2.3.5.11 subgroup

The 2.3.5.11 subgroup (a.k.a. yala in color notation) is a just intonation subgroup consisting of rational intervals where 2, 3, 5, and 11 are the only allowable prime factors, 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.

Regular temperaments

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): 7, 12, 15, 19, 22, 31, 34, 41, 46, 53, 65, 87, 118, 152, 224, 270, 335, 342, …

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.

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 (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, 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 (S9), 100/99 (S10), and 121/120 (S11), 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

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 rastmas and one syntonic comma (0.08 cents). Its boundary of usability begins at 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.