60edo
| ← 59edo | 60edo | 61edo → |
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
| 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) | |
Octave stretch
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. For other stretched and compressed tunings see #Nearby equal-step tunings.
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 |
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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 | 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 |
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Sagittal notation
This notation is a superset of the notations for EDOs 12 and 6.
Evo flavor
Revo flavor
Approximation to JI
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 301zpi | 59.920166 | 20.026647 | 7.046396 | 3.547352 | 1.131 | 15.932359 | 1201.598814 | 1.598814 | 10 | 10 |
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* |
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
Scales
- 5- to 10-tone scales in 60edo
- Amulet[idiosyncratic term] (approximated from 25edo, subset of magic): 5 2 5 5 2 5 7 5 5 2 5 7 5
- Approximations of gamelan scales:
- 5-tone pelog: 6 8 20 5 21
- 7-tone pelog: 6 8 12 8 5 14 7
- 5-tone slendro: 12 12 12 12 12
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 induces relatively large improvement to 5/1, 7/1 and 11/1 at the cost of worsening of 2/1 and 3/1. Each also causes the patent val to flip for 11/1 and 13/1.
| 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) | |
| 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) | |
- 139ed5
The tuning 139ed5 is 60edo but with slightly stretched octaves.
It induces relatively large improvement to 5/1, 7/1 and 11/1 at the cost of worsening of 2/1 and 13/1. It also causes the patent val for 11/1 to flip from 208 steps to 207 steps.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.73 | +2.36 | +0.00 | -1.19 | -1.92 | +9.56 | +6.17 | -5.98 | +4.04 | +3.64 | +8.45 |
| Relative (%) | +13.6 | +11.8 | +0.0 | -6.0 | -9.6 | +47.7 | +30.8 | -29.8 | +20.1 | +18.2 | +42.2 | |
| Steps (reduced) |
60 (60) |
95 (95) |
139 (0) |
168 (29) |
207 (68) |
222 (83) |
245 (106) |
254 (115) |
271 (132) |
291 (13) |
297 (19) | |
- 301zpi
The tuning 301zpi, the 301st zeta peak index, is 60edo but with slightly stretched octaves.
It induces relatively large improvement to 3/1, 5/1, 7/1, 11/1 and 17/1 at the cost of worsening of 2/1 and 13/1. It also causes the patent val for 11/1 to flip from 208 steps to 207 steps.
Like 60edo, 301zpi is distinctly consistent up to the 10-integer-limit.
| 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
| 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 induces a relatively large improvement to 11/1, at the cost of worsening of every smaller prime. It also causes the patent val for 7/1 to flip from 168 steps to 169.
| 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 induces a relatively large improvement to 7/1 and 11/1, at the cost of worsening of 2/1, 3/1 and 5/1. It also causes the patent val to flip for 5/1, 7/1 and 17/1.
| 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 induces a relatively large improvement to 7/1 and 11/1, at the cost of worsening of 2/1, 3/1 and 5/1. It also causes the patent val 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 i272dg (later letters in the alphabet are better). Given this, 272ed23 can be thought of as a 60edo clone tailor-made for catnip.
| 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.
Music
- 60edo improv (2025)