60ed6
| ← 59ed6 | 60ed6 | 61ed6 → |
60 equal divisions of the 6th harmonic (abbreviated 60ed6) is a nonoctave tuning system that divides the interval of 6/1 into 60 equal parts of about 51.7 ¢ each. Each step represents a frequency ratio of 61/60, or the 60th root of 6.
It is similar to 23edo, but with the octave (2/1) being compressed by 10.9 cents, and with the interval 6/1 being just, instead of 2/1 being just.
Harmonics
23edo’s 3/1, 5/1, 7/1 and 11/1 are all more than 20 cents away from just, causing them to exhibit very little consonance. 60ed6 improves upon all of their tunings, bringing all of them within 16 cents of just, and bringing 3, 5 and 7 within 11 cents of just. This dramatically increases the number of consonant intervals and chords available in the tuning.
The trade-off is that 60ed6’s octave - 2/1 - is significantly worse than 23edo. It has almost 11 cents of error, compared to 0. For some composers, 11 cents error on the octave may be unacceptable. But for others, it may be considered still close enough for consonance and octave equivalence to be well preserved, and they may see it a worthwhile sacrifice to unlock so many warm 11-limit harmonies.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.9 | +10.9 | +5.4 | -8.4 | -15.4 | +5.6 | +6.5 | +20.7 | +0.1 | +12.4 | +0.4 |
| Relative (%) | -21.1 | +21.1 | +10.5 | -16.2 | -29.7 | +10.8 | +12.5 | +40.1 | +0.3 | +24.1 | +0.7 | |
| Steps (reduced) |
23 (23) |
37 (37) |
54 (54) |
65 (5) |
80 (20) |
86 (26) |
95 (35) |
99 (39) |
105 (45) |
113 (53) |
115 (55) | |
23edo for comparison:
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -23.7 | -21.1 | +22.5 | +22.6 | -5.7 | -0.6 | +15.5 | -2.2 | +13.9 | +2.8 |
| Relative (%) | +0.0 | -45.4 | -40.4 | +43.1 | +43.3 | -11.0 | -1.2 | +29.8 | -4.2 | +26.6 | +5.3 | |
| Steps (reduced) |
23 (0) |
36 (13) |
53 (7) |
65 (19) |
80 (11) |
85 (16) |
94 (2) |
98 (6) |
104 (12) |
112 (20) |
114 (22) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 51.7 | 34/33, 35/34 |
| 2 | 103.4 | 18/17, 35/33 |
| 3 | 155.1 | 12/11, 23/21 |
| 4 | 206.8 | 35/31 |
| 5 | 258.5 | 29/25 |
| 6 | 310.2 | |
| 7 | 361.9 | 21/17 |
| 8 | 413.6 | 14/11, 19/15, 33/26 |
| 9 | 465.3 | 17/13 |
| 10 | 517 | 31/23, 35/26 |
| 11 | 568.7 | 25/18 |
| 12 | 620.4 | 10/7 |
| 13 | 672.1 | 25/17, 31/21 |
| 14 | 723.8 | 35/23 |
| 15 | 775.5 | |
| 16 | 827.2 | 21/13, 29/18 |
| 17 | 878.9 | |
| 18 | 930.6 | 12/7 |
| 19 | 982.3 | 30/17 |
| 20 | 1034 | 20/11 |
| 21 | 1085.7 | |
| 22 | 1137.4 | 29/15 |
| 23 | 1189.1 | |
| 24 | 1240.8 | |
| 25 | 1292.5 | 19/9 |
| 26 | 1344.2 | |
| 27 | 1395.9 | |
| 28 | 1447.6 | 30/13 |
| 29 | 1499.3 | |
| 30 | 1551 | |
| 31 | 1602.7 | |
| 32 | 1654.4 | 13/5 |
| 33 | 1706.1 | |
| 34 | 1757.8 | |
| 35 | 1809.5 | |
| 36 | 1861.2 | |
| 37 | 1912.9 | |
| 38 | 1964.6 | |
| 39 | 2016.3 | |
| 40 | 2068 | 33/10 |
| 41 | 2119.7 | 17/5 |
| 42 | 2171.4 | 7/2 |
| 43 | 2223.1 | |
| 44 | 2274.8 | 26/7 |
| 45 | 2326.5 | 23/6 |
| 46 | 2378.2 | |
| 47 | 2429.9 | |
| 48 | 2481.6 | 21/5 |
| 49 | 2533.3 | |
| 50 | 2585 | |
| 51 | 2636.7 | |
| 52 | 2688.4 | 33/7 |
| 53 | 2740.1 | 34/7 |
| 54 | 2791.8 | |
| 55 | 2843.5 | 31/6 |
| 56 | 2895.2 | |
| 57 | 2946.9 | 11/2 |
| 58 | 2998.6 | 17/3 |
| 59 | 3050.3 | 35/6 |
| 60 | 3102 | 6/1 |