37edo
| ← 36edo | 37edo | 38edo → |
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
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As a no-3s tuning, it is the 10th zeta peak edo. 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 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 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
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\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 111edo. 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 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 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 | 55/54, 56/55 | |||
| 2 | 64.86 | 27/26, 28/27 | |||
| 3 | 97.30 | 55/52 | 16/15 | ||
| 4 | 129.73 | 14/13 | 13/12, 15/14 | 12/11 | |
| 5 | 162.16 | 11/10 | 10/9, 12/11 | 13/12 | |
| 6 | 194.59 | 28/25 | 9/8, 10/9 | ||
| 7 | 227.03 | 8/7 | 9/8 | ||
| 8 | 259.46 | 7/6, 15/13 | |||
| 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, 32/25 | 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 | 15/11 | 18/13 | |
| 18 | 583.78 | 7/5 | 18/13 | ||
| 19 | 616.22 | 10/7 | 13/9 | ||
| 20 | 648.65 | 16/11 | 22/15 | 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, 25/16 | 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, 26/15 | |||
| 30 | 972.97 | 7/4 | 16/9 | ||
| 31 | 1005.41 | 25/14 | 16/9, 9/5 | ||
| 32 | 1037.84 | 20/11 | 9/5, 11/6 | ||
| 33 | 1070.27 | 13/7 | 24/13, 28/15 | 11/6 | |
| 34 | 1102.70 | 104/55 | 15/8 | ||
| 35 | 1135.14 | 27/14, 52/27 | |||
| 36 | 1167.57 | ||||
| 37 | 1200.00 | 2/1 |
Notation
| Degrees | Cents | Ups and Downs Notation | ||
|---|---|---|---|---|
| 0 | 0.00 | Perfect 1sn | P1 | D |
| 1 | 32.43 | Minor 2nd | m2 | Eb |
| 2 | 64.86 | Upminor 2nd | ^m2 | ^Eb |
| 3 | 97.30 | Downmid 2nd | v~2 | ^^Eb |
| 4 | 129.73 | Mid 2nd | ~2 | Ed |
| 5 | 162.16 | Upmid 2nd | ^~2 | vvE |
| 6 | 194.59 | Downmajor 2nd | vM2 | vE |
| 7 | 227.03 | Major 2nd | M2 | E |
| 8 | 259.46 | Minor 3rd | m3 | F |
| 9 | 291.89 | Upminor 3rd | ^m3 | ^F |
| 10 | 324.32 | Downmid 3rd | v~3 | ^^F |
| 11 | 356.76 | Mid 3rd | ~3 | Ft |
| 12 | 389.19 | Upmid 3rd | ^~3 | vvF# |
| 13 | 421.62 | Downmajor 3rd | vM3 | vF# |
| 14 | 454.05 | Major 3rd | M3 | F# |
| 15 | 486.49 | Perfect 4th | P4 | G |
| 16 | 518.92 | Up 4th, Dim 5th | ^4, d5 | ^G, Ab |
| 17 | 551.35 | Downmid 4th, Updim 5th | v~4, ^d5 | ^^G, ^Ab |
| 18 | 583.78 | Mid 4th, Downmid 5th | ~4, v~5 | Gt, ^^Ab |
| 19 | 616.22 | Mid 5th, Upmid 4th | ~5, ^~4 | Ad, vvG# |
| 20 | 648.65 | Upmid 5th, Downaug 5th | ^~5, vA4 | vvA, vG# |
| 21 | 681.08 | Down 5th, Aug 4th | v5, A4 | vA, G# |
| 22 | 713.51 | Perfect 5th | P5 | A |
| 23 | 745.95 | Minor 6th | m6 | Bb |
| 24 | 778.38 | Upminor 6th | ^m6 | ^Bb |
| 25 | 810.81 | Downmid 6th | v~6 | ^^Bb |
| 26 | 843.24 | Mid 6th | ~6 | Bd |
| 27 | 875.68 | Upmid 6th | ^~6 | vvB |
| 28 | 908.11 | Downmajor 6th | vM6 | vB |
| 29 | 940.54 | Major 6th | M6 | B |
| 30 | 972.97 | Minor 7th | m7 | C |
| 31 | 1005.41 | Upminor 7th | ^m7 | ^C |
| 32 | 1037.84 | Downmid 7th | v~7 | ^^C |
| 33 | 1070.27 | Mid 7th | ~7 | Ct |
| 34 | 1102.70 | Upmid 7th | ^~7 | vvC# |
| 35 | 1135.14 | Downmajor 7th | vM7 | vC# |
| 36 | 1167.57 | Major 7th | M7 | C# |
| 37 | 1200.00 | Perfect 8ve | P8 | D |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.5 | [86 -37⟩ | [⟨37 86]] | -0.619 | 0.619 | 1.91 |
| 2.5.7 | 3136/3125, 4194304/4117715 | [⟨37 86 104]] | -0.905 | 0.647 | 2.00 |
| 2.5.7.11 | 176/175, 1375/1372, 65536/65219 | [⟨37 86 104 128]] | -0.681 | 0.681 | 2.10 |
| 2.5.7.11.13 | 176/175, 640/637, 847/845, 1375/1372 | [⟨37 86 104 128 137]] | -0.692 | 0.610 | 1.88 |
- 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ETs doing better in these subgroups are 109, 581, 103, 124 and 93, respectively.
Rank-2 temperaments
| Generator | In patent val | In 37b val |
|---|---|---|
| 1\37 | ||
| 2\37 | Sycamore | |
| 3\37 | Passion | |
| 4\37 | Twothirdtonic | Negri |
| 5\37 | Porcupine / porcupinefish | |
| 6\37 | Didacus / roulette | |
| 7\37 | Shoe / semaja | Shoe / laconic / gorgo |
| 8\37 | Semaphore (37bd) | |
| 9\37 | Gariberttet | |
| 10\37 | Orgone | |
| 11\37 | Beatles | |
| 12\37 | Würschmidt (out-of-tune) | |
| 13\37 | Skwares (37dd) | |
| 14\37 | Ammonite | |
| 15\37 | Ultrapyth | |
| 16\37 | Undecimation | |
| 17\37 | Freivald, emka | |
| 18\37 | ||