2.3.5.7.11.13.19.29 subgroup
The 2.3.5.7.11.13.19.29 subgroup (a.k.a. yazalathanatwena in color notation, hereon referred to as the subgroup) consists of just intonation intervals such that the highest prime factor in all ratios is 29, but without 17 or 23. It is thus a subset of the 29-limit, or alternatively, it can be seen as the 13-limit with extra primes 19 and 29, or the 2.3.5.7.11.13.19 subgroup with 29 added in.
This subgroup is a rank-8 system, and can be modeled in a 7-dimensional lattice, with the primes 3, 5, 7, 11, 13, 19 and 29 represented by each dimension. The prime 2 does not appear in typical lattices because octave equivalence is presumed. If octave equivalence is not presumed, an eighth dimension is needed.
This subgroup is significant because it adds 19 as a counterpart of 21 in the 16::24 harmonics series segment, so that any fifth-bounded chord involving 21 over the root has a harmonic (i.e. frequency-scale) inverse involving 19, and it adds 29 as a counterpart of 27 in 24::32, so that any fourth-bounded chord involving 27 over the root of 3 has a harmonic inverse involving 29. In addition, {19, 21} mirrors {27, 29} in 16::32, implying that any fifth-bounded chord involving 19 or 21 over the root has a fourth-bounded equivalent involving 27 or 29. 19/16 and 29/16 are close to a minor third and a supraminor seventh, respectively, so they can provide a special "minor" flavors to otonal chords without clashing too much with the fundamental. Meanwhile, harmonics 17, 23, 25, and 31 may be considered to clash with the fundamental, being close to tonic and fifth when octave reduced, so people may wish to exclude them. Therefore the subgroup can be considered to complete the harmonic series up to the fifth octave without the more difficult ones near the edges.
Regular temperaments
Rank-1 temperaments (edos)
Edos which represents the subgroup better (monotonic in the tonality diamond of {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} and decreasing TE error): 41, 53j, 58h, 72, 77, 80, 94, 99ef, 113, 118f, 118, 121, 130, 140, 152fj, 159, 183, 190, 198, 217, 224, 243e, 270, 422, 472, 494h, 552, … 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, "53j" means taking the second closest approximation of harmonic 29. |