37edo

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

37 equal divisions of the octave (abbreviated 37edo or 37ed2), also called 37-tone equal temperament (37tet) or 37 equal temperament (37et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 37 equal parts of about 32.4 ¢ each. Each step represents a frequency ratio of 2^1/37, or the 37th root of 2.

Theory

37edo is the 10th no-3 zeta peak edo, containing very accurate approximations of harmonics 5, 7, 11 and 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 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 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.

Odd harmonics

Approximation of odd harmonics in 37edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+11.6	+2.9	+4.1	-9.3	+0.0	+2.7	+14.4	-7.7	-5.6	+15.7	-12.1
Error	relative (%)	+36	+9	+13	-29	+0	+8	+45	-24	-17	+48	-37
Steps (reduced)		59 (22)	86 (12)	104 (30)	117 (6)	128 (17)	137 (26)	145 (34)	151 (3)	157 (9)	163 (15)	167 (19)

Subsets and supersets

37edo is the 12th prime edo, following 31edo and coming before 41edo.

74edo, which doubles it, provides an alternative approximation to harmonic 3 that supports meantone. 111edo, which triples it, gives a very accurate approximation of harmonic 3, and manifests itself as a great higher-limit system. 296edo, which slices its step in eight, is a good 13-limit system.

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 Oceanfront temperament.

37edo can only barely be considered as "dual-fifth", because the sharp fifth is 12 cents sharp of 3/2, has a regular diatonic scale, and can be interpreted as somewhat accurate regular temperaments like archy and the aforementioned oceanfront. In contrast, the flat fifth is 21 cents flat and the only low-limit interpretation is as the very inaccurate mavila.

Since both fifths do not support meantone, 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
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

List of 37et rank two temperaments by badness

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, oceanfront
16\37	Undecimation
17\37	Freivald, emka, onzonic
18\37