29-limit
The 29-limit consists of just intonation intervals whose ratios contain no prime factors higher than 29. It is the 10th prime limit and is a superset of the 23-limit and a subset of the 31-limit. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the 11-limit as both include the prime ending a record prime gap.
The 29-limit is a rank-10 system, and can be modeled in a 9-dimensional lattice, with the primes 3 to 29 represented by each dimension. The prime 2 does not appear in the typical 29-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a tenth dimension is needed.
These things are contained by the 29-limit, but not the 23-limit:
- The 29-odd-limit;
- Mode 15 of the harmonic or subharmonic series.
Edo approximations
282edo is the smallest edo that is consistent to the 29-odd-limit. 1323edo is the smallest edo that is distinctly consistent to the 29-odd-limit. Intervals 29/16 and 32/29 are very accurately approximated by 7edo (1\7 for 32/29, 6\7 for 29/16).
Music
- Cloud Aliens (2021)