37edo: Difference between revisions
Undo (there's not a rank-2 ("linear") jacobin). This section does need some serious cleanup |
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 37 (prime) | | Prime factorization = 37 (prime) | ||
| Step size = 32.432¢ | | Step size = 32.432¢ | ||
| Fifth = 22\37 | | Fifth = 22\37 (713.514¢) | ||
| Major 2nd = 7\37 | | Major 2nd = 7\37 (227¢) | ||
| | | Semitones = 6:1 (195¢ : 32¢) | ||
| | | Consistency = 7 | ||
| Monotonicity = 15 | |||
}} | }} | ||
''' | 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 == | == Theory == | ||
{| | {{Primes in edo|37}} | ||
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]] 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. | |||
37edo is the 12th [[prime EDO]], following [[31edo]] and coming before [[41edo]]. | |||
=== Subgroups === | === Subgroups === | ||
37edo offers close approximations to [[Harmonic series|harmonics]] 5, 7, 11, and 13, and a usable approximation of 9 as well. | |||
* 12\37 = 389.2 cents | |||
12\37 = 389.2 cents | * 30\37 = 973.0 cents | ||
* 17\37 = 551.4 cents | |||
30\37 = 973.0 cents | * 26\37 = 843.2 cents | ||
* [6\37 = 194.6 cents] | |||
17\37 = 551.4 cents | |||
26\37 = 843.2 cents | |||
[6\37 = 194.6 cents] | |||
This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as | This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as [[111edo]]. In fact, on the larger [[k*N subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111edo, it tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as [[74edo]]. | ||
=== | === Dual fifths === | ||
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in | The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo: | ||
The flat fifth is 21\37 = 681.1 cents (37b val) | The flat fifth is 21\37 = 681.1 cents (37b val) | ||
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"major third" = 14\37 = 454.1 cents | "major third" = 14\37 = 454.1 cents | ||
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[ | If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The Biosphere|Biome]] temperament. | ||
Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in | Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo. | ||
37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below). | |||
== Intervals == | == Intervals == | ||