2.3.5.13 subgroup

The 2.3.5.13 subgroup (a.k.a. yatha in color notation) is a just intonation subgroup consisting of rational intervals where 2, 3, 5, and 13 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 13. This is an infinite set, and is still infinite even if we octave-reduce every interval in it. Some examples within the octave include 5/4, 3/2, 13/8, 13/10, 39/32 and so on.

It can be thought out as an extension of the familiar 5-limit with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing 7 and 11. It shares some qualities with the 2.3.5.11 subgroup, specifically considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which differ by the small comma of 352/351.

This subgroup is notable for containing the simplest JI representations of interseptimal intervals, which are halfway between two interval categories, with 15/13 being an ultramajor second/inframinor third, 13/10 being an ultramajor third/infrafourth, 20/13 being an ultrafifth/inframinor sixth, and 26/15 being an ultramajor sixth/inframinor seventh. Importantly, 15/13 is close to half of the perfect fourth, and 26/15 is close to half of the perfect twelfth, with two intervals of 15/13 falling short of 4/3 by 676/675, the island comma.

Regular temperaments

Rank-1 temperaments (edos)

The 2.3.5.13 subgroup is relatively well approximated by the following edos (decreasing TE error, bold ones do particularly well in this subgroup): 7, 10, 12, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270, 354, 388, 407, 441, …

Rank-2 temperaments

The 2.3.5.13 version of kleismic (sometimes called cata) provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~13/9, tempering out the marveltwin comma 325/324. Then ~26/15 is found at three generators, with two such intervals reaching ~3/1, tempering out 676/675. The interval at +4 generators is a third of a 9/8 whole tone, representing all of 25/24, 26/25, and 27/26. Good tunings of cata include 34edo and especially 53edo, with other tunings such as 87edo and 140edo being usable as well.

The schismic extension that adds prime 13 via tempering out 325/324 provides a more complex temperament, well represented with 41edo and 53edo, though 94edo is more optimized and can extend to other subgroups. Pythagorean tuning also works surprisingly well, where the diminished fourth (-8 fifths) 8192/6561 and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a restriction of the full 13-limit cassandra mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954 ¢ and -1.428 ¢ of error respectively.

Other approximations of schismic reach prime 13 through other means, such as hemischis, dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. Helenus reaches 13/8 through -33 fifths, but it is a worse mapping.

For those searching very high-accuracy temperaments, the 2.3.5.13 extension of egads (19 & 422) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which 1342edo offers practically perfect approximations.

Rank-3 temperaments

Marveltwin offers a very low-complexity approximation to the subgroup, reaching 16/13 through (10/9)², and condensing the subgroup into a 5-limit planar temperament.

{140625/140608}, the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, 407edo, 441edo, 494edo, 901edo, and of course 1342edo.