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.

Regular temperaments

Rank-1 temperaments (edos)

Edos which represents the subgroup better (monotonic in the no-17 19-odd-limit 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. Bold edos are records of TE 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 one of the best equal temperaments for this subgroup, achieving a record of relative error that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: 2190, 6079, 8269 and 8539. The last two coincidentally differ by 270 and are prime edos.

Rank-2 temperaments

Cassandra provides a very intuitive approximation to this subgroup using the chain of fifths, naturally mapping 19/16 to the minor third, and is well represented with 41edo and 53edo, though 94edo is more optimized.

For those searching higher-accuracy temperaments, cotoneum keeps the chain of fifths, but does not temper out the schisma. However, newt, which splits the perfect fifth in halves (tempering out 2401/2400) and finding the aberschisma -41 hemififths away, is much more efficient. Another similar temperament is gariwizmic, which instead of splitting the fifth, splits the octave in half. Newt and gariwizmic meet in 270edo.

Other non-chain-of-fifths temperaments that meet in 270edo, and are thus great candidates for the subgroup, include vulture, satin, and paramity.

Rank-3 temperaments

Cassaschismic relates several formal commas in this subgroup to reduce them to essentially a generic comma a generic aberschisma, making it significant for notation systems based on the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include lif and eir.