37edo: Difference between revisions

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~~The '''~~37 equal divisions of the octave''' ('''37edo'''), or the '''37(-tone) equal temperament''' ('''37tet''', '''37et''') when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave into 37 [[equal]] steps.

== Theory ==

~~{{Primes in~~ edo|~~37}}~~

37edo is the 10th no-3 zeta peak edo, containing very accurate approximations of harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]] and [[13/1|13]]. It has an extremely accurate harmonic 11, being only 0.03 cents sharp.

~~As a no-3s tuning, it is the 10th zeta peak edo, containing very precise 5/4, 7/4, and 11/8.~~ Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).

Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).

37edo has ~~an extremely precise~~ 11/8, ~~being only~~ 0~~.03 cents sharp~~.

37edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a [[generator]] of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] enneatonic [[mos]], which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16-note mos.

~~37edo is also a very accurate~~ equal tuning for [[undecimation]] temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] nonatonic MOS, which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.

=== Odd harmonics ===

37edo is the 12th [[prime ~~EDO~~]], following [[31edo]] and coming before [[41edo]].

=== Subsets and supersets ===

37edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]].

=== Subgroups ===

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 }}
 {{Infobox ET}}
-The '''37 equal divisions of the octave''' ('''37edo'''), or the '''37(-tone) equal temperament''' ('''37tet''', '''37et''') when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave into 37 [[equal]] steps.
+{{EDO intro|37}}
 == Theory ==
-{{Primes in edo|37}}
+edo is the 10th no-3 zeta peak edo, containing very accurate approximations of harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]] and [[13/1|13]]. It has an extremely accurate harmonic 11, being only 0.03 cents sharp.
-As a no-3s tuning, it is the 10th zeta peak edo, containing very precise 5/4, 7/4, and 11/8. Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).
+Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).
-edo has an extremely precise 11/8, being only 0.03 cents sharp.
+edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a [[generator]] of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] enneatonic [[mos]], which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16-note mos.
-edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] nonatonic MOS, which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.
+=== Odd harmonics ===
+{{Harmonics in equal|37}}
-edo is the 12th [[prime EDO]], following [[31edo]] and coming before [[41edo]].
+=== Subsets and supersets ===
+edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]].
 === Subgroups ===