37edo

← 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

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 has very accurate approximations of harmonics 5, 7, 11 and 13, making it a good choice for a no-threes approach. Harmonic 11 is particularly accurate, being only 0.03 cents sharp. A usable approximation of 9 is available at 6\37 (194.6 cents) as well, and the no-3 no-15 no-21 23-odd-limit is represented consistently.

This means 37edo is useful in a number of ways. It is 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, it not only shares the same tuning as 19-limit 111edo, but 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. The native perfect fifth at 22\37 (713.5 cents) can also be used, making it a sharp-tending full 13-limit system, and there is the alternative, very flat fifth at 21\37 (681.1 cents), which generates an antidiatonic scale.

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 (%)	+35.6	+8.9	+12.8	-28.7	+0.1	+8.4	+44.5	-23.6	-17.3	+48.4	-37.2
Steps (reduced)		59 (22)	86 (12)	104 (30)	117 (6)	128 (17)	137 (26)	145 (34)	151 (3)	157 (9)	163 (15)	167 (19)

As a tuning of other temperaments

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).

It is a good tuning of the 2.5.11.13 subgroup temperament barton, especially if it is desirable to avoid approximating the perfect fifth.

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.

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.

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).

Miscellaneous properties

37edo has the sharpest fifth of any edo that can possibly be diamond monotone in the 15-odd-limit. The sharpest mapping of 7/4 where 9/8 is mapped no wider than 8/7 is 30\37, and the sharpest possible mapping of 15/8 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13 ~13/12 to half of 7/6. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then 7/4 and 15/8 would have to be flatter. Then 5/4 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see Monotonicity limits of small EDOs). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is 11\19.

Intervals

Inconsistent intervals are in italics.

#	Cents	Approximate ratios of 2.27.5.7.11.13 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	0.0	1/1
1	32.4	55/54, 56/55
2	64.9	27/26, 28/27
3	97.3	128/121, 55/52	16/15
4	129.7	14/13	13/12, 15/14	12/11
5	162.2	11/10	10/9, 12/11	13/12
6	194.6	28/25			9/8, 10/9
7	227.0	8/7	9/8
8	259.5		7/6, 15/13
9	291.9	13/11, 32/27		6/5, 7/6
10	324.3		6/5, 11/9
11	356.8	16/13, 27/22			11/9
12	389.2	5/4
13	421.6	14/11, 32/25			9/7
14	454.1	13/10	9/7
15	486.5		4/3
16	518.9	27/20		4/3
17	551.4	11/8	15/11		18/13
18	583.8	7/5	18/13
19	616.2	10/7	13/9
20	648.6	16/11	22/15		13/9
21	681.1	40/27		3/2
22	713.5		3/2
23	745.9	20/13	14/9
24	778.4	11/7, 25/16			14/9
25	810.8	8/5
26	843.2	13/8, 44/27			18/11
27	875.7		5/3, 18/11
28	908.1	22/13, 27/16		5/3, 12/7
29	940.5		12/7, 26/15
30	973.0	7/4	16/9
31	1005.4	25/14			16/9, 9/5
32	1037.8	20/11	9/5, 11/6
33	1070.3	13/7	24/13, 28/15	11/6
34	1102.7	121/64, 104/55	15/8
35	1135.1	27/14, 52/27
36	1167.6
37	1200.0	2/1

Proposed interval names and solfèges

Table of proposed interval names and solfèges
#	Cents	Ups and downs notation
0	0.0	Perfect 1sn	P1	D
1	32.4	Minor 2nd	m2	Eb
2	64.9	Upminor 2nd	^m2	^Eb
3	97.3	Downmid 2nd	v~2	^^Eb
4	129.7	Mid 2nd	~2	Ed
5	162.2	Upmid 2nd	^~2	vvE
6	194.6	Downmajor 2nd	vM2	vE
7	227.0	Major 2nd	M2	E
8	259.5	Minor 3rd	m3	F
9	291.9	Upminor 3rd	^m3	^F
10	324.3	Downmid 3rd	v~3	^^F
11	356.8	Mid 3rd	~3	Ft
12	389.2	Upmid 3rd	^~3	vvF#
13	421.6	Downmajor 3rd	vM3	vF#
14	454.1	Major 3rd	M3	F#
15	486.5	Perfect 4th	P4	G
16	518.9	Up 4th, dim 5th	^4, d5	^G, Ab
17	551.4	Downmid 4th, updim 5th	v~4, ^d5	^^G, ^Ab
18	583.8	Mid 4th, downmid 5th	~4, v~5	Gt, ^^Ab
19	616.2	Mid 5th, upmid 4th	~5, ^~4	Ad, vvG#
20	648.6	Upmid 5th, downaug 5th	^~5, vA4	vvA, vG#
21	681.1	Down 5th, aug 4th	v5, A4	vA, G#
22	713.5	Perfect 5th	P5	A
23	745.9	Minor 6th	m6	Bb
24	778.4	Upminor 6th	^m6	^Bb
25	810.8	Downmid 6th	v~6	^^Bb
26	843.2	Mid 6th	~6	Bd
27	875.7	Upmid 6th	^~6	vvB
28	908.1	Downmajor 6th	vM6	vB
29	940.5	Major 6th	M6	B
30	973.0	Minor 7th	m7	C
31	1005.4	Upminor 7th	^m7	^C
32	1037.8	Downmid 7th	v~7	^^C
33	1070.3	Mid 7th	~7	Ct
34	1102.7	Upmid 7th	^~7	vvC#
35	1135.1	Downmajor 7th	vM7	vC#
36	1167.6	Major 7th	M7	C#
37	1200.0	Perfect 8ve	P8	D

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol															
Flat symbol															

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol														
Flat symbol														

Kite's ups and downs notation

37edo can also be notated using Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 44.

Evo and Revo flavors

Alternative Evo flavor

Evo-SZ flavor

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 37edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 37edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/8, 16/11	0.033	0.1
13/10, 20/13	0.160	0.5
7/5, 10/7	1.272	3.9
13/7, 14/13	1.431	4.4
13/11, 22/13	2.682	8.3
13/8, 16/13	2.716	8.4
11/10, 20/11	2.842	8.8
5/4, 8/5	2.875	8.9
11/7, 14/11	4.114	12.7
7/4, 8/7	4.147	12.8
7/6, 12/7	7.411	22.9
5/3, 6/5	8.683	26.8
13/12, 24/13	8.843	27.3
9/8, 16/9	9.315	28.7
11/9, 18/11	9.349	28.8
15/14, 28/15	10.287	31.7
11/6, 12/11	11.525	35.5
3/2, 4/3	11.559	35.6
15/13, 26/15	11.718	36.1
13/9, 18/13	12.031	37.1
9/5, 10/9	12.191	37.6
9/7, 14/9	13.462	41.5
15/11, 22/15	14.401	44.4
15/8, 16/15	14.434	44.5

15-odd-limit intervals in 37edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/8, 16/11	0.033	0.1
13/10, 20/13	0.160	0.5
7/5, 10/7	1.272	3.9
13/7, 14/13	1.431	4.4
13/11, 22/13	2.682	8.3
13/8, 16/13	2.716	8.4
11/10, 20/11	2.842	8.8
5/4, 8/5	2.875	8.9
11/7, 14/11	4.114	12.7
7/4, 8/7	4.147	12.8
7/6, 12/7	7.411	22.9
5/3, 6/5	8.683	26.8
13/12, 24/13	8.843	27.3
15/14, 28/15	10.287	31.7
11/6, 12/11	11.525	35.5
3/2, 4/3	11.559	35.6
15/13, 26/15	11.718	36.1
15/11, 22/15	14.401	44.4
15/8, 16/15	14.434	44.5
9/7, 14/9	18.970	58.5
9/5, 10/9	20.242	62.4
13/9, 18/13	20.401	62.9
11/9, 18/11	23.084	71.2
9/8, 16/9	23.117	71.3

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 equal temperaments 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*	Cents*	In patent val	In 37b val
1\37	32.4
2\37	64.9	Sycamore
3\37	97.3	Passion
4\37	129.7	Twothirdtonic	Negri (37bd, out-of-tune)
5\37	162.2	Porcupine / porcupinefish
6\37	194.6	Hemiwürschmidt / hemiwur	Hemithirds (37b, out-of-tune)
7\37	227.0	Semaja / gorgik	Gorgo (37b)
8\37	259.5		Semaphore (37bd, out-of-tune)
9\37	291.9	Quasitemp
10\37	324.3	Hyperkleismic	Superkleismic (37bc, out-of-tune)
11\37	356.8	Beatles
12\37	389.2	Würschmidt (out-of-tune)
13\37	421.6	Skwares (37dd, out-of-tune)
14\37	454.1	Ammonite
15\37	486.5	Ultrapyth
16\37	518.9	Undecimation	Shallowtone (37b)
17\37	551.4	Freivald, emka
18\37	583.8	Cotritone

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

37edo's primes 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking. Some compressed-octave 37edo tunings (least to most compressed) include 161zpi, 86ed5, 104ed7, 133ed12 or 96ed6.

Scales

MOS scales

Ammonite[21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1
Beatles[7]: 4 7 4 7 4 7 4
Beatles[10]: 4 3 4 4 3 4 4 4 3 4
Beatles[17]: 3 1 3 1 3 3 1 3 1 3 1 3 3 1 3 1 3
Ultrapyth[5] (quasi-equipentatonic): 7 8 7 8 7 (recommended mode: 8 7 7 8 7)
Ultrapyth[7]: 7 1 7 7 7 1 7
Ultrapyth[12]: 1 6 1 6 1 6 1 1 6 1 6 1
Ultrapyth[17]: 1 5 1 1 1 5 1 1 5 1 1 5 1 1 1 5 1 (great as a dual-fifth scale)
Ultrapyth[22]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1 (great as a dual-fifth scale)
Passion[9]: 13 3 3 3 3 3 3 3 3
Passion[12]: 3 3 3 3 3 3 4 3 3 3 3 3
Passion[25]: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 (great as a dual-fifth scale)
Porcupine[5]: 5 17 5 5 5
Porcupine[6]: 12 5 5 5 5 5
Porcupine[7]: 5 5 5 7 5 5 5
Porcupine[15]: 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2
Porcupine[22]: 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2
Twothirdtonic[7]: 13 4 4 4 4 4 4
Twothirdtonic[8]: 9 4 4 4 4 4 4 4
Twothirdtonic[10]: 4 4 4 4 1 4 4 4 4 4
Twothirdtonic[19]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1

Scales by individuals

Budjarn Lambeth's scales
Contains idiosyncratic terms. Opalised ammonite^{[idiosyncratic term]} (modmos of Ammonite[8]): 5 4 6 5 2 5 4 6 Antechinus^{[idiosyncratic term]} (nonoctave period) Gutierrez-Lambeth quasi-subharmonic pentatonic^{[idiosyncratic term]} (octave-reduced ver.: 5 3 13 9 7) Approximated pelog lima: 4 5 12 4 12 Flattened ionian pentatonic: 12 3 6 12 4 Flattened major: 6 6 3 6 6 6 4 Flattened major pentatonic: 6 6 9 6 10 Sharpened natural minor: 7 3 6 6 3 6 6 Sharpened harmonic minor: 7 3 6 6 3 9 3 Sharpened pentatonic minor: 10 6 6 9 6 Superharmonic minor pentatatonic I: 7 3 12 13 2 Superharmonic minor pentatatonic II: 10 6 6 13 2 Flattened hexatonic minor: 6 3 6 6 9 7 Flattened phrygian dominant: 2 9 4 6 3 6 7 Sharpened blues aeolian hexatonic: 10 6 3 3 3 12 Flattened blues aeolian pentatonic: 9 6 6 3 13 Sharpened blues aeolian pentatonic: 10 12 3 6 6 Sharpened blues dorian hexatonic: 10 6 6 6 3 6 Extrasharp blues dorian hexatonic: 10 6 6 6 4 5 Roughened augmented: 10 2 10 2 11 2 Flattened cosmic: 15 6 3 6 7 (approximated from 32afdo) Sharpened Hirajoshi: 7 3 12 3 12 Sharpened Akebono I: 7 3 12 6 9 Roughened Javanese pentachordal: 2 8 9 2 16 Sharpened underpass: 10 12 7 2 6 (approximated from 10afdo) The scales listed in: User:BudjarnLambeth/Quasipelog theory The scales listed in: Oceanfront scales (not all Budjarn's)

Equally spaced scales

37ed4 (every 2 steps): 2 2 2...
Square root of 13 over 10 (every 7 steps): 7 7 7...
Every 8 steps (see below)

Every 8 steps of 37edo


Degrees	Cents	Approximate Ratios of 6.7.11.20.27 subgroup	Additional Ratios
0	0.000	1/1
1	259.46	7/6
2	518.92	27/20
3	778.38	11/7
4	1037.84	20/11, 11/6
5	1297.30		19/9
6	1556.76	27/11
7	1816.22	20/7
8	2075.68	10/3
9	2335.14	27/7
10	2594.59	9/2
11	2854.05		26/5
12	3113.51	6/1
13	3372.97	7/1
14	3632.43
15	3891.89		19/2
16	4151.35	11/1
17	4410.81
18	4670.27
19	4929.73
20	5189.19	20/1
21	5448.65
22	5708.11	27/1