13-limit - Revision history

Lucius Chiaraviglio: Fix grammar

2026-05-29T15:08:50Z

Fix grammar

← Older revision		Revision as of 15:08, 29 May 2026
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	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.		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 ~~convenient~~ 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]].		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 approximation ==

FloraC: /* Edo approximation */ - bold (what's the criteria?)

2026-05-29T08:32:22Z

Edo approximation: - bold (what's the criteria?)

← Older revision		Revision as of 08:32, 29 May 2026
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	[[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]].		[[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 ~~(bold ones do particularly well in this subgroup)~~.		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. }}		{{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. }}

FloraC: Rework

2026-05-29T08:23:56Z

Rework

← Older revision		Revision as of 08:23, 29 May 2026
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	{{Prime limit navigation\|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. ~~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 [[3/1\|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.		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:		These things are contained by the 13-limit, but not the 11-limit:
	* The [[13-odd-limit\|13-]] and [[15-~~odd-limit\|15-]][[Odd limit\|~~odd-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.		* 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.		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".		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.		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 ~~naturally~~ rank-reduced into the 7-limit by tempering out 4096/4095 ~~and 2080/2079~~, resulting in [[~~Catalog of rank-4 temperaments#Tridecimal olympic\|~~olympic]], which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32. ~~Another~~ notable rank-~~reduction is~~ [[cassaschismic]], which rank-reduces olympic by equating 64/63 with the ~~pythagorean~~ comma~~, thus finding the 2.3.7.11 subgroup along a chain of ever so slightly sharp fifths~~.		The subgroup can be convenient 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 approximation ==

Eufalesio: Things

2026-05-28T19:18:10Z

Things

← Older revision		Revision as of 19:18, 28 May 2026
Line 10:		Line 10:
	In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.		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".		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".

	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.		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 naturally rank-reduced into the 7-limit by tempering out 4096/4095 and 2080/2079, resulting in [[Catalog of rank-4 temperaments#Tridecimal olympic\|olympic]], which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32. Another notable rank-reduction is [[cassaschismic]], which rank-reduces olympic by equating 64/63 with the pythagorean comma, thus finding the 2.3.7.11 subgroup along a chain of ever so slightly sharp fifths.

	== Edo approximation ==		== 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]].		[[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.		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 (bold ones do particularly well in this subgroup).

	{{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. }}		{{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. }}

FloraC: Note completion of the 4th octave

2026-05-19T11:11:59Z

Note completion of the 4th octave

← Older revision		Revision as of 11:11, 19 May 2026
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	These things are contained by the 13-limit, but not the 11-limit:		These things are contained by the 13-limit, but not the 11-limit:
	* The [[13-odd-limit\|13-]] and [[15-odd-limit\|15-]][[Odd limit\|odd-limit]];		* The [[13-odd-limit\|13-]] and [[15-odd-limit\|15-]][[Odd limit\|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.

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

FloraC: Consistency with other pages. Style

2026-04-19T10:21:22Z

Consistency with other pages. Style

← Older revision		Revision as of 10:21, 19 April 2026
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	{{Prime limit navigation\|13}}		{{Prime limit navigation\|13}}
	The '''13-limit''' (~~AKA~~ ''yazalatha'' in [[color notation]]) ~~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. 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 [[3/1\|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.		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.
Line 10:		Line 10:
	In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.		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".		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".

	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.		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.

Eufalesio: color notation

2026-04-19T09:46:30Z

color notation

← Older revision		Revision as of 09:46, 19 April 2026
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	{{Prime limit navigation\|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-limit''' (AKA ''yazalatha'' in [[color notation]]) 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 [[3/1\|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.		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.

FloraC: Cleanup & recategorize

2026-02-06T08:26:34Z

Cleanup & recategorize

← Older revision		Revision as of 08:26, 6 February 2026
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	* [[Augmented chords in just intonation, some]] (they are 13-limit)		* [[Augmented chords in just intonation, some]] (they are 13-limit)

	== ~~Notes~~ ==		== References ==

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

Overthink: link to odd limit instead of explaining here, + more links, move consonances sentence to a new earlier paragraph

2026-02-05T04:40:20Z

link to odd limit instead of explaining here, + more links, move consonances sentence to a new earlier paragraph

← Older revision		Revision as of 04:40, 5 February 2026
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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''' 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.		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:		These things are contained by the 13-limit, but not the 11-limit:
	* The [[13-odd-limit\|13-]] and [[15-odd-limit]];		* The [[13-odd-limit\|13-]] and [[15-odd-limit\|15-]][[Odd limit\|odd-limit]];
	* Mode 7 and 8 of the harmonic or subharmonic series.		* Mode 7 and 8 of the harmonic or subharmonic 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 via vanishing of [[364/363]], 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".

			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.

	== Edo approximation ==		== Edo approximation ==

Lucius Chiaraviglio: /* Music */ Add Claire Rose's ''fretless harp guitar study'' (2026); fix sort order

2026-01-27T12:45:28Z

Music: Add Claire Rose's ''fretless harp guitar study'' (2026); fix sort order

← Older revision		Revision as of 12:45, 27 January 2026
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	== Music ==		== 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]]		; [[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 ''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}}		* [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]]		; [[Ben Johnston]]
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	; [[Claudi Meneghin]]		; [[Claudi Meneghin]]
	* [http://web.archive.org/web/20160412025512/http://soonlabel.com/xenharmonic/archives/2089 ''Canon on a ground''] – in 2.11.13 subgroup		* [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]]		; [[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/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.		* [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]]		; [[Randy Wells]]

← Older revision		Revision as of 08:32, 29 May 2026
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	[[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]].		[[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 ~~(bold ones do particularly well in this subgroup)~~.		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. }}		{{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. }}

← Older revision		Revision as of 19:18, 28 May 2026
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	In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.		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".		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".

	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.		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 naturally rank-reduced into the 7-limit by tempering out 4096/4095 and 2080/2079, resulting in [[Catalog of rank-4 temperaments#Tridecimal olympic\|olympic]], which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32. Another notable rank-reduction is [[cassaschismic]], which rank-reduces olympic by equating 64/63 with the pythagorean comma, thus finding the 2.3.7.11 subgroup along a chain of ever so slightly sharp fifths.

	== Edo approximation ==		== 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]].		[[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.		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 (bold ones do particularly well in this subgroup).

	{{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. }}		{{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. }}

← Older revision		Revision as of 10:21, 19 April 2026
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	{{Prime limit navigation\|13}}		{{Prime limit navigation\|13}}
	The '''13-limit''' (~~AKA~~ ''yazalatha'' in [[color notation]]) ~~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. 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 [[3/1\|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.		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.
Line 10:		Line 10:
	In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential [[consonance]]s.		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".		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".

	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.		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.

← Older revision		Revision as of 09:46, 19 April 2026
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	{{Prime limit navigation\|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-limit''' (AKA ''yazalatha'' in [[color notation]]) 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 [[3/1\|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.		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.