# 60edo

 ← 59edo 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

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 21/60, 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.

## Intervals

Degrees Cents Approximate Ratios
in the 2.3.5.7.13.17 subgroup
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

Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
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)
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 it is distinct

## 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.

Graham Breed
Robin Perry
William Sethares
Randy Wells