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''' (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-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 [[3/1|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 approximations ==~~

These things are contained by the 13-limit, but not the 11-limit:

~~A list of~~ [[~~edo~~]]~~s which represent 13~~-limit ~~intervals better: {{EDOs| 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }}~~ and ~~so on~~.

* 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.

~~Another list~~ of ~~edos which tunes~~ the 13-limit ~~well relative to their size (~~[[~~TE relative error|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.

In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.

: ~~'''~~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.

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 ==

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{| class="wikitable"

|-

! Ratio

! Cents ~~Value~~

! Cents value

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

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

! Name

|-

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== 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}} ~~(404 error as of 2/5/2020)~~

* [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}} (~~404 error~~ as of 2/5/~~2020~~)

* [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)

== See also ==

* [[13-odd-limit]]

* [[Gallery of just intervals]]

* [[Tridecimal neutral seventh chord]]

* [[Augmented chords in just intonation, some]] (they are 13-limit)

== References ==

[[Category:13-limit| ]]

[[Category:Rank-6 temperaments]]

[[Category:Lists of intervals]]

[[Category:Listen]]

~~[[Category:Rank 6]]~~

@@ Line 1: / Line 1: @@
 {{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''' (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-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 [[3/1|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 approximations ==
+These things are contained by the 13-limit, but not the 11-limit:
-A list of [[edo]]s which represent 13-limit intervals better: {{EDOs| 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 }} and so on.
+* 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.
-Another list of edos which tunes the 13-limit well relative to their size ([[TE relative error|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.
+In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.
-: '''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.
+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 ==
@@ Line 15: / Line 27: @@
 {| class="wikitable"
+|-
 ! Ratio
-! Cents Value
+! Cents value
-! colspan="2" | [[Color name|Color Name]]
+! colspan="2" | [[Color name]]
 ! Name
 |-
@@ Line 106: / Line 119: @@
 == 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}} (404 error as of 2/5/2020)
+* [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}} (404 error as of 2/5/2020)
+* [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)
 == See also ==
-* [[13-odd-limit]]
 * [[Gallery of just intervals]]
+* [[Tridecimal neutral seventh chord]]
+* [[Augmented chords in just intonation, some]] (they are 13-limit)
+== References ==
 [[Category:13-limit| ]] <!-- main article -->
+[[Category:Rank-6 temperaments]]
 [[Category:Lists of intervals]]
 [[Category:Listen]]
-[[Category:Rank 6]]