23-limit: Difference between revisions

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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.

== 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:

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== 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}}

* [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)

~~== 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| ]]

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 {{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.
 == 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:
@@ Line 157: / Line 165: @@
 == 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}}
 * [https://www.youtube.com/watch?v=EOIFIl5D-JE ''Liebestraum No. 3''] (1850) – rendered by Randy Wells (2021)
@@ Line 174: / Line 188: @@
 * [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 -->