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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''' (a.k.a. ''yazalatha'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 13. It is the 6th [[prime limit]] and is a superset of the [[11-limit]] and a subset of the [[17-limit]]. An example of an interval in the 13-limit is [[40/39]], since 40 is {{nowrap| 2 × 2 × 2 × 5 }} and 39 is {{nowrap| 3 × 13 }}; a counterexample is [[34/33]], since 34 is {{nowrap| 2 × 17 }}, and [[17/1|17]] is a prime number higher than 13.  


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 [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] represented by each dimension. The prime [[2/1|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; this means it completes the 4th octave of those series.
 
In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.
 
The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to [[neutral (interval quality)|neutral]] intervals, but are further from true ([[hemipyth]]agorean) neutral intervals than [[2.3.11 subgroup]] intervals, and thus may be termed "subneutral" and "superneutral". Such intervals can be obtained by translating a Pythagorean interval by the tridecimal quartertone of [[1053/1024]].
 
As prime limits increase, ratios containing different primes over [[3/1|3]] in the numerator and denominator become more and more numerous. Such ratios include [[15/14]], [[14/13]], [[11/10]], [[15/13]], [[13/11]], [[14/11]], [[13/10]], [[15/11]], [[7/5]], and their [[octave complement]]s. An example of a way to use these intervals is to build {{w|tertian harmony|tertian}} triads such as [[10:13:15]], which consists of a 13/10 ultramajor third and a [[15/13]] inframinor third. Other examples include the [[neogothic major and minor]] triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit via vanishing of [[364/363]], but can also be used as they are.
 
The subgroup can be conveniently rank-reduced into the 7-limit without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]], resulting in the [[olympic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32. Other notable rank-reductions include [[orthoschismic]] and [[cassaschismic]], which rank-reduces olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]].
 
== 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"
! colspan="2" |[[Kite's color notation|Interval name]]
!Ratio
!Cents Value
|-
|-
|3uz2
! Ratio
|thuzo 2nd
! Cents value
|14/13
! colspan="2" | [[Color name]]
|128
! Name
|-
|-
|3o2
| 14/13
|tho 2nd
| 128.298
|13/12
| 3uz2
|139
| thuzo 2nd
| tridecimal supraminor second
|-
|-
|3uy2
| 13/12
|thuyo 2nd
| 138.573
|15/13
| 3o2
|248
| tho 2nd
| tridecimal subneutral second
|-
|-
|3o1u3
| 15/13
|tholu 3rd
| 247.741
|13/11
| 3uy2
|289
| thuyo 2nd
| tridecimal semifourth
|-
|-
|3u3
| 13/11
|thu 3rd
| 289.210
|16/13
| 3o1u3
|359
| tholu 3rd
| tridecimal minor third
|-
|-
|3og4
| 16/13
|thogu 4th
| 359.472
|13/10
| 3u3
|454
| thu 3rd
| tridecimal supraneutral third
|-
|-
|3u4
| 13/10
|thu 4th
| 454.214
|18/13
| 3og4
|563
| thogu 4th
| tridecimal naiadic
|-
|-
|3o5
| 18/13
|tho 5th
| 563.382
|13/9
| 3u4
|637
| thu 4th
| tridecimal sub-tritone
|-
|-
|3uy5
| 13/9
|thuyo 5th
| 636.618
|20/13
| 3o5
|746
| tho 5th
| tridecimal super-tritone
|-
|-
|3o6
| 20/13
|tho 6th
| 745.786
|13/8
| 3uy5
|841
| thuyo 5th
| tridecimal cocytic
|-
|-
|3u1o6
| 13/8
|thulo 6th
| 840.528
|22/13
| 3o6
|911
| tho 6th
| tridecimal subneutral sixth
|-
|-
|3og7
| 22/13
|thogu 7th
| 910.790
|26/15
| 3u1o6
|952
| thulo 6th
| tridecimal major sixth
|-
|-
|3u7
| 26/15
|thu 7th
| 952.259
|24/13
| 3og7
|1061
| thogu 7th
| tridecimal semitwelfth
|-
|-
|3or7
| 24/13
|thoru 7th
| 1061.427
|13/7
| 3u7
|1072
| thu 7th
| tridecimal supraneutral seventh
|-
| 13/7
| 1071.702
| 3or7
| thoru 7th
| tridecimal submajor seventh
|}
|}
See: [[Gallery of Just Intervals]]


.
== Music ==
; [[User:Tristanbay|Tristan Bay]]
* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
 
; [[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}}
 
; [[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
 
; [[Thomas Leroy Meier]]
* [https://www.instagram.com/p/DQnh9iykW8O/?hl=en ''WIP cover of Sheik's Theme by Koji Kondo''] (2025; original was 1996) - tuning adapted from {{w|Ibn Sina}}
 
; [[Claudi Meneghin]]
* [http://web.archive.org/web/20160412025512/http://soonlabel.com/xenharmonic/archives/2089 ''Canon on a ground''] – in 2.11.13 subgroup
 
; [https://youtube.com/@hojominori?si=gqJP3hzvup2YL0sz Hojo Minori]
* [https://www.youtube.com/watch?v=xSIS2lobkTk ''P`rismatic fut`URE''] (2025)
 
; [[Claire Rose]]
* [https://www.youtube.com/shorts/SpqznLRjGGA ''fretless harp guitar study] (2026)
 
; [[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.
 
; [[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
== References ==


=See also=
[[Category:13-limit| ]] <!-- main article -->
[[Harmonic_Limit|Harmonic limit]]     
[[Category:Rank-6 temperaments]]
[[Category:13-limit]]
[[Category:Lists of intervals]]
[[Category:limit]]
[[Category:Listen]]
[[Category:listen]]
[[Category:prime_limit]]
[[Category:rank_6]]
Retrieved from "https://en.xen.wiki/w/13-limit"