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 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 limit]]s due to a substantial increment in its harmonic contents. Specifically, these things are contained by the 23-limit, but not the 19-limit:

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 {{Prime limit navigation|23}}
 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 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 limit]]s due to a substantial increment in its harmonic contents. Specifically, these things are contained by the 23-limit, but not the 19-limit: