2.3.5.13 subgroup

This page is a stub. You can help the Xenharmonic Wiki by expanding it.

The 2.3.5.13 subgroup 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 still infinite even if we restrict consideration to a single octave. 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 can be similar to the 2.3.5.11 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)

It is relatively well approximated by the following edos [bold ones edos that do particularly well in this subgroup]: 7, 15, 19, 24, 27, 31, 34, 46, 50, 53, 80, 87, 94, 96, 130, 140, 171, 217, 224, 270...

Rank-2 temperaments

Cata provides a fairly low complexity approximation to the subgroup, using a slightly flat ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens.

Pythagorean tuning also works surprisingly well, where the diminished fourth (-8 fifths) 8192/6561 and the triple augmented fourth (+20 fifths) 3486784401/2147483648 sound extremely close to 5/4 and 13/8 respectively, wherein tempering the schisma and tridecapyth comma provide a fairly more complex but 3/2-telic microtemperament, of which 53edo offers an almost perfect approximation. Pure fifths and octaves on the other hand, offer 5 and 13 with -1.954c and +1.428c of error.

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.

For those searching very high accuracy temperaments, the 2.3.5.13 extension of Egads (19&422) provides a highly complex, but 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 a practically perfect approximation.

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.