← 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
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

Notation

Ups and downs notation

60edo can be notated with ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
           
 
             
 
      
Flat symbol            
 
                      

Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

Semitones 0 15 25 35 45 1 1+15 1+25 1+35 1+45 2 2+15 2+25
Sharp symbol
 
 
 
 
 
 
 
 
 
 
 
 
 
Flat symbol
 
 
 
 
 
 
 
 
 
 
 
 

Sagittal notation

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

Evo flavor

 Sagittal notationPeriodic table of EDOs with sagittal notation45927/4505646/45

Revo flavor

 Sagittal notationPeriodic table of EDOs with sagittal notation45927/4505646/45

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 1\60 20.0 81/80 Bicommatic (60e)
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)
Quintile (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)
Decile (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

 

 

Octave stretch or compression

What follows is a comparison of compressed- and stretched-octave 60edo tunings.

60edo can benefit from slightly stretching the octave, especially when using it as a no-11 17-limit equal temperament. With the right amount of stretch we can find better harmonics 3, 5, and 7 at the expense of somewhat less accurate approximations of 2 and 13. Tunings such as 95edt or 155ed6 are great demonstrations of this.

303zpi
  • Step size: 19.913 ¢, octave size: 1194.78 ¢

Compressing the octave of 60edo by around 5 ¢ results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.75 ¢. The tuning 303zpi does this. So does 223ed13 whose octave is identical within 0.03 ¢.

Approximation of harmonics in 303zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.22 +9.69 +9.47 +1.51 +4.47 -3.53 +4.25 -0.53 -3.71 -9.41 -0.75
Relative (%) -26.2 +48.7 +47.6 +7.6 +22.5 -17.7 +21.4 -2.6 -18.6 -47.3 -3.8
Step 60 96 121 140 156 169 181 191 200 208 216
Approximation of harmonics in 303zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.07 -8.75 -8.71 -0.97 -6.36 -5.75 +0.21 -8.93 +6.16 +5.28 +7.97 -5.97
Relative (%) +0.4 -43.9 -43.8 -4.9 -31.9 -28.9 +1.1 -44.9 +31.0 +26.5 +40.0 -30.0
Step 223 229 235 241 246 251 256 260 265 269 273 276
169ed7
  • Step size: 19.958 ¢, octave size: 1197.50 ¢

Compressing the octave of 60edo by around 2.5 ¢ results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94 ¢. The tuning 169ed7 does this.

Approximation of harmonics in 169ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.97 -8.24 -7.93 +4.43 +7.73 +0.00 +8.03 +3.46 +0.46 -5.07 +3.76
Relative (%) -19.9 -41.3 -39.8 +22.2 +38.8 +0.0 +40.3 +17.4 +2.3 -25.4 +18.9
Steps
(reduced)
60
(60)
95
(95)
120
(120)
140
(140)
156
(156)
169
(0)
181
(12)
191
(22)
200
(31)
208
(39)
216
(47)
Approximation of harmonics in 169ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.73 -3.97 -3.81 +4.07 -1.22 -0.51 +5.56 -3.50 -8.24 -9.04 -6.26 -0.20
Relative (%) +23.7 -19.9 -19.1 +20.4 -6.1 -2.5 +27.9 -17.6 -41.3 -45.3 -31.4 -1.0
Steps
(reduced)
223
(54)
229
(60)
235
(66)
241
(72)
246
(77)
251
(82)
256
(87)
260
(91)
264
(95)
268
(99)
272
(103)
276
(107)
302zpi
  • Step size: 19.962 ¢, octave size: 1197.72 ¢

Compressing the octave of 60edo by around 2 ¢ results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84 ¢. The tuning 202zpi does this. So does the tuning 208ed11 whose octave is identical within 0.3 ¢.

Approximation of harmonics in 302zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.28 -5.57 -4.56 +8.37 -7.85 +4.75 -6.84 +8.83 +6.09 +0.78 +9.84
Relative (%) -11.4 -27.9 -22.8 +41.9 -39.3 +23.8 -34.3 +44.2 +30.5 +3.9 +49.3
Step 60 95 120 140 155 169 180 191 200 208 216
Approximation of harmonics in 302zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.96 +2.47 +2.80 -9.12 +5.70 +6.55 -7.20 +3.81 -0.81 -1.50 +1.39 +7.56
Relative (%) -44.9 +12.4 +14.0 -45.7 +28.5 +32.8 -36.1 +19.1 -4.1 -7.5 +7.0 +37.9
Step 222 229 235 240 246 251 255 260 264 268 272 276

302zpi is particularly well suited to catnip temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the patent vals, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.

60edo
  • Step size: 20.000 ¢, octave size: 1200.00 ¢

Pure-octaves 60edo approximates all harmonics up to 16 within 8.83 ¢.

Approximation of harmonics in 60edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 -1.96 +0.00 -6.31 -1.96 -8.83 +0.00 -3.91 -6.31 +8.68 -1.96
Relative (%) +0.0 -9.8 +0.0 -31.6 -9.8 -44.1 +0.0 -19.6 -31.6 +43.4 -9.8
Steps
(reduced)
60
(0)
95
(35)
120
(0)
139
(19)
155
(35)
168
(48)
180
(0)
190
(10)
199
(19)
208
(28)
215
(35)
Approximation of harmonics in 60edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.53 -8.83 -8.27 +0.00 -4.96 -3.91 +2.49 -6.31 +9.22 +8.68 -8.27 -1.96
Relative (%) -2.6 -44.1 -41.3 +0.0 -24.8 -19.6 +12.4 -31.6 +46.1 +43.4 -41.4 -9.8
Steps
(reduced)
222
(42)
228
(48)
234
(54)
240
(0)
245
(5)
250
(10)
255
(15)
259
(19)
264
(24)
268
(28)
271
(31)
275
(35)
215ed12
  • Step size: 20.009 ¢, octave size: 1200.55 ¢

Stretching the octave of 215ed12 by around half a cent results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 9.44 ¢. The tuning 215ed12 does this.

Approximation of harmonics in 215ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.55 -1.09 +1.09 -5.05 -0.55 -7.30 +1.64 -2.18 -4.50 -9.44 +0.00
Relative (%) +2.7 -5.5 +5.5 -25.2 -2.7 -36.5 +8.2 -10.9 -22.5 -47.2 +0.0
Steps
(reduced)
60
(60)
95
(95)
120
(120)
139
(139)
155
(155)
168
(168)
180
(180)
190
(190)
199
(199)
207
(207)
215
(0)
Approximation of harmonics in 215ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.49 -6.75 -6.14 +2.18 -2.73 -1.64 +4.81 -3.96 -8.39 -8.89 -5.81 +0.55
Relative (%) +7.5 -33.7 -30.7 +10.9 -13.6 -8.2 +24.0 -19.8 -41.9 -44.4 -29.0 +2.7
Steps
(reduced)
222
(7)
228
(13)
234
(19)
240
(25)
245
(30)
250
(35)
255
(40)
259
(44)
263
(48)
267
(52)
271
(56)
275
(60)
60et, 13-limit WE tuning / 155ed6
  • Step size: 20.013 ¢, octave size: 1200.78 ¢

Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this. So does 155ed6 whose octaves differ by only 0.02 ¢.

Approximation of harmonics in 60et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.78 -0.72 +1.56 -4.51 +0.06 -6.64 +2.34 -1.44 -3.73 -8.63 +0.84
Relative (%) +3.9 -3.6 +7.8 -22.5 +0.3 -33.2 +11.7 -7.2 -18.6 -43.1 +4.2
Step 60 95 120 139 155 168 180 190 199 207 215
Approximation of harmonics in 60et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.36 -5.86 -5.23 +3.12 -1.77 -0.66 +5.80 -2.95 -7.36 -7.85 -4.75 +1.62
Relative (%) +11.8 -29.3 -26.1 +15.6 -8.8 -3.3 +29.0 -14.7 -36.8 -39.2 -23.7 +8.1
Step 222 228 234 240 245 250 255 259 263 267 271 275
95edt
  • Step size: 20.021 ¢, octave size: 1201.23 ¢

Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06 ¢. The tuning 95edt does this.

Approximation of harmonics in 95edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.23 +0.00 +2.47 -3.45 +1.23 -5.37 +3.70 +0.00 -2.22 -7.06 +2.47
Relative (%) +6.2 +0.0 +12.3 -17.2 +6.2 -26.8 +18.5 +0.0 -11.1 -35.3 +12.3
Steps
(reduced)
60
(60)
95
(0)
120
(25)
139
(44)
155
(60)
168
(73)
180
(85)
190
(0)
199
(9)
207
(17)
215
(25)
Approximation of harmonics in 95edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.04 -4.13 -3.45 +4.94 +0.09 +1.23 +7.73 -0.98 -5.37 -5.82 -2.70 +3.70
Relative (%) +20.2 -20.6 -17.2 +24.7 +0.4 +6.2 +38.6 -4.9 -26.8 -29.1 -13.5 +18.5
Steps
(reduced)
222
(32)
228
(38)
234
(44)
240
(50)
245
(55)
250
(60)
255
(65)
259
(69)
263
(73)
267
(77)
271
(81)
275
(85)
301zpi
  • Step size: 20.027 ¢, octave size: 1201.62 ¢

Stretching the octave of 60edo by around 1.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48 ¢. The tuning 301zpi does this.

Approximation of harmonics in 301zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.62 +0.61 +3.24 -2.56 +2.23 -4.29 +4.86 +1.22 -0.94 -5.73 +3.85
Relative (%) +8.1 +3.0 +16.2 -12.8 +11.1 -21.4 +24.3 +6.1 -4.7 -28.6 +19.2
Step 60 95 120 139 155 168 180 190 199 207 215
Approximation of harmonics in 301zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.47 -2.67 -1.95 +6.48 +1.66 +2.84 +9.37 +0.68 -3.68 -4.11 -0.96 +5.47
Relative (%) +27.3 -13.3 -9.7 +32.4 +8.3 +14.2 +46.8 +3.4 -18.4 -20.5 -4.8 +27.3
Step 222 228 234 240 245 250 255 259 263 267 271 275
139ed5
  • Step size: 20.045 ¢, octave size: 1202.73 ¢

Stretching the octave of 60edo by a little under ¢ results in improved primes 5, 7 and 11, but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 9.56 ¢. The tuning 139ed5 does this.

Approximation of harmonics in 139ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.73 +2.36 +5.45 +0.00 +5.09 -1.19 +8.18 +4.72 +2.73 -1.92 +7.81
Relative (%) +13.6 +11.8 +27.2 +0.0 +25.4 -6.0 +40.8 +23.5 +13.6 -9.6 +39.0
Steps
(reduced)
60
(60)
95
(95)
120
(120)
139
(0)
155
(16)
168
(29)
180
(41)
190
(51)
199
(60)
207
(68)
215
(76)
Approximation of harmonics in 139ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.56 +1.53 +2.36 -9.14 +6.17 +7.45 -5.98 +5.45 +1.17 +0.81 +4.04 -9.51
Relative (%) +47.7 +7.6 +11.8 -45.6 +30.8 +37.1 -29.8 +27.2 +5.8 +4.0 +20.1 -47.4
Steps
(reduced)
222
(83)
228
(89)
234
(95)
239
(100)
245
(106)
250
(111)
254
(115)
259
(120)
263
(124)
267
(128)
271
(132)
274
(135)
35edf
  • Step size: 20.056 ¢, octave size: 1203.35 ¢

Stretching the octave of 60edo by a little over 3 ¢ results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00 ¢. The tuning 35edf does this.

Approximation of harmonics in 35edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.35 +3.35 +6.70 +1.45 +6.70 +0.56 -10.00 +6.70 +4.80 +0.24 -10.00
Relative (%) +16.7 +16.7 +33.4 +7.2 +33.4 +2.8 -49.9 +33.4 +23.9 +1.2 -49.9
Steps
(reduced)
60
(25)
95
(25)
120
(15)
139
(34)
155
(15)
168
(28)
179
(4)
190
(15)
199
(24)
207
(32)
214
(4)
Approximation of harmonics in 35edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.18 +3.91 +4.80 -6.65 +8.73 -10.00 -3.33 +8.15 +3.91 +3.60 +6.86 -6.65
Relative (%) -40.8 +19.5 +23.9 -33.2 +43.5 -49.9 -16.6 +40.7 +19.5 +17.9 +34.2 -33.2
Steps
(reduced)
221
(11)
228
(18)
234
(24)
239
(29)
245
(0)
249
(4)
254
(9)
259
(14)
263
(18)
267
(22)
271
(26)
274
(29)

Scales

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.

 

Music

Graham Breed
Bryan Deister
Robin Perry
William Sethares
Randy Wells