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The 13-prime-limit refers to a constraint on selecting just intonation intervals such that the highest [[prime_numbers|prime number]] in all ratios is 13. Thus, [[40/39|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|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|23/13]] is not within the 13-limit, since [[23-limit|23]] is a prime number higher than 13).
{{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 a superset of the [[11-limit]] and a subset of the [[17-limit]].  


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 need.
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|Edo]]s good for 13-limit are 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 183, 207, 217, 224, 270, 494, 851, 1075, 1282, 1578, 2159, 2190, 2684, 3265, 3535, 4573, 5004, 5585, 6079, 8269, 8539, 13854, 14124, 16808, 20203, 22887, 28742, 32007, 37011, 50434, 50928, 51629, 54624, 56202, 59467, 64471, 65052, ... .
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 ([[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.  
 
'''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 ==
== Intervals ==
Here are all the 15-odd-limit intervals of 13:
Here are all the 15-odd-limit intervals of 13:
{| class="wikitable"
{| class="wikitable"
!Ratio
!Cents Value
! colspan="2" |[[Kite's color notation|Interval name]]
|-
|-
|14/13
! Ratio
|128
! Cents value
|3uz2
! colspan="2" | [[Color name]]
|thuzo 2nd
! Name
|-
| 14/13
| 128.298
| 3uz2
| thuzo 2nd
| tridecimal supraminor second
|-
|-
|13/12
| 13/12
|139
| 138.573
|3o2
| 3o2
|tho 2nd
| tho 2nd
| tridecimal subneutral second
|-
|-
|15/13
| 15/13
|248
| 247.741
|3uy2
| 3uy2
|thuyo 2nd
| thuyo 2nd
| tridecimal semifourth
|-
|-
|13/11
| 13/11
|289
| 289.210
|3o1u3
| 3o1u3
|tholu 3rd
| tholu 3rd
| tridecimal minor third
|-
|-
|16/13
| 16/13
|359
| 359.472
|3u3
| 3u3
|thu 3rd
| thu 3rd
| tridecimal supraneutral third
|-
|-
|13/10
| 13/10
|454
| 454.214
|3og4
| 3og4
|thogu 4th
| thogu 4th
| tridecimal naiadic
|-
|-
|18/13
| 18/13
|563
| 563.382
|3u4
| 3u4
|thu 4th
| thu 4th
| tridecimal sub-tritone
|-
|-
|13/9
| 13/9
|637
| 636.618
|3o5
| 3o5
|tho 5th
| tho 5th
| tridecimal super-tritone
|-
|-
|20/13
| 20/13
|746
| 745.786
|3uy5
| 3uy5
|thuyo 5th
| thuyo 5th
| tridecimal cocytic
|-
|-
|13/8
| 13/8
|841
| 840.528
|3o6
| 3o6
|tho 6th
| tho 6th
| tridecimal subneutral sixth
|-
|-
|22/13
| 22/13
|911
| 910.790
|3u1o6
| 3u1o6
|thulo 6th
| thulo 6th
| tridecimal major sixth
|-
|-
|26/15
| 26/15
|952
| 952.259
|3og7
| 3og7
|thogu 7th
| thogu 7th
| tridecimal semitwelfth
|-
|-
|24/13
| 24/13
|1061
| 1061.427
|3u7
| 3u7
|thu 7th
| thu 7th
| tridecimal supraneutral seventh
|-
|-
|13/7
| 13/7
|1072
| 1071.702
|3or7
| 3or7
|thoru 7th
| thoru 7th
| tridecimal submajor seventh
|}
|}
See: [[Gallery of Just Intervals]]


.
== Music ==
 
; [[E8 Heterotic]]
* [https://www.youtube.com/watch?v=cUR3MsI-mWM ''Justification''] (2022)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=ratGb2qTStQ ''Bicycle Wheels''] (2023)
 
; [[Dave Hill]]
* [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]]
* ''String Quartet No. 5'' (1979) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-5 Bandcamp] | [https://www.youtube.com/watch?v=jOpQwiEB4g0 YouTube] – performed by Kepler Quartet
* ''String Quartet No. 7'' (1984)
** "Movt. 1" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-scurrying-forceful-intense Bandcamp] | [https://www.youtube.com/watch?v=-TdFgtAf5Cg YouTube]
** "Movt. 2" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-eerie Bandcamp] | [https://www.youtube.com/watch?v=Tq9cjvgnbAY YouTube]
** "Movt. 3" – [https://newworldrecords.bandcamp.com/track/string-quartet-no-7-with-solemnity Bandcamp] | [https://www.youtube.com/watch?v=jgFQAGyF0Gw YouTube]
:: performed by Kepler Quartet
 
; [[Kaiveran Lugheidh]]
* [https://soundcloud.com/vale-10/unlicensed-copy ''Unlicensed Copy''] (2017) – mostly 7-limit with some erstwhile 13-based chromaticisms
 
; [[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]]
* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=wbZXArV5ffw ''Dying Visions of a Lonesome Machine''] (2021)
* [https://www.youtube.com/watch?v=bA6wr07PiYE ''Avenoir''] (2022)
* [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)
* [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)


=Music=
== See also ==
[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|Dave Hill]]
* [[Gallery of just intervals]]
* [[Tridecimal neutral seventh chord]]
* [[Augmented chords in just intonation, some]] (they are 13-limit)


[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
== Notes ==


=See also=
[[Category:13-limit| ]] <!-- main article -->
[[Harmonic_Limit|Harmonic limit]]     
[[Category:Lists of intervals]]
[[Category:13-limit]]
[[Category:Listen]]
[[Category:limit]]
[[Category:Rank 6]]
[[Category:listen]]
[[Category:prime_limit]]
[[Category:rank_6]]
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