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.
The 29-limit intervals of the 2.3.29 subgroup are submajor and supraminor, with 29/27 being a supraminor second, 32/29 a submajor second, 29/24 a supraminor third, and 36/29 a submajor third, with their octave complements classified accordingly. While supraminor and submajor intervals occur in lower limits, such as 14/13, 11/10, and 17/14, these combine multiple primes higher than 3, unlike the 29-limit ones. The 29th harmonic is thus quite simple to classify by diatonic classification, and has a characteristic interval quality like harmonics 5, 7, etc. Primes 17 and 23 are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when octave reduced respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.
However, the 29-limit approaches the point where consonance stops being registered, and intervals become very close to each other, such as 29/28 only being wider than 30/29 by 841/840, a comma of 2.06 ¢. This difference is unnoticeable melodically, and very difficult to hear harmonically.
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. The intervals 29/16 and 32/29 are very accurately approximated by 7edo (1\7 for 32/29, 6\7 for 29/16).
Edos with increasingly better approximations of the 29-limit (monotonicity limit ≥ 29 and decreasing TE error): 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 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, "99ef" means taking the second closest approximations of harmonics 11 and 13. |
Music
- Spring Your Miracle (2026)
- Cloud Aliens (2021)