Equivalence continuum: Difference between revisions

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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''1, and the second interval, which we may call the targeted interval, is ''q''2, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap| {{subsup|''q''|1|''n''}} ~ ''q''2 }}, 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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This guarantees that in the corresponding temperament, ''n'' equals the order of the last formal prime in the comma, and equals the number of steps to obtain the interval class of the second formal prime in the generator chain.

=== 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''2/''q''1)''m'' ~ ''q''2. Here the stacked interval is ''q''2/''q''1, and the targeted interval remains ''q''2. For instance, the inversion of the syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)''m'' ~ 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''1 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.

== Geometric interpretation ==

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'''{{nowrap|''n'' − ''r''}}, identifying '''R'''{{nowrap|''n'' − ''r''}} with the '''R'''-vector space {{nowrap|ker(''T'') &otimes; '''R'''}}.

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* [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''''x'' − '''u'''''y''}} {{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~~]].''~~

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.

* [[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 [[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]])

* [[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:

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** 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]])

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[[Category:Math]]

[[Category:Regular temperament theory]]

~~[[Category:Equivalence continua| ]] ~~

@@ Line 1: / Line 1: @@
-{{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.
@@ Line 13: / Line 13: @@
 This guarantees that in the corresponding temperament, ''n'' equals the order of the last formal prime in the comma, and equals the number of steps to obtain the interval class of the second formal prime in the generator chain.
+=== 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 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.
 == Geometric interpretation ==
+{{Inacc}}
 Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'',&nbsp;''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'''}}.
@@ Line 33: / Line 40: @@
 * [[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.
 * [[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 [[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]])
-* [[2.3.7&nbsp;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&nbsp;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&nbsp;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:
@@ Line 69: / Line 76: @@
 ** the [[breedsmic–syntonic equivalence continuum]] ([[squares]])
-* [[2.3.5.11&nbsp;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]])
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 [[Category:Math]]
 [[Category:Regular temperament theory]]
-[[Category:Equivalence continua| ]] <!-- main article -->