Equivalence continuum: Difference between revisions

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An '''equivalence continuum''' is the space of all rank-''k'' temperaments on a specified [[JI subgroup]] that are supported by a specified temperament of a lower rank on the same subgroup (such as an [[edo]] viewed as a temperament on said subgroup). This term was first used by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref><ref>[https://en.xen.wiki/index.php?title=Temperament_orphanage&oldid=27177 Xenharmonic Wiki | ''Temperament orphanage''] – first occurrence on this wiki, same date as the thread above. </ref>.

~~Examples:~~

* The [[father-3 equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[3edo]].

* The [[Archytas-diatonic equivalence continuum]] is the 2.3.7 rank-2 equivalence continuum of [[5edo]].

* The [[syntonic-chromatic equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[7edo]].

* The [[schismic-Pythagorean equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[12edo]].

* The [[syntonic-Archytas equivalence continuum]] is a 7-limit rank-2 equivalence continuum of [[12edo]].

* The [[marvel-syntonic equivalence continuum]] and the [[breedsmic-syntonic equivalence continuum]] are 7-limit rank-2 equivalence continua of [[31edo]].

~~See [[:Category:Equivalence continua|Category:Equivalence continua]] for more.~~

== Geometric interpretation ==

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* [[hemithirds]] tempers out 1029/1024 = '''u'''''z'' = (0, 0, 1) and 3136/3125 = 2'''u'''''x'' + '''u'''''y'' = (2, 1, 0). This corresponds to '''v''' = (1, −2, 0).

* [[miracle]] tempers out 1029/1024 = '''u'''''z'' = (0, 0, 1) and 225/224 = '''u'''''x'' − '''u'''''y'' = (1, −1, 0). This corresponds to '''v''' = (1, 1, 0).

== Examples ==

''See also [[:Category:Equivalence continua|Category:Equivalence continua]].''

All equivalence continua currently on the wiki are rank-''n'+1 continua of (rank-''n'+1) temperaments within a rank-''n''+2 subgroup that are supported by a rank-''n'' system.

* [[5-limit]] rank-2 continua include:

** the [[father-3 equivalence continuum]] ([[3edo]])

** the [[syntonic-diatonic equivalence continuum]] ([[5edo]])

** the [[syntonic-chromatic equivalence continuum]] ([[7edo]])

** the [[schismic-Pythagorean equivalence continuum]] ([[12edo]])

** the [[syntonic-kleismic equivalence continuum]] ([[19edo]])

** the [[superpyth-22 equivalence continuum]] ([[22edo]])

** the [[syntonic-31 equivalence continuum]] ([[31edo]])

** the [[diaschismic-gothmic equivalence continuum]] ([[34edo]])

** the [[schismic-countercommatic equivalence continuum]] ([[41edo]])

** the [[schismic-Mercator equivalence continuum]] ([[53edo]])

** the [[ennealimmal-enneadecal equivalence continuum]] ([[171edo]])

** the [[tarot equivalence continuum]] ([[1848edo]])

* [[2.3.7 subgroup]] rank-2 continua include:

** the [[Archytas-diatonic equivalence continuum]] ([[5edo]])

* [[2.5.7 subgroup]] rank-2 continua include:

** the [[jubilismic-augmented equivalence continuum]] ([[6edo]])

** the [[augmented-cloudy equivalence continuum]] ([[15edo]])

** the [[rainy-didacus equivalence continuum]] ([[31edo]])

* [[3.5.7 subgroup]] rank-2 continua include:

** the [[sensamagic-gariboh equivalence continuum]] ([[13edt]])

* [[7-limit]] rank-3 continua include:

** the [[syntonic-Archytas equivalence continuum]] ([[dominant (temperament)|dominant]])

** the [[marvel-syntonic equivalence continuum]] ([[septimal meantone]])

** the [[breedsmic-syntonic equivalence continuum]] ([[squares]])

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

** the [[syntonic-rastmic equivalence continuum]] ([[mohaha]])

== Notes ==

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 An '''equivalence continuum''' is the space of all rank-''k'' temperaments on a specified [[JI subgroup]] that are supported by a specified temperament of a lower rank on the same subgroup (such as an [[edo]] viewed as a temperament on said subgroup). This term was first used by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref><ref>[https://en.xen.wiki/index.php?title=Temperament_orphanage&oldid=27177 Xenharmonic Wiki | ''Temperament orphanage''] – first occurrence on this wiki, same date as the thread above. </ref>.
-Examples:
-* The [[father-3 equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[3edo]].
-* The [[Archytas-diatonic equivalence continuum]] is the 2.3.7 rank-2 equivalence continuum of [[5edo]].
-* The [[syntonic-chromatic equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[7edo]].
-* The [[schismic-Pythagorean equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[12edo]].
-* The [[syntonic-Archytas equivalence continuum]] is a 7-limit rank-2 equivalence continuum of [[12edo]].
-* The [[marvel-syntonic equivalence continuum]] and the [[breedsmic-syntonic equivalence continuum]] are 7-limit rank-2 equivalence continua of [[31edo]].
-See [[:Category:Equivalence continua|Category:Equivalence continua]] for more.
 == Geometric interpretation ==
@@ Line 29: / Line 19: @@
 * [[hemithirds]] tempers out 1029/1024 = '''u'''<sub>''z''</sub> = (0, 0, 1) and 3136/3125 = 2'''u'''<sub>''x''</sub> + '''u'''<sub>''y''</sub> = (2, 1, 0). This corresponds to '''v''' = (1, &minus;2, 0).
 * [[miracle]] tempers out 1029/1024 = '''u'''<sub>''z''</sub> = (0, 0, 1) and 225/224 = '''u'''<sub>''x''</sub> &minus; '''u'''<sub>''y''</sub> = (1, &minus;1, 0). This corresponds to '''v''' = (1, 1, 0).
+== Examples ==
+''See also [[:Category:Equivalence continua|Category:Equivalence continua]].''
+All equivalence continua currently on the wiki are rank-''n'+1 continua of (rank-''n'+1) temperaments within a rank-''n''+2 subgroup that are supported by a rank-''n'' system.
+* [[5-limit]] rank-2 continua include:
+** the [[father-3 equivalence continuum]] ([[3edo]])
+** the [[syntonic-diatonic equivalence continuum]] ([[5edo]])
+** the [[syntonic-chromatic equivalence continuum]] ([[7edo]])
+** the [[schismic-Pythagorean equivalence continuum]] ([[12edo]])
+** the [[syntonic-kleismic equivalence continuum]] ([[19edo]])
+** the [[superpyth-22 equivalence continuum]] ([[22edo]])
+** the [[syntonic-31 equivalence continuum]] ([[31edo]])
+** the [[diaschismic-gothmic equivalence continuum]] ([[34edo]])
+** the [[schismic-countercommatic equivalence continuum]] ([[41edo]])
+** the [[schismic-Mercator equivalence continuum]] ([[53edo]])
+** the [[ennealimmal-enneadecal equivalence continuum]] ([[171edo]])
+** the [[tarot equivalence continuum]] ([[1848edo]])
+* [[2.3.7 subgroup]] rank-2 continua include:
+** the [[Archytas-diatonic equivalence continuum]] ([[5edo]])
+* [[2.5.7 subgroup]] rank-2 continua include:
+** the [[jubilismic-augmented equivalence continuum]] ([[6edo]])
+** the [[augmented-cloudy equivalence continuum]] ([[15edo]])
+** the [[rainy-didacus equivalence continuum]] ([[31edo]])
+* [[3.5.7 subgroup]] rank-2 continua include:
+** the [[sensamagic-gariboh equivalence continuum]] ([[13edt]])
+* [[7-limit]] rank-3 continua include:
+** the [[syntonic-Archytas equivalence continuum]] ([[dominant (temperament)|dominant]])
+** the [[marvel-syntonic equivalence continuum]] ([[septimal meantone]])
+** the [[breedsmic-syntonic equivalence continuum]] ([[squares]])
+* [[2.3.5.11 subgroup]] rank-3 continua include:
+** the [[syntonic-rastmic equivalence continuum]] ([[mohaha]])
 == Notes ==