Equivalence continuum: Difference between revisions

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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.

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.

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.

For example, in the [[syntonic–chromatic equivalence continuum]], a number of [[81/80|syntonic commas]] is equated with the [[2187/2048|Pythagorean chromatic semitone]]: {{nowrap| (81/80)''n'' ~ 2187/2048 }}, 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. By specifying different values of ''n'', we obtain temperaments such as [[porcupine]], [[tetracot]], [[amity]], and so on.

~~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>.

The 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>.

== Choice of basis ==

It can be shown that different choices of intervals can lead to essentially identical continua, where the related individual temperaments are the same. For instance, in the syntonic–chromatic equivalence continuum, if the stacked interval ''q''1 is the syntonic comma, it does not matter if the targeted interval ''q''2, a chromatic semitone, is Pythagorean (2187/2048), major (135/128), or classical (25/24), as they only differ by whole multiples of the syntonic comma. For consistency, the following scheme is established as the default choice for stacked and targeted intervals for equivalence continua of rank-2 temperaments:

* The stacked interval ''q''1 should have the least nonzero absolute value of order in the last formal prime. Typically it is ±1, but in case that is impossible, it is ±2, ±3, ….

* The targeted interval ''q''2 should have order 0 in the last formal prime. In particular, for continua of 2.3.5, 2.3.7, 2.3.11, …, it should be a 3-limit interval.

* The comma tempered out in the temperament corresponding to {{nowrap| ''n'' {{=}} 1 }} should be smaller in size than ''q''2.

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.

== Geometric interpretation ==

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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. 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.
+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.
-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.
+For example, in the [[syntonic–chromatic equivalence continuum]], a number of [[81/80|syntonic commas]] is equated with the [[2187/2048|Pythagorean chromatic semitone]]: {{nowrap| (81/80)<sup>''n''</sup> ~ 2187/2048 }}, 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. By specifying different values of ''n'', we obtain temperaments such as [[porcupine]], [[tetracot]], [[amity]], and so on.
-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>.
+The 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>.
+== Choice of basis ==
+It can be shown that different choices of intervals can lead to essentially identical continua, where the related individual temperaments are the same. For instance, in the syntonic–chromatic equivalence continuum, if the stacked interval ''q''<sub>1</sub> is the syntonic comma, it does not matter if the targeted interval ''q''<sub>2</sub>, a chromatic semitone, is Pythagorean (2187/2048), major (135/128), or classical (25/24), as they only differ by whole multiples of the syntonic comma. For consistency, the following scheme is established as the default choice for stacked and targeted intervals for equivalence continua of rank-2 temperaments:
+* The stacked interval ''q''<sub>1</sub> should have the least nonzero absolute value of order in the last formal prime. Typically it is ±1, but in case that is impossible, it is ±2, ±3, ….
+* The targeted interval ''q''<sub>2</sub> should have order 0 in the last formal prime. In particular, for continua of 2.3.5, 2.3.7, 2.3.11, …, it should be a 3-limit interval.
+* The comma tempered out in the temperament corresponding to {{nowrap| ''n'' {{=}} 1 }} should be smaller in size than ''q''<sub>2</sub>.
+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.
 == Geometric interpretation ==