13-limit: Difference between revisions

(5 intermediate revisions by 2 users not shown)

Line 14:

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 121:

; [[User:Tristanbay|Tristan Bay]]

* [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/8gZD_Za1JPQ ''8:9:10:11:12:13:14 improv''] (2026) — on modern lyre with backing track made using the same instrument in different ways

; [[E8 Heterotic]]

Line 164:

Line 167:

* [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 ==

@@ Line 14: / Line 14: @@
 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 121: / Line 121: @@
 ; [[User:Tristanbay|Tristan Bay]]
 * [https://youtu.be/ouUV2Uwr2qI ''Junp''] – in [[User:Tristanbay/Margo Scale|a 2.3.11/7.13/7 subgroup JI scale]]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/8gZD_Za1JPQ ''8:9:10:11:12:13:14 improv''] (2026) — on modern lyre with backing track made using the same instrument in different ways
 ; [[E8 Heterotic]]
@@ Line 164: / Line 167: @@
 * [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 ==

13-limit: Difference between revisions

Lucius Chiaraviglio (talk | contribs)