37edo
| ← 36edo | 37edo | 38edo → |
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 | |||
| 3 | 97.30 | ||||
| 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 | ||||
| 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 | ||