13-limit: Difference between revisions

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The '''13-limit''' or 13-prime-limit consists of [[just intonation]] ~~intervals~~ such that the highest [[prime number]] in all ratios is 13. Thus, [[40/39]] would be ~~allowable~~, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but 34/33 would not ~~be allowable~~, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, [[3/2]] is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. [[23/13]] is not within the 13-limit, since [[23-limit|23]] 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 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-~~prime~~-~~limit~~ 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.

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.

== Edo approximations ==

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 {{Prime limit navigation|13}}
-The '''13-limit''' or 13-prime-limit consists of [[just intonation]] intervals such that the highest [[prime number]] in all ratios is 13. Thus, [[40/39]] would be allowable, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but 34/33 would not be allowable, since 34 is 2 × 17, and [[17-limit|17]] is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, [[3/2]] is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. [[23/13]] is not within the 13-limit, since [[23-limit|23]] 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 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-prime-limit 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.
+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.
 == Edo approximations ==