60edo

The 60 equal divisions of the octave (60edo), or the 60-tone equal temperament (60tet), 60 equal temperament (60et) when viewed from a regular temperament perspective, divides the octave into 60 equal parts of exactly 20 cents each.

Theory

Since 60 = 5×12, 60edo belongs to the family of edos which contain 12edo, and like the other small edos of this kind, it tempers out the Pythagorean comma, 531441/524288 = [-19 12⟩. In the 5-limit, it tempers out both the magic comma, 3125/3072, and the amity comma, 1600000/1594323, and supplies the optimal patent val for 5-limit magic, tempering out 3125/3072. In the 7-limit it tempers out 875/864, 245/243, 225/224 and 10976/10935, and supports magic, compton and tritonic temperaments. In the 11-limit, the 60e val scores lower in badness than the patent val, and makes for an excellent tritonic tuning. It tempers out 121/120 and 441/440, whereas the patent val tempers out 100/99, 385/384 and 540/539. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for 13-limit tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 bleu temperament.

60edo is the 9th highly melodic EDO.

Approximation of odd harmonics in 60edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.96	-6.31	-8.83	-3.91	+8.68	-0.53	-8.27	-4.96	+2.49	+9.22	-8.27
	Relative (%)	-9.8	-31.6	-44.1	-19.6	+43.4	-2.6	-41.3	-24.8	+12.4	+46.1	-41.4
Steps (reduced)		95 (35)	139 (19)	168 (48)	190 (10)	208 (28)	222 (42)	234 (54)	245 (5)	255 (15)	264 (24)	271 (31)

Intervals

Degrees	Cents	Approximate Ratios in the 2.3.5.13.17 subgroup	Additional Ratios of 7 and 11 (tending flat, i.e. 60e val)
0	0	1/1
1	20	81/80	49/48
2	40		50/49, 64/63, 33/32
3	60	25/24	28/27, 36/35
4	80
5	100	17/16, 18/17
6	120	16/15	15/14, 14/13
7	140	13/12
8	160		12/11, 11/10
9	180	10/9
10	200	9/8
11	220	17/15
12	240	15/13	8/7
13	260		7/6
14	280	20/17	13/11, 33/28
15	300	32/27
16	320	6/5
17	340	39/32	11/9, 17/14
18	360	16/13	27/22, 21/17
19	380	5/4
20	400	81/64
21	420		33/26, 14/11
22	440		9/7, 22/17
23	460	13/10, 17/13	21/16
24	480
25	500	4/3
26	520	27/20
27	540		11/8, 15/11
28	560	18/13
29	580		7/5
30	600	17/12, 24/17
31	620		10/7
32	640	13/9
33	660		16/11, 22/15
34	680	40/27
35	700	3/2
36	720
37	740	20/13, 26/17	32/21
38	760		14/9, 17/11
39	780		52/33, 11/7
40	800	160/81
41	820	8/5
42	840	13/8	44/27, 34/21
43	860	64/39	18/11, 28/17
44	880	5/3
45	900	27/16
46	920	17/10	22/13, 56/33
47	940		12/7
48	960	26/15	7/4
49	980	30/17
50	1000	16/9
51	1020	9/5
52	1040		11/6, 20/11
53	1060
54	1080	15/8	28/15, 13/7
55	1100	17/9, 32/17
56	1120
57	1140	48/25	27/14, 35/18
58	1160		49/25, 63/32, 64/33
59	1180	160/81	96/49
60	1200	2/1

Regular temperament properties

Since multiple vals provide good temperaments, they are listed.

Subgroup	Comma List	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	3125/3072, 1600000/1594323	[⟨60 95 139]]	1.32	1.11
2.3.5.7	225/224, 245/243, 64827/64000	[⟨60 95 139 168]]	1.78	1.25
2.3.5.7.11	100/99, 225/224, 385/384, 3969/3872	[⟨60 95 139 168 208]]	0.91	2.05
2.3.5.7.11	121/120, 225/224, 245/243, 441/440	[⟨60 95 139 168 207]] (60e)	2.08	1.27
2.3.5.7.11.13	100/99, 105/104, 225/224, 385/384, 2457/2420	[⟨60 95 139 168 208 222]]	0.79	1.89
2.3.5.7.11.13	105/104, 121/120, 196/195, 275/273, 325/324	[⟨60 95 139 168 207 222]] (60e)	1.75	1.36

Music

• Rojqoq (So-Called Peace) play by William Sethares

• Skating On Thin Ice play

• My Mother Said So play

• Black Salt - White Pepper play

• NGC 300play, all by Robin Perry

• Dingshi and Gene's Jitterbug (ogg) Gene's Jitterbug (mp3) Score by Graham Breed

Images

Robin Perry guitar