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 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.

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.

* Mode 7 and 8 of the harmonic or subharmonic series; this means it completes the 4th octave of those series.

~~The~~ 13-limit ~~intervals~~ of the ~~[[2.3.~~13 ~~subgroup]], such~~ as [[~~13/12]] and [[16/13~~]]~~, are close to neutral intervals, but are further from true (hemipythagorean) neutral intervals than [[2.3.11 subgroup]] intervals, and may thus be termed "subneutral" and "superneutral"~~.

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

The 13~~- or 15-odd~~-limit ~~consists~~ of ~~intervals whose numerators~~ and ~~denominators~~, ~~when all factors of two have been removed~~, are ~~less~~ than ~~or equal to 13 or 15~~, ~~respectively~~. 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. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance~~]]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. ~~Another example is~~ the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit but can also be used as they are.

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 ==

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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]]

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* [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]]

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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

; [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.

~~; [[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]]

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

== ~~Notes~~ ==

== 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 {{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''' (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.
 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.
+* Mode 7 and 8 of the harmonic or subharmonic series; this means it completes the 4th octave of those series.
-The 13-limit intervals of the [[2.3.13 subgroup]], such as [[13/12]] and [[16/13]], are close to neutral intervals, but are further from true (hemipythagorean) neutral intervals than [[2.3.11 subgroup]] intervals, and may thus be termed "subneutral" and "superneutral".
+In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.
-The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. 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. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]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. Another example is the neogothic major and minor triads of [[22:28:33]] and [[22:26:33]], which can be tempered to the 13-odd-limit but can also be used as they are.
+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 ==
@@ Line 115: / 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]]
@@ Line 125: / Line 131: @@
 * [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]]
@@ Line 145: / Line 148: @@
 ; [[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.
-; [[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]]
@@ Line 164: / Line 170: @@
 * [[Augmented chords in just intonation, some]] (they are 13-limit)
-== Notes ==
+== References ==
 [[Category:13-limit| ]] <!-- main article -->
+[[Category:Rank-6 temperaments]]
 [[Category:Lists of intervals]]
 [[Category:Listen]]
-[[Category:Rank 6]]