60edo
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. 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, using the 60d val.
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) | |
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.
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 | 33/28 |
| 15 | 300 | 32/27 | 13/11 |
| 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 | 33/26 |
| 21 | 420 | 14/11 | |
| 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 | 11/7 | |
| 40 | 800 | 128/81 | 52/33 |
| 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 | 22/13 |
| 46 | 920 | 17/10 | 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
Stein–Zimmermann–Gould notation
Stein–Zimmermann–Gould notation uses sharps and flats with arrows:
| 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 | | | | | | | | | | | | |
Kite's ups and downs notation
60edo can also be notated with Kite's 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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Sagittal notation
This notation is a superset of the notations for edos 12 and 6.
Evo flavor
Revo flavor
Approximation to JI
Interval mappings
The following tables show how 15-odd-limit intervals are represented in 60edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/8, 16/13 | 0.528 | 2.6 |
| 15/14, 28/15 | 0.557 | 2.8 |
| 13/12, 24/13 | 1.427 | 7.1 |
| 3/2, 4/3 | 1.955 | 9.8 |
| 9/5, 10/9 | 2.404 | 12.0 |
| 11/7, 14/11 | 2.492 | 12.5 |
| 7/5, 10/7 | 2.512 | 12.6 |
| 15/11, 22/15 | 3.049 | 15.2 |
| 13/9, 18/13 | 3.382 | 16.9 |
| 9/8, 16/9 | 3.910 | 19.6 |
| 5/3, 6/5 | 4.359 | 21.8 |
| 9/7, 14/9 | 4.916 | 24.6 |
| 11/10, 20/11 | 5.004 | 25.0 |
| 13/10, 20/13 | 5.786 | 28.9 |
| 5/4, 8/5 | 6.314 | 31.6 |
| 7/6, 12/7 | 6.871 | 34.4 |
| 11/9, 18/11 | 7.408 | 37.0 |
| 15/13, 26/15 | 7.741 | 38.7 |
| 15/8, 16/15 | 8.269 | 41.3 |
| 13/7, 14/13 | 8.298 | 41.5 |
| 11/8, 16/11 | 8.682 | 43.4 |
| 7/4, 8/7 | 8.826 | 44.1 |
| 13/11, 22/13 | 9.210 | 46.0 |
| 11/6, 12/11 | 9.363 | 46.8 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/8, 16/13 | 0.528 | 2.6 |
| 15/14, 28/15 | 0.557 | 2.8 |
| 13/12, 24/13 | 1.427 | 7.1 |
| 3/2, 4/3 | 1.955 | 9.8 |
| 9/5, 10/9 | 2.404 | 12.0 |
| 7/5, 10/7 | 2.512 | 12.6 |
| 13/9, 18/13 | 3.382 | 16.9 |
| 9/8, 16/9 | 3.910 | 19.6 |
| 5/3, 6/5 | 4.359 | 21.8 |
| 9/7, 14/9 | 4.916 | 24.6 |
| 13/10, 20/13 | 5.786 | 28.9 |
| 5/4, 8/5 | 6.314 | 31.6 |
| 7/6, 12/7 | 6.871 | 34.4 |
| 15/13, 26/15 | 7.741 | 38.7 |
| 15/8, 16/15 | 8.269 | 41.3 |
| 13/7, 14/13 | 8.298 | 41.5 |
| 11/8, 16/11 | 8.682 | 43.4 |
| 7/4, 8/7 | 8.826 | 44.1 |
| 13/11, 22/13 | 9.210 | 46.0 |
| 11/6, 12/11 | 10.637 | 53.2 |
| 11/9, 18/11 | 12.592 | 63.0 |
| 11/10, 20/11 | 14.996 | 75.0 |
| 15/11, 22/15 | 16.951 | 84.8 |
| 11/7, 14/11 | 17.508 | 87.5 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/8, 16/13 | 0.528 | 2.6 |
| 15/14, 28/15 | 0.557 | 2.8 |
| 13/12, 24/13 | 1.427 | 7.1 |
| 3/2, 4/3 | 1.955 | 9.8 |
| 9/5, 10/9 | 2.404 | 12.0 |
| 11/7, 14/11 | 2.492 | 12.5 |
| 7/5, 10/7 | 2.512 | 12.6 |
| 15/11, 22/15 | 3.049 | 15.2 |
| 13/9, 18/13 | 3.382 | 16.9 |
| 9/8, 16/9 | 3.910 | 19.6 |
| 5/3, 6/5 | 4.359 | 21.8 |
| 9/7, 14/9 | 4.916 | 24.6 |
| 11/10, 20/11 | 5.004 | 25.0 |
| 13/10, 20/13 | 5.786 | 28.9 |
| 5/4, 8/5 | 6.314 | 31.6 |
| 7/6, 12/7 | 6.871 | 34.4 |
| 11/9, 18/11 | 7.408 | 37.0 |
| 15/13, 26/15 | 7.741 | 38.7 |
| 15/8, 16/15 | 8.269 | 41.3 |
| 13/7, 14/13 | 8.298 | 41.5 |
| 7/4, 8/7 | 8.826 | 44.1 |
| 11/6, 12/11 | 9.363 | 46.8 |
| 13/11, 22/13 | 10.790 | 54.0 |
| 11/8, 16/11 | 11.318 | 56.6 |
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
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 155ed6, 95edt or 301zpi make good options for this.
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
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 (2025-05-16)
- 60edo improv (2025-11-22)