31-limit

Prime limit

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← 29-limit  31-limit  37-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; this means it completes the 5th octave of those series.

The 31-limit intervals of the 2.3.31 subgroup are mainly interseptimal, with 32/31 being a quartertone, 36/31 and 31/27 being semifourths, and 31/24 being a naiadic, with their octave complements classified accordingly. While interseptimal intervals are abundant in lower limits, 31 is the first prime where such intervals occur without combining primes higher than 3. As such, the intervals of 31 are difficult to classify diatonically, but add a contrasting character to the lower limits, in particular 11 and 13, which are mainly neutral in quality.

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

Randy Wells

• Soul Box (2020)
• Echo and Narcissus (2020)
• Theme for a primate-like creature (2024)