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>.

An '''equivalence continuum''' comprises all the [[regular temperament|temperaments]] where a number of a certain interval is equated with another interval. It 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.

For example, in the [[syntonic–chromatic equivalence continuum]], a number of [[81/80|syntonic commas]] is equated with the [[2187/2048|Pythagorean chromatic semitone]], and this describes all temperaments supported by [[7edo|7et]] since that is the unique temperament that [[tempering out|tempers out]] both and hence all the combinations thereof.

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>.

== Geometric interpretation ==

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

** the [[~~augmented–chromatic~~ equivalence continuum]] ([[3edo]])

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

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

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

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-{{inacc}}
+{{Inacc}}
-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>.
+An '''equivalence continuum''' comprises all the [[regular temperament|temperaments]] where a number of a certain interval is equated with another interval. It 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.
+For example, in the [[syntonic–chromatic equivalence continuum]], a number of [[81/80|syntonic commas]] is equated with the [[2187/2048|Pythagorean chromatic semitone]], and this describes all temperaments supported by [[7edo|7et]] since that is the unique temperament that [[tempering out|tempers out]] both and hence all the combinations thereof.
+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>.
 == Geometric interpretation ==
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 * [[5-limit]] rank-2 continua include:
-** the [[augmented–chromatic equivalence continuum]] ([[3edo]])
+** the [[father–3 equivalence continuum]] ([[3edo]])
 ** the [[syntonic–diatonic equivalence continuum]] ([[5edo]])
 ** the [[syntonic–chromatic equivalence continuum]] ([[7edo]])