23-limit

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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.

Edo approximations

A list of edos with progressively better tunings for 23-limit intervals: 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 (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.

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

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

24/23 .. 73.681¢ .. 23u1 .. twethu 1sn

23/22 .. 76.956¢ .. 23o1u2 .. twetholu 2nd

23/21 .. 157.493¢ .. 23or2 .. twethoru 2nd

26/23 .. 212.253¢ .. 23u3o2 .. twethutho 2nd

23/20 .. 241.961¢ .. 23og3 .. twethogu 3rd

23/19 .. 330.761¢ .. 23o19u3 .. twethonu 3rd

28/23 .. 340.552¢ .. 23uz3 .. twethuzo 3rd

23/18 .. 424.364¢ .. 23o4 .. twetho 4th

30/23 .. 459.994¢ .. 23uy3 .. twethuyo 3rd

23/17 .. 523.319¢ .. 23o17u4 .. twethosu 4th

32/23 .. 571.726¢ .. 23u4 .. twethu 4th

23/16 .. 628.274¢ .. 23o5 .. twetho 5th

34/23 .. 676.681¢ .. 23u17o5 .. twethuso 5th

23/15 .. 740.006¢ .. 23og6 .. twethogu 6th

36/23 .. 775.636¢ .. 23u5 .. twethu 5th

23/14 .. 859.448¢ .. 23or6 .. twethoru 6th

38/23 .. 869.239¢ .. 23u19o6 .. twethuno 6th

40/23 .. 958.039¢ .. 23uy6 .. twethuyo 6th

23/13 .. 987.747¢ .. 23o3u7 .. twethothu 7th

42/23 ..1042.507¢ .. 23uz7 .. twethuzo 7th

44/23 .. 1123.044¢ .. 23u1o7 .. twethulo 7th

23/12 .. 1126.391¢ .. 23o8 .. twetho 8ve

See also