2.3.5.13 subgroup
The 2.3.5.13 subgroup (AKA 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.
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
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. It is well represented by 34- and 53edo, with 87edo being an almost perfect approximation.
The schismic extension that adds prime 13 via tempering out the marveltwin comma 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 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.