60edo

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60edo

61edo →

Prime factorization	22 × 3 × 5
Step size	20¢
Fifth	35\60 (700¢) (→7\12)
Semitones (A1:m2)	5:5 (100¢ : 100¢)
Consistency limit	9
Distinct consistency limit	9
Special properties	Highly composite

60 equal divisions of the octave (abbreviated 60edo or 60ed2), also called 60-tone equal temperament (60tet) or 60 equal temperament (60et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 60 equal parts of exactly 20 ¢ each. Each step represents a frequency ratio of 2^1/60, or the 60th root of 2.

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 225/224, 245/243, 875/864, and 10976/10935, and supports magic, compton and tritonic temperaments. In the 11-limit, the 60e val ⟨60 95 139 168 207] scores lower in badness than the patent val ⟨60 95 139 168 208] 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 subgroup temperament, it is also excellent for the 2.3.7.11.13-subgroup bleu temperament.

Odd harmonics

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)

Subsets and supersets

60edo is the 9th highly composite edo, with subset edos 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. In addition, it is of largest consistency among highly composite edos for its size, being consistent in the 9-odd-limit, and all such edos all the way to 27720edo are consistent in only at most 7-odd-limit.

A step of 60edo is exactly 9 dexls, or exactly 41 minas.

Notation

Ups and downs notation

60edo can be notated using ups and downs notation using Helmholtz-Ellis accidentals:

Semitones	0	1⁄5	2⁄5	3⁄5	4⁄5	1	1+1⁄5	1+2⁄5	1+3⁄5	1+4⁄5	2	2+1⁄5	2+2⁄5
Sharp Symbol
Flat Symbol

If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with three arrows.

Sagittal notation

This notation is a superset of the notations for EDOs 12 and 6.

Evo flavor

Revo flavor

Intervals

Degrees	Cents	Approximate ratios in the 2.3.5.7.13.17 subgroup	Additional ratios of 11 (tending flat, 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	21/20
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	8/7, 15/13
13	260	7/6
14	280	20/17	13/11, 33/28
15	300	32/27
16	320	6/5
17	340	39/32, 17/14	11/9
18	360	16/13, 21/17	27/22
19	380	5/4
20	400	81/64
21	420		14/11, 33/26
22	440	9/7	22/17
23	460	21/16, 13/10, 17/13
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	32/21, 20/13, 26/17
38	760	14/9	17/11
39	780		52/33, 11/7
40	800	128/81
41	820	8/5
42	840	13/8, 34/21	44/27
43	860	64/39, 28/17	18/11
44	880	5/3
45	900	27/16
46	920	17/10	22/13, 56/33
47	940	12/7
48	960	7/4, 26/15
49	980	30/17
50	1000	16/9
51	1020	9/5
52	1040		11/6, 20/11
53	1060	24/13
54	1080	15/8, 28/15, 13/7
55	1100	17/9, 32/17
56	1120	40/21
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

Multiple vals are listed since they all provide good temperaments.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	3125/3072, 531441/524288	[⟨60 95 139]]	+1.32	1.11	5.56
2.3.5.7	225/224, 245/243, 64827/64000	[⟨60 95 139 168]]	+1.78	1.25	6.23
2.3.5.7.13	105/104, 196/195, 245/243, 8281/8192	[⟨60 95 139 168 222]]	+1.45	1.29	6.46
2.3.5.7.11	121/120, 225/224, 245/243, 441/440	[⟨60 95 139 168 207]] (60e)	+2.08	1.27	6.33
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	6.80
2.3.5.7.11	100/99, 225/224, 385/384, 3087/3025	[⟨60 95 139 168 208]] (60)	+0.91	2.05	10.22
2.3.5.7.11.13	100/99, 105/104, 144/143, 196/195, 1352/1331	[⟨60 95 139 168 208 222]] (60)	+0.79	1.89	9.44

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	7\60	140.0	13/12	Quintannic (60e)
1	13\60	260.0	7/6	Superpelog (7-limit, 60bbccdd)
1	17\60	340.0	39/32	Houborizic (60) / houbor (60e)
1	19\60	380.0	5/4	Magic (60) / witchcraft (60e)
1	29\60	580.0	7/5	Tritonic (60e) / tritoni (60)
2	7\60	140.0	13/12	Fifive / fifives (60)
2	19\60 (11\60)	380.0 (220.0)	5/4 (25/22)	Astrology (60de) / divination (60e)
2	13\60	260.0	7/6	Bamity (11-limit, 60e)
3	7\60	140.0	243/224	Septichrome
5	19\60 (5\60)	380.0 (100.0)	5/4 (256/245)	Warlock
5	25\60 (1\60)	500.0 (20.0)	4/3 (81/80)	Pental (60)
6	17\60 (3\60)	340.0 (60.0)	375/308 (1760/1701)	Semiseptichrome (11-limit, 60e)
10	25\60 (1\60)	500.0 (20.0)	4/3 (91/90)	Decal (60e) Decic (60) / splendecic (60e) / prodecic (60e)
12	19\60 (1\60)	380.0 (20.0)	5/4 (81/80)	Compton / comptone (60e)
12	12\60 (2\60)	240.0 (40.0)	8/7 (40/39)	Catnip (60cf)
15	25\60 (3\60)	500.0 (20.0)	4/3 (126/125)	Pentadecal (60) / quindecal (60e)
20	25\60 (2\60)	500.0 (20.0)	4/3 (99/98)	Degrees (60e)

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

Diagrams

Scales

• 5- to 10-tone scales in 60edo

Nearby equal-step tunings

There are a few other useful equal-step tunings which occur close to 60edo in step size:

207ed11, 168ed7

The tunings 207ed11 and 168ed7 are almost identical. Each is 60edo but with slightly stretched octaves.

Each causes relatively large improvement to 5/1, 7/1 and 11/1 at the cost of moderate worsening of 2/1 and 3/1.

Each also causes the vals to flip for 11/1 and 13/1.

Approximation of prime harmonics in 207ed11

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+3.28	+3.24	+1.29	+0.36	+0.00	-8.44	+8.44	-3.63	+6.54	+6.33	-8.85
	Relative (%)	+16.4	+16.2	+6.4	+1.8	+0.0	-42.1	+42.1	-18.1	+32.6	+31.6	-44.1
Steps (reduced)		60 (60)	95 (95)	139 (139)	168 (168)	207 (0)	221 (14)	245 (38)	254 (47)	271 (64)	291 (84)	296 (89)

Approximation of prime harmonics in 168ed7

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+3.15	+3.04	+0.99	+0.00	-0.44	-8.92	+7.92	-4.17	+5.96	+5.71	-9.49
	Relative (%)	+15.7	+15.1	+4.9	+0.0	-2.2	-44.5	+39.5	-20.8	+29.7	+28.5	-47.3
Steps (reduced)		60 (60)	95 (95)	139 (139)	168 (0)	207 (39)	221 (53)	245 (77)	254 (86)	271 (103)	291 (123)	296 (128)

301zpi

The tuning 301zpi, the 301st zeta peak index, is 60edo but with slightly stretched octaves.

It causes relatively large improvement to 3/1, 5/1, 7/1, 11/1 and 17/1 at the cost of relatively small worsening of 2/1 and relatively large worsening of 13/1.

It also causes the val for 11/1 to flip from 208 steps to 207 steps.

301zpi is both consistent and distinctly consistent up to the 10-integer-limit, which is unusually high for a two digit edo or three digit zpi.

Approximation of prime harmonics in 301zpi

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+1.60	+0.58	-2.61	-4.35	-5.80	+5.39	+1.57	+9.28	-1.05	-1.82	+2.88
	Relative (%)	+8.0	+2.9	-13.0	-21.7	-29.0	+26.9	+7.9	+46.3	-5.3	-9.1	+14.4
Step		60	95	139	168	207	222	245	255	271	291	297

60edo

Approximation of prime harmonics in 60edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.96	-6.31	-8.83	+8.68	-0.53	-4.96	+2.49	-8.27	-9.58	-5.04
	Relative (%)	+0.0	-9.8	-31.6	-44.1	+43.4	-2.6	-24.8	+12.4	-41.4	-47.9	-25.2
Steps (reduced)		60 (0)	95 (35)	139 (19)	168 (48)	208 (28)	222 (42)	245 (5)	255 (15)	271 (31)	291 (51)	297 (57)

255ed19

The tuning 255ed19 is 60edo but with slightly compressed octaves.

It causes a relatively large improvement to 11/1, at the cost of relatively small worsening of every smaller prime.

It also causes the val for 7/1 to flip from 168 steps to 169.

Approximation of prime harmonics in 255ed19

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-0.59	-2.88	-7.67	+9.53	+6.65	-2.69	-7.34	+0.00	+9.07	+7.57	-7.93
	Relative (%)	-2.9	-14.4	-38.4	+47.7	+33.3	-13.5	-36.7	+0.0	+45.4	+37.9	-39.7
Steps (reduced)		60 (60)	95 (95)	139 (139)	169 (169)	208 (208)	222 (222)	245 (245)	255 (0)	272 (17)	292 (37)	297 (42)

208ed11

The tuning 208ed11 is 60edo but with slightly compressed octaves.

It causes a relatively large improvement to 7/1 and 11/1, at the cost of moderate worsening of 2/1, 3/1 and 5/1.

It also causes the vals to flip for 5/1, 7/1 and 17/1.

Approximation of prime harmonics in 208ed11

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-2.50	-5.92	+7.84	+4.12	+0.00	-9.79	+4.78	-8.16	+0.37	-1.77	+2.53
	Relative (%)	-12.5	-29.7	+39.3	+20.6	+0.0	-49.1	+23.9	-40.9	+1.9	-8.8	+12.7
Steps (reduced)		60 (60)	95 (95)	140 (140)	169 (169)	208 (0)	222 (14)	246 (38)	255 (47)	272 (64)	292 (84)	298 (90)

272ed23

The tuning 272ed23 is 60edo but with slightly compressed octaves.

It causes a relatively large improvement to 7/1 and 11/1, at the cost of moderate worsening of 2/1, 3/1 and 5/1.

It also causes the vals to flip for 5/1, 7/1, 13/1 and 17/1.

These characteristics, particularly the flipped 5/1 and 13/1, are preferred over pure-octaves 60edo for catnip temperament specifically. They change catnip’s warts from 60cf to 272dg (later letters in the alphabet are better).

Given this, 272ed23 can be thought of as a 60edo clone tailor-made for catnip.

Approximation of prime harmonics in 272ed23

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-2.59	-6.05	+7.65	+3.89	-0.28	+9.86	+4.44	-8.51	+0.00	-2.17	+2.12
	Relative (%)	-13.0	-30.3	+38.3	+19.5	-1.4	+49.4	+22.2	-42.6	+0.0	-10.8	+10.6
Steps (reduced)		60 (60)	95 (95)	140 (140)	169 (169)	208 (208)	223 (223)	246 (246)	255 (255)	272 (0)	292 (20)	298 (26)

Instruments

Due to its highly composite nature, 60edo has an unusually high number of ways it can be subdivided. This means it has multiple good skip-fretting systems which can be used to create stringed instruments with playable fret spacings that still span the full gamut. Probably the best of these is tuning a 20edo guitar to major thirds, as demonstrated by Robin Perry in the image below. This is very closely related to the Kite Guitar, with tuning accuracy slightly worse in the 11-limit, but far better when ratios of 13, 17 & 19 are added.

• Lumatone mapping for 60edo
• Skip fretting system 60 2 29
• Skip fretting system 60 3 19
• Skip fretting system 60 4 17

Music

Graham Breed

• Dingshi
• Gene's Jitterbug (ogg) (mp3) Score

Robin Perry

• Skating On Thin Ice play
• My Mother Said So play
• Black Salt - White Pepper play
• NGC 300 play

William Sethares

• Rojqoq (So-Called Peace) play

Randy Wells

• Absinthe Green
• The Well of Sensitivity