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 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.

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 ==

[[~~EDO~~]]s which represent 13-limit intervals better (with decreasing [[TE ~~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.

[[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.

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.

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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|-

! Ratio

! Cents ~~Value~~

! Cents value

! colspan="2" | [[Color name~~|Color Name~~]]

! colspan="2" | [[Color name]]

! Name

|-

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; [[Claudi Meneghin]]

* [http://web.archive.org/web/20160412025512/http://soonlabel.com/xenharmonic/archives/2089 ''Canon on a ground''] – in 2.11.13 subgroup

; [[Gene Ward Smith]]

* [https://archive.org/details/ThrenodyForTheVictimsOfWolfgangAmadeusMozart ''Threnody for the Victims of Wolfgang Amadeus Mozart''] (archived 2010) – 13-limit JI in [[6079edo]] tuning

* [https://archive.org/details/RoughDiamond ''Rough Diamond''] (archived 2010) a.k.a. ''Diamond in the Rough''<ref>[http://lumma.org/tuning/gws/gene.html xenharmony.org mirror | ''Gene's Music'']</ref> – symphonic con brio using the Partch 13-odd-limit tonality diamond as a scale.

; [[User:Tristanbay|Tristan Bay]]

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* [[Gallery of just intervals]]

* [[Tridecimal neutral seventh chord]]

== Notes ==

[[Category:13-limit| ]]

@@ Line 2: / Line 2: @@
 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.
+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 ==
-[[EDO]]s which represent 13-limit intervals better (with decreasing [[TE 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.
+[[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.
-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.
+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 ==
@@ Line 17: / Line 17: @@
 |-
 ! Ratio
-! Cents Value
+! Cents value
-! colspan="2" | [[Color name|Color Name]]
+! colspan="2" | [[Color name]]
 ! Name
 |-
@@ Line 130: / Line 130: @@
 ; [[Claudi Meneghin]]
 * [http://web.archive.org/web/20160412025512/http://soonlabel.com/xenharmonic/archives/2089 ''Canon on a ground''] – in 2.11.13 subgroup
+; [[Gene Ward Smith]]
+* [https://archive.org/details/ThrenodyForTheVictimsOfWolfgangAmadeusMozart ''Threnody for the Victims of Wolfgang Amadeus Mozart''] (archived 2010) – 13-limit JI in [[6079edo]] tuning
+* [https://archive.org/details/RoughDiamond ''Rough Diamond''] (archived 2010) a.k.a. ''Diamond in the Rough''<ref>[http://lumma.org/tuning/gws/gene.html xenharmony.org mirror | ''Gene's Music'']</ref> – symphonic con brio using the Partch 13-odd-limit tonality diamond as a scale.
 ; [[User:Tristanbay|Tristan Bay]]
@@ Line 144: / Line 148: @@
 * [[Gallery of just intervals]]
 * [[Tridecimal neutral seventh chord]]
+== Notes ==
 [[Category:13-limit| ]] <!-- main article -->