23-limit: Difference between revisions

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In 23-limit [[~~Just_intonation|~~Just Intonation]], all ratios contain no prime factors higher than 23.

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:

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

[[~~94edo|~~94edo]] is the first [[~~EDO|~~EDO]] to be consistent in the 23-limit. The smallest EDO where the 23-limit is distinctly consistent, meaning each element of the tonality diamond is distinguished, is [[282edo~~|282edo~~]].

[[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: [[Harmonic_Limit|Harmonic Limit]], [[19-limit|19-limit]], [[17-limit|17-limit]]

@@ Line 1: / Line 1: @@
-In 23-limit [[Just_intonation|Just Intonation]], all ratios contain no prime factors higher than 23.
+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:
@@ Line 47: / Line 47: @@
 /12 .. 1126.391¢ .. 23o8 .. twetho 8ve
-[[94edo|94edo]] is the first [[EDO|EDO]] to be consistent in the 23-limit. The smallest EDO where the 23-limit is distinctly consistent, meaning each element of the tonality diamond is distinguished, is [[282edo|282edo]].
+[[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: [[Harmonic_Limit|Harmonic Limit]], [[19-limit|19-limit]], [[17-limit|17-limit]]