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

These things are contained by the 13-limit, but not the 11-limit:

* The [[13-odd-limit|13-]] and [[15-odd-limit]];

* Mode 7 and 8 of the harmonic or subharmonic series.

== Edo approximation ==

[[Edo]]s which represent 13-limit intervals better (~~with~~ decreasing [[TE error]]): {{EDOs| 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on.

[[Edo]]s which represent 13-limit intervals better ([[monotonicity limit]] ≥ 13 and decreasing [[TE error]]): {{EDOs| 15, 17c, 19, 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

Here is a list of edos which tunes the 13-limit well relative to their size ({{nowrap|[[TE relative error]] < 5.5%}}): {{EDOs| 31, 41, 46, 53, 58, 72, 87, 94, 103, 111, 121, 130, 140, 152f, 159, 183, 190, 198, 212, 217, 224, 270, 282, 296, 301, 311, 320, 328, 342f, 354, 364, 369f, 373, 383, 400, 414, 422, 431, 441, 460, 472, 494 }}, and so on.

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== Music ==

; [[E8 Heterotic]]

* [https://~~youtu~~.be/cUR3MsI-mWM~~?si=-dIKc8d5kFln8H6T~~ ''Justification''] (2022)

* [https://www.youtube.com/watch?v=cUR3MsI-mWM ''Justification''] (2022)

; [[Francium]]

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* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Venusian Cataclysms'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play]{{dead link}}

* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Chord Progression on the Harmonic Overtone Series'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play]{{dead link}}

; [https://youtube.com/@hojominori?si=gqJP3hzvup2YL0sz Hojo Minori]

* [https://www.youtube.com/watch?v=xSIS2lobkTk ''P`rismatic fut`URE''] (2025)

; [[Ben Johnston]]

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== See also ==

* [[13-odd-limit]]

* [[Gallery of just intervals]]

* [[Tridecimal neutral seventh chord]]

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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 factor]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is {{nowrap|2 × 2 × 2 × 5}} and 39 is {{nowrap|3 × 13}}, but [[34/33]] would not, since 34 is {{nowrap|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''' or 13-prime-limit consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. Thus, [[40/39]] would be within the 13-limit, since 40 is {{nowrap|2 × 2 × 2 × 5}} and 39 is {{nowrap|3 × 13}}, but [[34/33]] would not, since 34 is {{nowrap|2 × 17}}, and [[17-limit|17]] is a prime number higher than 13. The 13-limit is the 6th [[prime limit]] and is 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.
+These things are contained by the 13-limit, but not the 11-limit:
+* The [[13-odd-limit|13-]] and [[15-odd-limit]];
+* Mode 7 and 8 of the harmonic or subharmonic series.
 == Edo approximation ==
-[[Edo]]s which represent 13-limit intervals better (with decreasing [[TE error]]): {{EDOs| 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on.
+[[Edo]]s which represent 13-limit intervals better ([[monotonicity limit]] ≥ 13 and decreasing [[TE error]]): {{EDOs| 15, 17c, 19, 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
 Here is a list of edos which tunes the 13-limit well relative to their size ({{nowrap|[[TE relative error]] < 5.5%}}): {{EDOs| 31, 41, 46, 53, 58, 72, 87, 94, 103, 111, 121, 130, 140, 152f, 159, 183, 190, 198, 212, 217, 224, 270, 282, 296, 301, 311, 320, 328, 342f, 354, 364, 369f, 373, 383, 400, 414, 422, 431, 441, 460, 472, 494 }}, and so on.
@@ Line 107: / Line 111: @@
 == Music ==
 ; [[E8 Heterotic]]
-* [https://youtu.be/cUR3MsI-mWM?si=-dIKc8d5kFln8H6T ''Justification''] (2022)
+* [https://www.youtube.com/watch?v=cUR3MsI-mWM ''Justification''] (2022)
 ; [[Francium]]
@@ Line 116: / Line 121: @@
 * [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Venusian Cataclysms'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/02_hill_venusian-cataclysms.mp3 play]{{dead link}}
 * [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Chord Progression on the Harmonic Overtone Series'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play]{{dead link}}
+; [https://youtube.com/@hojominori?si=gqJP3hzvup2YL0sz Hojo Minori]
+* [https://www.youtube.com/watch?v=xSIS2lobkTk ''P`rismatic fut`URE''] (2025)
 ; [[Ben Johnston]]
@@ Line 145: / Line 153: @@
 == See also ==
-* [[13-odd-limit]]
 * [[Gallery of just intervals]]
 * [[Tridecimal neutral seventh chord]]