23-limit: Difference between revisions

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{{Prime limit navigation|23}}
{{Prime limit navigation|23}}
The '''23-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 23. It is the 9th [[prime limit]] and is thus a superset of the [[19-limit]] and a subset of the [[29-limit]]. The prime 23 is significant as being the start of a record prime gap ending at 29, the previous record prime gap being the one corresponding to the [[7-limit]]. Thus, it is arguably a potential ideal stopping point for [[prime limit]]s due to it corresponding to the full [[27-odd-limit]] and thus corresponding to the [[14afdo|14th]] mode [[Overtone scale|of the harmonic series]], which is to say that all of the first 28 [[harmonic]]s are in the 23-limit.
The '''23-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 23. It is the 9th [[prime limit]] and is a superset of the [[19-limit]] and a subset of the [[29-limit]].  


== Edo approximations ==
The 23-limit is a rank-9 system, and can be modeled in a 8-dimensional lattice, with the primes 3, 5, 7, 11, 13, 17, 19, and 23 represented by each dimension. The prime 2 does not appear in the typical 23-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a ninth dimension is needed.  
A list of [[edo]]s with progressively better tunings for 23-limit intervals: {{EDOs| 80, 87, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460 }} and so on.  


Another list of edos which provides relatively good tunings for 23-limit intervals ([[TE simple badness]] < 5%): {{EDOs| 94, 190g, 193, 217, 243e, 270, 282, 311, 328h, 373g, 388, 422, 436, 460, 525, 540, 566g, 581, 624, 639h, 643i, 653, 692i, 718, 742i, 764(h), 814, 860, 882, 908, 935, 954h, 997, 1012, 1046dgh, 1075, 1106, 1125, 1178, 1205g, 1224, 1236(h), 1258, 1282, 1308, 1323, 1357efhi, 1385, 1395, 1419, 1448(g), 1506hi, 1578, 1600, 1646, 1672h, 1677e, 1696, 1718, 1730(g), 1759, 1768gi, 1817hi, 1821ef, 1889, 1920, 1966, 2000, 2038, 2041, 2072, 2087h, 2103, 2113, 2132eh, 2159, 2217, 2231, 2243e, 2270i, 2311, 2320, 2414, 2460 }} and so on.
The prime 23 is significant as being the start of a record prime gap ending at 29, the previous record prime gap being the one corresponding to the [[7-limit]]. Thus, it is arguably a potential ideal stopping point for [[prime limit]]s due to a substantial increment in its harmonic contents. Specifically, these things are contained by the 23-limit, but not the 19-limit:
* The [[23-odd-limit|23-]], [[25-odd-limit|25-]], and [[27-odd-limit]];
* Mode 12, 13, and 14 of the harmonic or subharmonic series.  


: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "121i" means taking the second closest approximation of harmonics 23.
== Edo approximation ==
Here is a list of [[edo]]s with progressively better tunings for 23-limit intervals ([[monotonicity limit]] ≥ 23 and decreasing [[TE error]]): {{EDOs| 58hi, 62, 68e, 72, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
 
Here is a list of edos which provides relatively good tunings for 23-limit intervals ([[TE relative error]] < 5%): {{EDOs| 94, 190g, 193, 217, 243e, 270, 282, 311, 328h, 373g, 388, 422, 436, 460, 525, 540, 566g, 581, 624, 639h, 643i, 653, 692i, 718, 742i, 764(h), 814, 860, 882, 908, 935, 954h, 997, 1012, 1046dgh, 1075, 1106, 1125, 1178, 1205g, 1224, 1236(h), 1258, 1282, 1308, 1323, 1357efhi, 1385, 1395, 1419, 1448(g), 1506hi, 1578, 1600, 1646, 1672h, 1677e, 1696, 1718, 1730(g), 1759, 1768gi, 1817hi, 1821ef, 1889, 1920, 1966, 2000, 2038, 2041, 2072, 2087h, 2103, 2113, 2132eh, 2159, 2217, 2231, 2243e, 2270i, 2311, 2320, 2414, 2460 }} and so on.
 
: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "58hi" means taking the second closest approximations of harmonics 19 and 23.


[[94edo]] is the first [[edo]] to be consistent in the [[23-odd-limit]]. The smallest edo where the [[23-odd-limit]] is distinctly consistent, meaning each element of the tonality diamond is distinguished, is [[282edo]], although [[311edo]] may be preferred for excellent consistency in much larger odd limits, and thus is a good choice if you want the 23-odd-limit to be distinctly consistent and the 27-odd-limit (and higher) to be consistent.
[[94edo]] is the first [[edo]] to be consistent in the [[23-odd-limit]]. The smallest edo where the [[23-odd-limit]] is distinctly consistent, meaning each element of the tonality diamond is distinguished, is [[282edo]], although [[311edo]] may be preferred for excellent consistency in much larger odd limits, and thus is a good choice if you want the 23-odd-limit to be distinctly consistent and the 27-odd-limit (and higher) to be consistent.


== 23-odd-limit intervals ==
== 23-odd-limit intervals ==
[[File:Some 23-limit otonal chords.png|thumb|15 pentads and 1 hexad, with 23 as the highest odd harmonic, avoiding steps smaller than 23/21.]]
Ratios of 23 in the 23-odd-limit are:
Ratios of 23 in the 23-odd-limit are:


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== Music ==
== Music ==
; [[Domin]]
* [https://www.youtube.com/watch?v=RzHkfoa4m3g ''Uttuagn''] (2024)
; [[Francium]]
* "GAY SAPIENS" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/5vILBQgWJduJf2ctGGbyUv Spotify] | [https://francium223.bandcamp.com/track/gay-sapiens Bandcamp] | [https://www.youtube.com/watch?v=DHiwdGuZRII YouTube]
; {{W|Franz Liszt}}
; {{W|Franz Liszt}}
* [https://www.youtube.com/watch?v=EOIFIl5D-JE ''Liebestraum No. 3''] (1850) – rendered by Randy Wells (2021)
* [https://www.youtube.com/watch?v=EOIFIl5D-JE ''Liebestraum No. 3''] (1850) – rendered by Randy Wells (2021)
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* [https://www.youtube.com/watch?v=jpHylRu6XLM ''ser0tonin circuits in a neural network''] (2023)
* [https://www.youtube.com/watch?v=jpHylRu6XLM ''ser0tonin circuits in a neural network''] (2023)


== See also ==
; {{W|Noah Dean DaSilva Jordan}}
* [[23-odd-limit]]
* [https://open.spotify.com/album/2OGG4tT7INfj7iBeN09KDJ Gracias a Dios] (2023) for solo jarana (series 23/22, 23/21....23/12)


[[Category:23-limit| ]] <!-- main article -->
[[Category:23-limit| ]] <!-- main article -->

Latest revision as of 16:20, 20 August 2025

← 19-limit  23-limit  29-limit →

The 23-limit consists of just intonation intervals whose ratios contain no prime factors higher than 23. It is the 9th prime limit and is a superset of the 19-limit and a subset of the 29-limit.

The 23-limit is a rank-9 system, and can be modeled in a 8-dimensional lattice, with the primes 3, 5, 7, 11, 13, 17, 19, and 23 represented by each dimension. The prime 2 does not appear in the typical 23-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a ninth dimension is needed.

The prime 23 is significant as being the start of a record prime gap ending at 29, the previous record prime gap being the one corresponding to the 7-limit. Thus, it is arguably a potential ideal stopping point for prime limits due to a substantial increment in its harmonic contents. Specifically, these things are contained by the 23-limit, but not the 19-limit:

  • The 23-, 25-, and 27-odd-limit;
  • Mode 12, 13, and 14 of the harmonic or subharmonic series.

Edo approximation

Here is a list of edos with progressively better tunings for 23-limit intervals (monotonicity limit ≥ 23 and decreasing TE error): 58hi, 62, 68e, 72, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460 and so on. For a more comprehensive list, see Sequence of equal temperaments by error.

Here is a list of edos which provides relatively good tunings for 23-limit intervals (TE relative error < 5%): 94, 190g, 193, 217, 243e, 270, 282, 311, 328h, 373g, 388, 422, 436, 460, 525, 540, 566g, 581, 624, 639h, 643i, 653, 692i, 718, 742i, 764(h), 814, 860, 882, 908, 935, 954h, 997, 1012, 1046dgh, 1075, 1106, 1125, 1178, 1205g, 1224, 1236(h), 1258, 1282, 1308, 1323, 1357efhi, 1385, 1395, 1419, 1448(g), 1506hi, 1578, 1600, 1646, 1672h, 1677e, 1696, 1718, 1730(g), 1759, 1768gi, 1817hi, 1821ef, 1889, 1920, 1966, 2000, 2038, 2041, 2072, 2087h, 2103, 2113, 2132eh, 2159, 2217, 2231, 2243e, 2270i, 2311, 2320, 2414, 2460 and so on.

Note: wart notation is used to specify the val chosen for the edo. In the above list, "58hi" means taking the second closest approximations of harmonics 19 and 23.

94edo is the first edo to be consistent in the 23-odd-limit. The smallest edo where the 23-odd-limit is distinctly consistent, meaning each element of the tonality diamond is distinguished, is 282edo, although 311edo may be preferred for excellent consistency in much larger odd limits, and thus is a good choice if you want the 23-odd-limit to be distinctly consistent and the 27-odd-limit (and higher) to be consistent.

23-odd-limit intervals

15 pentads and 1 hexad, with 23 as the highest odd harmonic, avoiding steps smaller than 23/21.

Ratios of 23 in the 23-odd-limit are:

Ratio Cents Value Color Name Interval Name
24/23 73.681¢ 23u1 twethu 1sn lesser vicesimotertial semitone
23/22 76.956¢ 23o1u2 twetholu 2nd greater vicesimotertial semitone
23/21 157.493¢ 23or2 twethoru 2nd large vicesimotertial neutral second
26/23 212.253¢ 23u3o2 twethutho 2nd vicesimotertial whole tone
23/20 241.961¢ 23og3 twethogu 3rd vicesimotertial inframinor third
23/19 330.761¢ 23o19u3 twethonu 3rd vicesimotertial supraminor third
28/23 340.552¢ 23uz3 twethuzo 3rd vicesimotertial neutral third
23/18 424.364¢ 23o4 twetho 4th vicesimotertial diminished fourth
30/23 459.994¢ 23uy3 twethuyo 3rd vicesimotertial ultramajor third
23/17 523.319¢ 23o17u4 twethosu 4th vicesimotertial acute fourth
32/23 571.726¢ 23u4 twethu 4th vicesimotertial narrow tritone
23/16 628.274¢ 23o5 twetho 5th vicesimotertial wide tritone
34/23 676.681¢ 23u17o5 twethuso 5th vicesimotertial grave fifth
23/15 740.006¢ 23og6 twethogu 6th vicesimotertial ultraminor sixth
36/23 775.636¢ 23u5 twethu 5th vicesimotertial augmented fifth
23/14 859.448¢ 23or6 twethoru 6th vicesimotertial neutral sixth
38/23 869.239¢ 23u19o6 twethuno 6th vicesimotertial submajor sixth
40/23 958.039¢ 23uy6 twethuyo 6th vicesimotertial ultramajor sixth
23/13 987.747¢ 23o3u7 twethothu 7th vicesimotertial minor seventh
42/23 1042.507¢ 23uz7 twethuzo 7th small vicesimotertial neutral seventh
44/23 1123.044¢ 23u1o7 twethulo 7th vicesimotertial major seventh
23/12 1126.391¢ 23o8 twetho 8ve vicesimotertial major seventh

Trivia

Music

Domin
Francium
Franz Liszt
Randy Wells
Noah Dean DaSilva Jordan
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