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.

Music

Randy Wells

• 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

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