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{{Prime limit navigation|23}}
{{Prime limit navigation|23}}
In 23-limit [[just intonation]], all ratios contain no prime factors higher than 23. 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 it corresponding to the full [[27-odd-limit]] and thus corresponding to mode 14 of the harmonic series, which is to say that all of the first 28 harmonics 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 relative error|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.
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.
== 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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| 23og3
| 23og3
| twethogu 3rd
| twethogu 3rd
| vicesimotertial subminor third
| vicesimotertial inframinor third
|-
|-
| [[23/19]]
| [[23/19]]
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| 23uy6
| 23uy6
| twethuyo 6th
| twethuyo 6th
| vicesimotertial supermajor sixth
| vicesimotertial ultramajor sixth
|-
|-
| [[23/13]]
| [[23/13]]
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* Unlike most other prime limits, the smallest [[superparticular ratio]] of [[Harmonic class|HC23]] is larger than the smallest one of HC19. 23 is the first prime limit to show this phenomenon. The ratio is, in fact, larger than the second smallest one of HC19. See [[List of superparticular intervals]].  
* Unlike most other prime limits, the smallest [[superparticular ratio]] of [[Harmonic class|HC23]] is larger than the smallest one of HC19. 23 is the first prime limit to show this phenomenon. The ratio is, in fact, larger than the second smallest one of HC19. See [[List of superparticular intervals]].  


== See also ==
== Music ==
* [[Harmonic limit]]
; [[Domin]]
* [[23-odd-limit]]
* [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}}
* [https://www.youtube.com/watch?v=EOIFIl5D-JE ''Liebestraum No. 3''] (1850) – rendered by Randy Wells (2021)
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=FRy9CePMcJs ''Siren (Waterhouse)''] (2021)
* [https://www.youtube.com/watch?v=Jq_q51vVn4w ''Wilson/Nelson''] (2021)
* [https://www.youtube.com/watch?v=8Jk0KHGawgA ''Laudanum''] (2022)
* [https://www.youtube.com/watch?v=q43e8FhJcRI ''Koi Hito''] (2022)
* [https://www.youtube.com/watch?v=WP5rsOHRWpw ''Hazybrew''] (2022)
* [https://www.youtube.com/watch?v=Qb6IGHzuAtM ''Roses and Wolfsbane''] (2022)
* [https://www.youtube.com/watch?v=ahMoa0R4EjE ''2049: A Love Song''] (2022)
* [https://www.youtube.com/watch?v=NG5XJxV2d2M ''We're a Midwest Emo Intro Waiting to Happen''] (2023)
* [https://www.youtube.com/watch?v=mBlrFTf9Bec ''Phonograph Needle on an Ice Giant's Rings''] (2023)
* [https://www.youtube.com/watch?v=kgovRP0UAfw ''A Cosmic Turtle Grazing Upon Stellar Elements''] (2023)
* [https://www.youtube.com/watch?v=m9QaxFOlnYg ''A Hycean World''] (2023)
* [https://www.youtube.com/watch?v=jpHylRu6XLM ''ser0tonin circuits in a neural network''] (2023)
 
; {{W|Noah Dean DaSilva Jordan}}
* [https://open.spotify.com/album/2OGG4tT7INfj7iBeN09KDJ Gracias a Dios] (2023) for solo jarana (series 23/22, 23/21....23/12)


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