2.3.5.7.11.13.19 subgroup
The 2.3.5.7.11.13.19 subgroup (a.k.a. yazalathana in color notation) consists of just intonation intervals such that the highest prime factor in all ratios is 19, but without 17. It is thus a subset of the 19-limit, or alternatively, it can be seen as the 13-limit with an extra prime 19.
This subgroup is a rank-7 system, and can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13 and 19 represented by each dimension. The prime 2 does not appear in typical lattices because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed.
The subgroup can be conveniently rank-reduced into the 5-limit without much loss in accuracy by tempering out 2080/2079 and 4096/4095 and 1216/1215, resulting in the cassaschismic temperament, which equates 36/35 with 1053/1024 and (64/63)2 with 33/32, and 64/63 with the Pythagorean comma. Other notable rank-reductions include neonewt and garibaldi/cassandra; newt splits the fifth in half (tempering out 2401/2400) and finding the aberschisma at -41 hemififths; and garibaldi combines the pythagorean comma, 64/63 and 81/80 into one general comma, that when doubled acts as ~33/32 and ~1053/1024; this tempers out 225/224 and 352/351.
270edo is arguably the equal best temperament for this subgroup, achieving a record of absolute, relative error, and logflat badness that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions; 8539edo and 8269edo, which concidentally differ by 270 and are prime edos.
Edo approximation
Edos which represents the subgroup better (monotonic, and decreasing TE error): 27e, 31, 34dh, 38df, 41f*, 41, 50, 53, 58h, 72, 87, 94, 103h, 111, 121, 130, 152f, 190, 217, 224, 270, 552, 581… and so on. For a more comprehensive list, see Sequence of equal temperaments by error. Bold temperaments are records of relative error.
| Note: | Wart notation is used to specify the val chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. |