37edo

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Deutsch

← 36edo 37edo 38edo →
Prime factorization 37 (prime)
Step size 32.4324 ¢ 
Fifth 22\37 (713.514 ¢)
Semitones (A1:m2) 6:1 (194.6 ¢ : 32.43 ¢)
Dual sharp fifth 22\37 (713.514 ¢)
Dual flat fifth 21\37 (681.081 ¢)
Dual major 2nd 6\37 (194.595 ¢)
Consistency limit 7
Distinct consistency limit 7

37edo is a scale derived from dividing the octave into 37 equal steps. It is the 12th prime edo, following 31edo and coming before 41edo.

Theory

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 +11.6 +2.9 +4.1 +0.0 +2.7 -7.7 -5.6 -12.1
relative (%) 0 +36 +9 +13 +0 +8 -24 -17 -37
nearest edomapping 37 22 12 30 17 26 3 9 19

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 porcupinefish, which is about as accurate as "13-limit porcupine" 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 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 37-edo 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.

Subgroups

37edo offers close approximations to harmonics 5, 7, 11, and 13 [and a usable approximation of 9 as well].

12\37 = 389.2 cents

30\37 = 973.0 cents

17\37 = 551.4 cents

26\37 = 843.2 cents

[6\37edo = 194.6 cents]

This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger 3*37 subgroup 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, 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 74et.

The Two Fifths

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 sharp fifth is 22\37 = 713.5 cents

21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6

"minor third" = 10\37 = 324.3 cents

"major third" = 11\37 = 356.8 cents

22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1

"minor third" = 8\37 = 259.5 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 Biome temperament.

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

Degrees Cents Approximate Ratios
of 2.5.7.11.13.27 subgroup 		Additional Ratios of 3
with a sharp 3/2 		Additional Ratios of 3
with a flat 3/2 		Additional Ratios of 9
with 194.59¢ 9/8
0 0.00 1/1
1 32.43
2 64.86 27/26, 28/27
3 97.30 55/52
4 129.73 14/13 13/12 12/11
5 162.16 11/10 10/9, 12/11 13/12
6 194.59 9/8, 10/9
7 227.03 8/7 9/8
8 259.46 7/6
9 291.89 13/11, 32/27 6/5, 7/6
10 324.32 6/5, 11/9
11 356.76 16/13, 27/22 11/9
12 389.19 5/4
13 421.62 14/11 9/7
14 454.05 13/10 9/7
15 486.49 4/3
16 518.92 27/20 4/3
17 551.35 11/8 18/13
18 583.78 7/5 18/13
19 616.22 10/7 13/9
20 648.65 16/11 13/9
21 681.08 40/27 3/2
22 713.51 3/2
23 745.95 20/13 14/9
24 778.38 11/7 14/9
25 810.81 8/5
26 843.24 13/8, 44/27 18/11
27 875.68 5/3, 18/11
28 908.11 22/13, 27/16 5/3, 12/7
29 940.54 12/7
30 972.97 7/4 16/9
31 1005.41 16/9, 9/5
32 1037.84 11/6 9/5, 11/6
33 1070.27 13/7 24/13 11/6
34 1102.70 104/55
35 1135.14 27/14, 52/27
36 1167.57
37 1200.00 2/1

Just approximation

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 37et.

3-limit 5-limit 7-limit 11-limit 13-limit no-3 11-limit no-3 13-limit no-3 17-limit no-3 19-limit no-3 23-limit
Octave stretch (¢) -3.65 -2.85 -2.50 -2.00 -1.79 -0.681 -0.692 -0.265 -0.0386 +0.299
Error absolute (¢) 3.64 3.18 2.82 2.71 2.52 0.681 0.610 1.11 1.17 1.41
relative (%) 11.24 9.82 8.70 8.37 7.78 2.10 1.88 3.41 3.59 4.35
  • 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ET that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.

Scales

Linear temperaments

Generator "Sharp 3/2" temperaments "Flat 3/2" temperaments (37b val)
1\37
2\37 Sycamore
3\37 Passion
4\37 Twothirdtonic Negri
5\37 Porcupine/porcupinefish
6\37 Roulette
7\37 Semaja Gorgo/Laconic
8\37 Semiphore
9\37 Gariberttet
10\37 Orgone
11\37 Beatles
12\37 Würschmidt (out-of-tune)
13\37 Squares
14\37 Ammonite
15\37 Ultrapyth, not superpyth
16\37 Not mavila (this is "undecimation")
17\37 Emka
18\37

Music

Links

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