37edo: Difference between revisions

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== Theory ==

~~=== RTT ===~~

37edo is the 10th [[no-threes subgroup temperaments|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.

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

37edo is the 10th [[No-threes subgroup temperaments|no-3]] ~~zeta~~ [[~~Zeta~~ peak edo~~|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.

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]]).

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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.

In the no-3 [[13-odd-limit]], [[37edo]] maintains the smallest relative error of any ~~EDO~~ until [[851edo]], and the smallest absolute error until [[103edo]].

In the no-3 [[13-odd-limit]], 37edo maintains the smallest relative error of any edo until [[851edo]], and the smallest absolute error until [[103edo]]{{clarify}}.

=== Odd harmonics ===

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=== Rank-2 temperaments ===

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

* [[List of 37et rank two temperaments by badness]]

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 == Theory ==
-=== RTT ===
+edo is the 10th [[no-threes subgroup temperaments|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.
-''See [[regular temperament]] for more about what all this means and how to use it.''
-edo is the 10th [[No-threes subgroup temperaments|no-3]] zeta [[Zeta peak edo|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.
 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]]).
@@ Line 18: / Line 15: @@
 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.
-In the no-3 [[13-odd-limit]], [[37edo]] maintains the smallest relative error of any EDO until [[851edo]], and the smallest absolute error until [[103edo]].
+In the no-3 [[13-odd-limit]], 37edo maintains the smallest relative error of any edo until [[851edo]], and the smallest absolute error until [[103edo]]{{clarify}}. <!-- what is the metric being used? -->
 === Odd harmonics ===
@@ Line 689: / Line 686: @@
 === Rank-2 temperaments ===
-''See [[regular temperament]] for more about what all this means and how to use it.''
 * [[List of 37et rank two temperaments by badness]]