23-limit: Difference between revisions

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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.

Ratios of 23 in the 23-odd limit include:

== Edo approximations ==

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.

[[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

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

[[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.

== See also ==

@@ Line 1: / Line 1: @@
 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.
-Ratios of 23 in the 23-odd limit include:
+== Edo approximations ==
+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.
+[[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:
 /23 .. 73.681¢ .. 23u1 .. twethu 1sn
@@ Line 46: / Line 54: @@
 /12 .. 1126.391¢ .. 23o8 .. twetho 8ve
-[[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.
 == See also ==