23-limit

Prime limit

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← 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

• Unlike most other prime limits, the smallest superparticular ratio of 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.

Music

Domin

• Uttuagn (2024)

Francium

• "GAY SAPIENS" from CAPSLOCK (2024) – Spotify | Bandcamp | YouTube

Franz Liszt

• Liebestraum No. 3 (1850) – rendered by Randy Wells (2021)

Randy Wells

• Siren (Waterhouse) (2021)
• Wilson/Nelson (2021)
• Laudanum (2022)
• Koi Hito (2022)
• Hazybrew (2022)
• Roses and Wolfsbane (2022)
• 2049: A Love Song (2022)
• We're a Midwest Emo Intro Waiting to Happen (2023)
• Phonograph Needle on an Ice Giant's Rings (2023)
• A Cosmic Turtle Grazing Upon Stellar Elements (2023)
• A Hycean World (2023)
• ser0tonin circuits in a neural network (2023)

Noah Dean DaSilva Jordan

• Gracias a Dios (2023) for solo jarana (series 23/22, 23/21....23/12)