View source for 13-limit

{{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-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 ([[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 ==
Here are all the 15-odd-limit intervals of 13:

{| class="wikitable"
|-
! Ratio
! Cents value
! colspan="2" | [[Color name]]
! Name
|-
| 14/13
| 128.298
| 3uz2
| thuzo 2nd
| tridecimal supraminor second
|-
| 13/12
| 138.573
| 3o2
| tho 2nd
| tridecimal subneutral second
|-
| 15/13
| 247.741
| 3uy2
| thuyo 2nd
| tridecimal semifourth
|-
| 13/11
| 289.210
| 3o1u3
| tholu 3rd
| tridecimal minor third
|-
| 16/13
| 359.472
| 3u3
| thu 3rd
| tridecimal supraneutral third
|-
| 13/10
| 454.214
| 3og4
| thogu 4th
| tridecimal naiadic
|-
| 18/13
| 563.382
| 3u4
| thu 4th
| tridecimal sub-tritone
|-
| 13/9
| 636.618
| 3o5
| tho 5th
| tridecimal super-tritone
|-
| 20/13
| 745.786
| 3uy5
| thuyo 5th
| tridecimal cocytic
|-
| 13/8
| 840.528
| 3o6
| tho 6th
| tridecimal subneutral sixth
|-
| 22/13
| 910.790
| 3u1o6
| thulo 6th
| tridecimal major sixth
|-
| 26/15
| 952.259
| 3og7
| thogu 7th
| tridecimal semitwelfth
|-
| 24/13
| 1061.427
| 3u7
| thu 7th
| tridecimal supraneutral seventh
|-
| 13/7
| 1071.702
| 3or7
| thoru 7th
| tridecimal submajor seventh
|}

== 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)

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

== Notes ==

[[Category:13-limit| ]] <!-- main article -->
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 6]]