29-limit

Prime limit

---

← 23-limit  29-limit  31-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

Gordon Wery

• lucky lopside (2025)

Domin

• Ixnge Uotbnatsz (2.3.7.13.29 JI) (2026)
• Ixnge Blulg (2.3.7.13.29 JI) (2025)
• Ixnge Euiia (2.3.7.13.29 JI) (dark ambient) (2025)
• Exngo Onkha (2.3.7.13.29 JI) (2025)
• Ixnge Bnga (2.3.7.13.29 JI) (2025)
• Angagn Yeein (2.3.7.13.29 JI) (2025)
• Exngo Rhkue (2.3.7.13.29 JI) (2025)
• Oxnga Cryxs (2.3.7.13.29 JI) (2025)
• Angagn Dgaa (2.3.7.13.29 JI) (2025)
• Exngo Auerkt (2.3.7.13.29 JI) (2025)
• Angagn Rgar (2.3.7.13.29 JI) (2025)
• Ixnge Ioikra (2.3.7.13.29 JI) (2025)
• Ixnge Sauej (2.3.7.13.29 JI) (2025)
• Oxnga Otrahn (2.3.7.13.29 JI) (2025)
• Angagn Ehji (2.3.7.13.29 JI) (2025)
• Angagn Leish (2.3.7.13.29 JI) (2025)
• Angagn Aazx (2.3.7.13.29 JI) (2025)
• Angagn Ount (2.3.7.13.29 JI) (2025)
• Axngo Zilt (2.3.7.13.29 JI) (2025)
• Oxnga Żdżarg (2.3.7.13.29 JI) (2025)
• Exngo Akkaya (2.3.7.13.29 JI) (2025)
• Oxnga Shkarr (2.3.7.13.29 JI) (2025)
• Axngo Knett (2.3.7.13.29 JI) (2025)
• Axngo Ndge (2.3.7.13.29 JI) (2026)

Francium

• Spring Your Miracle (2026)

Randy Wells

• Cloud Aliens (2021)