37edo

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37edo

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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

• MOS Scales of 37edo
• roulette6
• roulette7
• roulette13
• roulette19
• Shoe
• 37ED4
• The Square Root of 13/10

Linear temperaments

• List of 37et rank two temperaments by badness

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

• Toccata Bianca 37edo by Aaron Krister Johnson
• Shorn Brown play and Jellybear play by Andrew Heathwaite
• The Kog Sisters by Joe Monzo
• Porcupine Lullaby by Ray Perlner

Links

• 37edo at Tonalsoft