23-limit: Difference between revisions

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]].  

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 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:

* Mode 12, 13, and 14 of the harmonic or subharmonic series.  

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.

[[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:

| [[23/19]]

| 330.761¢

| 340.552¢

| 424.364¢

* 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]].  

* [https://www.youtube.com/watch?v=RzHkfoa4m3g ''Uttuagn''] (2024)

* "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]

* [https://www.youtube.com/watch?v=EOIFIl5D-JE ''Liebestraum No. 3''] (1850) – rendered by Randy Wells (2021)

* [https://www.youtube.com/watch?v=FRy9CePMcJs ''Siren (Waterhouse)''] (2021)

* [https://www.youtube.com/watch?v=Jq_q51vVn4w ''Wilson/Nelson''] (2021)

* [https://open.spotify.com/album/2OGG4tT7INfj7iBeN09KDJ Gracias a Dios] (2023) for solo jarana (series 23/22, 23/21....23/12)

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