2026
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	* [[7-limit]] rank-3 continua include:		* [[7-limit]] rank-3 continua include:
			** the [[semaphore–chromatic equivalence continuum]] ([[decimal]])
	** the [[syntonic–Archytas equivalence continuum]] ([[dominant (temperament)\|dominant]])		** the [[syntonic–Archytas equivalence continuum]] ([[dominant (temperament)\|dominant]])
	** the [[marvel–syntonic equivalence continuum]] ([[septimal meantone]])		** the [[marvel–syntonic equivalence continuum]] ([[septimal meantone]])
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	* [[11-limit]] rank-4 continua include:		* [[11-limit]] rank-4 continua include:
			** the [[keemic equivalence continuum]] ([[Keemic\|undecimal keemic]])
	** the [[marvel–keenanismic equivalence continuum]] ([[Marvel\|undecimal marvel]])		** the [[marvel–keenanismic equivalence continuum]] ([[Marvel\|undecimal marvel]])

Xenllium at 23:04, 30 January 2026

2026-01-30T23:04:21Z

← Older revision		Revision as of 23:04, 30 January 2026
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	** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])		** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])
	** the [[rastmic–ptolemismic equivalence continuum]] ([[tetracot]])		** the [[rastmic–ptolemismic equivalence continuum]] ([[tetracot]])

			* [[2.3.7.11 subgroup\|2.3.7.11-subgroup]] rank-3 continua include:
			** the [[Archytas–mothwellsmic equivalence continuum]] ([[supra]])

			* [[11-limit]] rank-4 continua include:
			** the [[marvel–keenanismic equivalence continuum]] ([[Marvel\|undecimal marvel]])

	== References ==		== References ==

Xenllium at 22:56, 30 January 2026

2026-01-30T22:56:53Z

← Older revision		Revision as of 22:56, 30 January 2026
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	* [[7-limit]] rank-3 continua include:		* [[7-limit]] rank-3 continua include:
	** the [[syntonic–Archytas equivalence continuum]] ([[dominant (temperament)\|dominant]])		** the [[syntonic–Archytas equivalence continuum]] ([[dominant (temperament)\|dominant]])
	** the [[~~Marvel–syntonic~~ equivalence continuum]] ([[septimal meantone]])		** the [[marvel–syntonic equivalence continuum]] ([[septimal meantone]])
	** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])		** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])
	** the [[~~Marvel–sensamagic~~ equivalence continuum]] ([[Magic\|septimal magic]])		** the [[marvel–sensamagic equivalence continuum]] ([[Magic\|septimal magic]])
	** the [[~~Starling–sensamagic~~ equivalence continuum]] ([[Sensi\|septimal sensi]])		** the [[starling–sensamagic equivalence continuum]] ([[Sensi\|septimal sensi]])
			** the [[marvel–breedsmic equivalence continuum]] ([[Miracle\|septimal miracle]])
			** the [[ragismic–breedsmic equivalence continuum]] ([[Ennealimmal\|septimal ennealimmal]])

	* [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] rank-3 continua include:		* [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] rank-3 continua include:
			** the [[syntonic–ptolemismic equivalence continuum]] ([[calliope]])
	** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])		** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])
			** the [[rastmic–ptolemismic equivalence continuum]] ([[tetracot]])

	== References ==		== References ==

Xenllium: /* List of equivalence continua */

2026-01-09T12:45:07Z

List of equivalence continua

← Older revision		Revision as of 12:45, 9 January 2026
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	** the [[Marvel–syntonic equivalence continuum]] ([[septimal meantone]])		** the [[Marvel–syntonic equivalence continuum]] ([[septimal meantone]])
	** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])		** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])
			** the [[Marvel–sensamagic equivalence continuum]] ([[Magic\|septimal magic]])
			** the [[Starling–sensamagic equivalence continuum]] ([[Sensi\|septimal sensi]])

	* [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] rank-3 continua include:		* [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] rank-3 continua include:

FloraC: /* List of equivalence continua */ + justifications for most of these entries

2025-07-28T16:17:12Z

List of equivalence continua: + justifications for most of these entries

FloraC: Move the messagebox to the specific section that deserves it

2025-03-22T09:30:47Z

Move the messagebox to the specific section that deserves it

← Older revision		Revision as of 09:30, 22 March 2025
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	~~{{Inacc}}~~
	An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.		An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.

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	== Geometric interpretation ==		== Geometric interpretation ==
			{{Inacc}}

	Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'') associated with a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' is the space of [[Mathematical theory of saturation\|saturated]] ({{nowrap\|''n − k''}})-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-({{nowrap\|''n − r''}}) lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian {{nowrap\|'''G''' {{=}} '''Gr'''(''n − k'', ''n − r'')}} of ({{nowrap\|''n − k''}})-dimensional vector subspaces of '''R'''<sup>{{nowrap\|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap\|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap\|ker(''T'') &otimes; '''R'''}}.		Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'') associated with a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' is the space of [[Mathematical theory of saturation\|saturated]] ({{nowrap\|''n − k''}})-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-({{nowrap\|''n − r''}}) lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian {{nowrap\|'''G''' {{=}} '''Gr'''(''n − k'', ''n − r'')}} of ({{nowrap\|''n − k''}})-dimensional vector subspaces of '''R'''<sup>{{nowrap\|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap\|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap\|ker(''T'') &otimes; '''R'''}}.

FloraC: Undo revision 185662 by Domin (talk). This is not novelty. It's a very insightful way to classify temperaments

2025-03-13T10:51:02Z

Undo revision 185662 by Domin (talk). This is not novelty. It's a very insightful way to classify temperaments

← Older revision		Revision as of 10:51, 13 March 2025
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	~~{{novelty}}~~

	{{Inacc}}		{{Inacc}}
	An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.		An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.

Domin at 10:26, 13 March 2025

2025-03-13T10:26:35Z

← Older revision		Revision as of 10:26, 13 March 2025
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			{{novelty}}

	{{Inacc}}		{{Inacc}}
	An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.		An '''equivalence continuum''' comprises all the [[regular temperament\|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap\| {{subsup\|''q''\|1\|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup.

FloraC: Section title; -category (category: Equivalence continua is a set category)

2025-03-12T13:13:48Z

Section title; -category (category: Equivalence continua is a set category)

← Older revision		Revision as of 13:13, 12 March 2025
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	=== Inversion ===		=== Inversion ===
	A continuum can be inverted by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}, with temperaments in it characterized by the relation (''q''<sub>2</sub>/''q''<sub>1</sub>)<sup>''m''</sup> ~ ''q''<sub>2</sub>. Here the stacked interval is ''q''<sub>2</sub>/''q''<sub>1</sub>, and the targeted interval remains ''q''<sub>2</sub>. For instance, the inversion of syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)<sup>''m''</sup> ~ 2187/2048.		A continuum can be inverted by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}, with temperaments in it characterized by the relation (''q''<sub>2</sub>/''q''<sub>1</sub>)<sup>''m''</sup> ~ ''q''<sub>2</sub>. Here the stacked interval is ''q''<sub>2</sub>/''q''<sub>1</sub>, and the targeted interval remains ''q''<sub>2</sub>. For instance, the inversion of the syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)<sup>''m''</sup> ~ 2187/2048.

	This ''m''-continuum, like the ''n''-continuum, also meets the requirements for a possible default choice, and raises the question which one should be the ''n''-continuum and which one should be the ''m''-continuum. In principle, we take the ''n''-continuum as the main continuum and the ''m''-continuum supplementary. If one of the candidate stacked intervals is simpler ''and'' smaller, we set it to ''q''<sub>1</sub> of the ''n''-continuum so that more useful temperaments are included in it. However, the simpler interval is sometimes the larger one, in which case the choice could be made on a heuristic basis.		This ''m''-continuum, like the ''n''-continuum, also meets the requirements for a possible default choice, and raises the question which one should be the ''n''-continuum and which one should be the ''m''-continuum. In principle, we take the ''n''-continuum as the main continuum and the ''m''-continuum supplementary. If one of the candidate stacked intervals is simpler ''and'' smaller, we set it to ''q''<sub>1</sub> of the ''n''-continuum so that more useful temperaments are included in it. However, the simpler interval is sometimes the larger one, in which case the choice could be made on a heuristic basis.
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	* [[Miracle]] tempers out {{nowrap\|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap\|{{=}} (0, 0, 1)}} and {{nowrap\|225/224 {{=}} '''u'''<sub>''x''</sub> − '''u'''<sub>''y''</sub>}} {{nowrap\|{{=}} (1, −1, 0)}}. This corresponds to {{nowrap\|'''v''' {{=}} (1, 1, 0)}}.		* [[Miracle]] tempers out {{nowrap\|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap\|{{=}} (0, 0, 1)}} and {{nowrap\|225/224 {{=}} '''u'''<sub>''x''</sub> − '''u'''<sub>''y''</sub>}} {{nowrap\|{{=}} (1, −1, 0)}}. This corresponds to {{nowrap\|'''v''' {{=}} (1, 1, 0)}}.

	== ~~Examples~~ ==		== List of equivalence continua ==
	''See also ~~[[:Category:Equivalence continua~~\|Category:Equivalence continua~~]].''~~		{{See also\| Category: Equivalence continua }}

	All equivalence continua currently on the wiki are rank-{{nowrap\|(''n'' + 1)}} continua of rank-{{nowrap\|(''n'' + 1)}} temperaments within a rank-{{nowrap\|(''n'' + 2)}} subgroup that are supported by a rank-''n'' system.		All equivalence continua currently on the wiki are rank-{{nowrap\|(''n'' + 1)}} continua of rank-{{nowrap\|(''n'' + 1)}} temperaments within a rank-{{nowrap\|(''n'' + 2)}} subgroup that are supported by a rank-''n'' system.
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Equivalence continua\| ]] <!-- main article -->~~

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	* [[5-limit]] rank-2 continua include:		* [[5-limit]] rank-2 continua include:
	** the [[father–3 equivalence continuum]] ([[3edo]])		** the [[father–3 equivalence continuum]] ([[3edo]], a 3- and 5-limit record edo)
	** the [[syntonic–diatonic equivalence continuum]] ([[5edo]])		** the [[syntonic–diatonic equivalence continuum]] ([[5edo]], a 3- and 5-limit record edo)
	** the [[syntonic–chromatic equivalence continuum]] ([[7edo]])		** the [[syntonic–chromatic equivalence continuum]] ([[7edo]], a 3- and 5-limit record edo)
	** the [[schismic–Pythagorean equivalence continuum]] ([[12edo]])		** the [[schismic–Pythagorean equivalence continuum]] ([[12edo]], a 3- and 5-limit record edo)
	** the [[syntonic–kleismic equivalence continuum]] ([[19edo]])		** the [[syntonic–kleismic equivalence continuum]] ([[19edo]], a 5-limit record edo)
	** the [[superpyth–22 equivalence continuum]] ([[22edo]])		** the [[superpyth–22 equivalence continuum]] ([[22edo]])
	** the [[syntonic–31 equivalence continuum]] ([[31edo]])		** the [[syntonic–31 equivalence continuum]] ([[31edo]], a 5-limit record edo)
	** the [[diaschismic–gothmic equivalence continuum]] ([[34edo]])		** the [[diaschismic–gothmic equivalence continuum]] ([[34edo]], a 5-limit record edo)
	** the [[schismic–countercommatic equivalence continuum]] ([[41edo]])		** the [[schismic–countercommatic equivalence continuum]] ([[41edo]], a 3-limit record edo)
	** the [[schismic–Mercator equivalence continuum]] ([[53edo]])		** the [[schismic–Mercator equivalence continuum]] ([[53edo]], a 3- and 5-limit record edo)
	** the [[ennealimmal–enneadecal equivalence continuum]] ([[171edo]])		** the [[ennealimmal–enneadecal equivalence continuum]] ([[171edo]], a 5-limit record edo)
	** the [[tarot equivalence continuum]] ([[1848edo]])		** the [[tarot equivalence continuum]] ([[1848edo]])

	* [[2.3.7~~ ~~subgroup]] rank-2 continua include:		* [[2.3.7 subgroup\|2.3.7-subgroup]] rank-2 continua include:
	** the [[~~Archytas–chromatic~~ equivalence continuum]] ([[~~7edo~~]])		** the [[Archytas–diatonic equivalence continuum]] ([[5edo]], a 3-limit and 2.3.7-subgroup record edo)
	** the [[~~Archytas–diatonic~~ equivalence continuum]] ([[~~5edo~~]])		** the [[Archytas–chromatic equivalence continuum]] ([[7edo]], a 3-limit record edo)

	* [[2.5.7~~ ~~subgroup]] rank-2 continua include:		* [[2.5.7 subgroup\|2.5.7-subgroup]] rank-2 continua include:
	** the [[jubilismic–augmented equivalence continuum]] ([[6edo]])		** the [[jubilismic–augmented equivalence continuum]] ([[6edo]], a 2.5.7-subgroup record edo)
	** the [[augmented–cloudy equivalence continuum]] ([[15edo]])		** the [[augmented–cloudy equivalence continuum]] ([[15edo]], a 2.5.7-subgroup record edo)
	** the [[rainy–didacus equivalence continuum]] ([[31edo]])		** the [[rainy–didacus equivalence continuum]] ([[31edo]], a 2.5.7-subgroup record edo)

	* [[3.5.7~~ ~~subgroup]] rank-2 continua include:		* [[3.5.7 subgroup\|3.5.7-subgroup]] rank-2 continua include:
	** the [[sensamagic–gariboh equivalence continuum]] ([[13edt]])		** the [[sensamagic–gariboh equivalence continuum]] ([[13edt]], a 3.5.7-subgroup record edo)

	* [[7-limit]] rank-3 continua include:		* [[7-limit]] rank-3 continua include:
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	** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])		** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])

	* [[2.3.5.11~~ ~~subgroup]] rank-3 continua include:		* [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] rank-3 continua include:
	** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])		** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]])