13-limit: Difference between revisions

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The '''13~~-prime~~-limit''' ~~refers to a constraint on selecting~~ 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]] 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-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.

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== Music ==

* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Venusian Cataclysms] [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play] by [[Dave Hill]] {{dead link}} (404 error as of 2/5/2020)

* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Chord Progression on the Harmonic Overtone Series] [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play] by Dave Hill {{dead link}} (404 error as of 2/5/2020)

== See also ==

* [[Harmonic limit]]

* [[13-odd-limit]]

* [[Gallery of ~~Just Intervals~~]]

* [[Gallery of just intervals]]

[[Category:Just intonation]]

[[Category:Limit]]

[[Category:Prime limit]]

[[Category:13-limit]]

[[Category:13-limit| ]]

[[Category:Interval collection]]

[[Category:Rank 6]]

[[Category:Listen]]

~~[[Category:Rank 6]]~~

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-The '''13-prime-limit''' refers to a constraint on selecting 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]] 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-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.
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 == Music ==
 * [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Venusian Cataclysms] [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play] by [[Dave Hill]] {{dead link}} (404 error as of 2/5/2020)
 * [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Chord Progression on the Harmonic Overtone Series] [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play] by Dave Hill  {{dead link}} (404 error as of 2/5/2020)
 == See also ==
 * [[Harmonic limit]]
 * [[13-odd-limit]]
-* [[Gallery of Just Intervals]]
+* [[Gallery of just intervals]]
+[[Category:Just intonation]]
 [[Category:Limit]]
 [[Category:Prime limit]]
-[[Category:13-limit]]
+[[Category:13-limit| ]] <!-- main article -->
+[[Category:Interval collection]]
+[[Category:Rank 6]]
 [[Category:Listen]]
-[[Category:Rank 6]]