37edo

Deutsch

37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th prime edo, following 31edo and coming before 41edo.

Theory

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	28/27, 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

Selected just intervals

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19	prime 23
Error	absolute (¢)	0.0	+11.56	+2.88	+4.15	+0.03	+2.72	-7.66	-5.62	-12.06
	relative (%)	0.0	+35.6	+8.9	+12.8	+0.1	+8.4	-23.6	-17.3	-37.2

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
10\37
11\37	Beatles
12\37	Würschmidt (out-of-tune)
13\37
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

Links

• 37edo at Tonalsoft