13-limit: Difference between revisions

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The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime number]] in all ~~ratios~~ is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but [[34/33]] would not, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13.

The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime number]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but [[34/33]] would not, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is thus a superset of the [[11-limit]] and a subset of the [[17-limit]].

The 13-limit is a [[Rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.

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== See also ==

* [[Harmonic limit]]

* [[13-odd-limit]]

* [[Gallery of just intervals]]

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 {{Prime limit navigation|13}}
-The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime number]] in all ratios is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but [[34/33]] would not, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13.
+The '''13-limit''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime number]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but [[34/33]] would not, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is thus a superset of the [[11-limit]] and a subset of the [[17-limit]].
 The 13-limit is a [[Rank and codimension|rank-6]] system, and can be modeled in a 5-dimensional [[lattice]], with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a sixth dimension is needed.
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 == See also ==
-* [[Harmonic limit]]
 * [[13-odd-limit]]
 * [[Gallery of just intervals]]