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 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''' 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 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~~ approximation ==

== EDO approximation ==

~~Here is a list of~~ [[~~edo~~]]s which represent 13-limit intervals better (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 (with decreasing [[Tenney-Euclidean error|Tenney–Euclidean 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.

Here is a list of edos which tunes the 13-limit well relative to their size ([[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.

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.

: '''Note''': [[~~wart~~ notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

'''Note:''' [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==

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{| class="wikitable"

|-

! Ratio

! Cents Value

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; [[User:Tristanbay|Tristan Bay]]

* [https://youtu.be/ouUV2Uwr2qI ''Junp''] - in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]

* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]

; [[Randy Wells]]

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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 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''' 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 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 approximation ==
+== EDO approximation ==
-Here is a list of [[edo]]s which represent 13-limit intervals better (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 (with decreasing [[Tenney-Euclidean error|Tenney&ndash;Euclidean 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.
-Here is a list of edos which tunes the 13-limit well relative to their size ([[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.
+Here is a list of edos which tunes the 13-limit well relative to their size ({{nowrap|[[TE relative error]] &lt; 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.
-: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.
+'''Note:''' [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.
 == Intervals ==
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 {| class="wikitable"
+|-
 ! Ratio
 ! Cents Value
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 ; [[User:Tristanbay|Tristan Bay]]
-* [https://youtu.be/ouUV2Uwr2qI ''Junp''] - in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
+* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
 ; [[Randy Wells]]