2.3.5.11 subgroup: Difference between revisions
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The '''2.3.5.11 subgroup''' is a [[just intonation subgroup]] consisting of [[ | The '''2.3.5.11 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, and 13 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on. | ||
In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected by the small comma of [[352/351]]. | In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected by the small comma of [[352/351]]. | ||
== Regular temperaments == | == Regular temperaments == | ||
=== Rank-1 temperaments (edos) === | === Rank-1 temperaments (edos) === | ||
It is relatively well approximated by the following edos | It is relatively well approximated by the following edos (bold ones do particularly well in this subgroup): [[7edo|7]], [[15edo|15]], [[22edo|22]], [[24edo|'''24''']], [[31edo|31]], [[38edo|38]], [[41edo|41]], [[46edo|46]], [[65edo|'''65''']], [[72edo|'''72''']], [[80edo|80]], [[87edo|'''87''']], [[94edo|94]], [[96edo|96]], [[118edo|'''118''']], [[130edo|130]], [[137edo|137]], [[159edo|'''159''']], [[183edo|183]], [[217edo|217]], [[224edo|224]], [[270edo|'''270''']], [[311edo|311]], … | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[ | [[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction. | ||
[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. [[65edo]] is the intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup. | [[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. [[65edo]] is the intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup. | ||
=== Rank-3 temperaments === | === Rank-3 temperaments === | ||
[[ | [[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[243/242|rastmas]] and one [[81/80|syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others. | ||