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.
Regular temperaments
Rank-1 temperaments (edos)
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 edos 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. |
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.
Rank-2 temperaments
Cassandra provides a very intuitive extension using the chain of fifths, naturally extending 19/16 to the minor third. Well represented with 41edo and 53edo, though 94edo is more optimized and can extend to other subgroups, and is specially prominent in 94edo, extending to the full 23-limit.
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, 7/4 at +25 gens, and 13/8 at −13 gens, which is optimal in 130edo, albeit 19/16 is worsely tuned because the fifth is flatter.
For those searching higher accuracy temperaments, gariwizmic also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one semioctave, 7/4 at −14 fifths, 11/8 at +23 fifths and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at 270edo.
Other non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the subgroup are vulture, cotoneum, newt, and ennealimmal. Cotoneum, well represented by 217edo, has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent, albeit prime 11 is not that good, though prime 19 is decent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the minisma, equating the 36/35 generator to the 1053/1024.