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.

== Edo ~~approximations~~ ==

== Edo approximation ==

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.

Here is a list of [[edo]]s with progressively better tunings for 23-limit intervals (decreasing [[TE error]]): {{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.

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, "121i" means taking the second closest approximation of harmonics 23.

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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.
-== Edo approximations ==
+== Edo approximation ==
-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.
+Here is a list of [[edo]]s with progressively better tunings for 23-limit intervals (decreasing [[TE error]]): {{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.
+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, "121i" means taking the second closest approximation of harmonics 23.