Equivalence continuum: Difference between revisions
m →Examples: improvement/rework |
→List of equivalence continua: + justifications for most of these entries |
||
| (7 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
An '''equivalence continuum''' is the | 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]]: {{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. | |||
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. | |||
=== 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 == | == Geometric interpretation == | ||
{{Inacc}} | |||
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'''<sup>{{nowrap|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap|ker(''T'') ⊗ '''R'''}}. | 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'''<sup>{{nowrap|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap|ker(''T'') ⊗ '''R'''}}. | ||
| Line 21: | 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)}}. | * [[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)}}. | ||
== | == List of 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. | 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: | * [[5-limit]] rank-2 continua include: | ||
** the [[ | ** 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 [[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]]) | ** the [[tarot equivalence continuum]] ([[1848edo]]) | ||
* [[2.3.7 | * [[2.3.7 subgroup|2.3.7-subgroup]] rank-2 continua include: | ||
** the [[ | ** the [[Archytas–diatonic equivalence continuum]] ([[5edo]], a 3-limit and 2.3.7-subgroup record edo) | ||
** the [[ | ** the [[Archytas–chromatic equivalence continuum]] ([[7edo]], a 3-limit record edo) | ||
* [[2.5.7 | * [[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 | * [[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: | * [[7-limit]] rank-3 continua include: | ||
| Line 57: | Line 76: | ||
** the [[breedsmic–syntonic equivalence continuum]] ([[squares]]) | ** the [[breedsmic–syntonic equivalence continuum]] ([[squares]]) | ||
* [[2.3.5.11 | * [[2.3.5.11 subgroup|2.3.5.11-subgroup]] rank-3 continua include: | ||
** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]]) | ** the [[syntonic–rastmic equivalence continuum]] ([[mohaha]]) | ||
| Line 65: | Line 84: | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||