Equivalence continuum: Difference between revisions

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~~{{novelty}}~~

~~{{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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== 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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* [[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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-{{novelty}}
-{{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 22: / Line 20: @@
 == 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'''}}.
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 * [[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:
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 ** 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]])