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 [[rational interval]]s where 2, 3, 5, and 11 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 11. 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. | The '''2.3.5.11 subgroup''' (a.k.a. ''yala'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, and 11 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 11. 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) === | ||
The 2.3.5.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''7''', 12, 15, 19, 22, 31, 34, 41, 46, 53, '''65''', 87, 118, '''152''', 224, 270, 335, '''342''', … }} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
[[ | Since [[7edo]] provides a relatively accurate approximation of the 2.3.5.11 subgroup for its size, temperaments in this subgroup tend to work well with the 7-form. | ||
[[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 | [[Porcupine]] (15 & 22) provides a simple yet high-damage approximation to the subgroup. It is generated by a submajor second of around 163 cents, representing [[10/9]], [[11/10]], and [[12/11]], tempering out [[55/54]], [[100/99]], and [[121/120]]. Two of these reach [[6/5]]~[[11/9]], and three of them reach the perfect fourth [[4/3]]. It finds 11/8 at –4 generators and 5/4 at –5 generators. The interval at –7 generators represents several important ratios, such as [[25/24]], [[33/32]], [[45/44]], and [[81/80]]. | ||
[[Mohaha]] (24 & 31) can be considered an extension to [[meantone]] that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], tempering out [[243/242]]. Here [[11/8]]~[[15/11]] is found as a semi-augmented fourth, and [[11/10]]~[[12/11]] is found as a neutral second, meaning [[121/120]] is also tempered out. The neutral third is very close to 11/9 as a result of the flat fifth, though this means the 11th harmonic is tuned considerably flat. | |||
[[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]] {{nowrap| (41 & 53) }} mapping, tempering out [[2200/2187]], and another by finding 11/8 at the quadruple-diminished seventh (–30 fifths) through the [[helenus]] mapping, tempering out [[8019/8000]]. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction, and the optimal tuning for prime 11 is closer to the optimal tuning for 5. | |||
[[Gravity]] (7 & 58) 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. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique 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 [[ | [[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 [[rastma]]s and one [[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. | ||
[[Category:Just intonation subgroups|#]] | |||
[[Category:Rank-4 temperaments|#]] | |||
[[Category:11-limit|#]] | |||