31-limit
The 31-limit consists of just intonation intervals whose ratios contain no prime factors higher than 31. It is the 11th prime limit and is a superset of the 29-limit and a subset of the 37-limit.
The 31-limit is a rank-11 system, and can be modeled in a 10-dimensional lattice, with the primes 3 to 31 represented by each dimension. The prime 2 does not appear in the typical 31-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, an eleventh dimension is needed.
Like the 23-limit, the 31-limit implies a substantial increment in its harmonic contents compared to previous prime limit. Specifically, these things are contained by the 31-limit, but not the 29-limit:
- The 31-, 33-, and 35-odd-limit;
- Mode 16, 17, and 18 of the harmonic or subharmonic series.
Edo approximations
311edo is the smallest edo that is consistent to the 31-odd-limit. 1600edo is the smallest edo that is distinctly consistent to the 31-odd-limit.
Edos with increasingly better approximations of the 31-limit (monotonicity limit ≥ 31 and decreasing TE error): 99efk, 121ik, 130, 140, 149, 152fgj, 159, 183, 190g, 217, 243e, 270, 311, 388, 422, 525, 566gj, 571, 624jk, 639hj, 643ijk, 653, 692ik, 718, 742i, 863efgjk, 882, 908, 954hj, 1205g, 1289, 1308, 1395, 1578, etc. For a more comprehensive list, see Sequence of equal temperaments by error.
| Note: | Wart notation is used to specify the val chosen for the edo. In the above list, "99efk" means taking the second closest approximations of harmonics 11, 13, and 31. |
Music
- Soul Box (2020)
- Echo and Narcissus (2020)
- Theme for a primate-like creature (2024)
See also
- No-twos 31-limit
- User:Contribution/31-limit – Includes an enormous table of 31-limit intervals