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 [[3/1|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|15-]][[Odd limit|~~odd-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.

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.

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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 {{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 [[3/1|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|15-]][[Odd limit|odd-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.
 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.
+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 ==