29-limit: Difference between revisions
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These things are contained by the 29-limit, but not the 23-limit: | These things are contained by the 29-limit, but not the 23-limit: | ||
* The [[29-odd-limit]]; | * The [[29-odd-limit]]; | ||
* Mode 15 of the harmonic or subharmonic series. | * 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. While lower-limit supraminor and submajor intervals exist, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1|29th harmonic]] is thus quite simple to classify by [[5L 2s|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1|5]], [[7/1]], etc. Note that primes [[17/1|17]] and [[23/1|23]] are not so friendly in terms of interval categorization, and many 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{{c}}. This difference is [[JND|unnoticeable]] melodically, and very difficult to hear harmonically. | |||
== Edo approximations == | == Edo approximations == | ||
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; [[Randy Wells]] | ; [[Randy Wells]] | ||
* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021) | * [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021) | ||
[[Category:29-limit| ]] <!-- main article --> | [[Category:29-limit| ]] <!-- main article --> | ||
Revision as of 23:05, 16 January 2026
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. While lower-limit supraminor and submajor intervals exist, 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, 7/1, etc. Note that primes 17 and 23 are not so friendly in terms of interval categorization, and many 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. 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)