13-limit: Difference between revisions

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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.

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 a 7-limit basis 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, among other properties. Other notable rank-reductions include [[orthoschismic]] and [[septischismic]], which rank-reduce olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]] and thus rank-reducing the 13-limit into a 2.3.7 basis or a 5-limit basis.

== Edo approximation ==

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* [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)

* [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)

; [[Austin White]]

* [https://www.youtube.com/watch?v=jM6_u8cK9fE ''Break for string trio''] (2026) — has link to score in video description; score notated in [[HEJI]]

== See also ==

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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.
-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 a 7-limit basis 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, among other properties. Other notable rank-reductions include [[orthoschismic]] and [[septischismic]], which rank-reduce olympic by equating 81/80 or 64/63 respectively with the [[Pythagorean comma]] and thus rank-reducing the 13-limit into a 2.3.7 basis or a 5-limit basis.
 == Edo approximation ==
@@ Line 164: / Line 164: @@
 * [https://www.youtube.com/watch?v=rBS2gGTostA ''I Was a Teenage Boltzmann Brain''] (2022)
 * [https://www.youtube.com/watch?v=NwsMMnOTcQ4 ''Atlas Apassionata''] (2022)
+; [[Austin White]]
+* [https://www.youtube.com/watch?v=jM6_u8cK9fE ''Break for string trio''] (2026) — has link to score in video description; score notated in [[HEJI]]
 == See also ==