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.
Theory
60ed6 can be viewed as 23edo with the octave being compressed by 10.9 cents, and with the 6th harmonic being just, instead of the octave being just.
23edo's harmonics 3, 5, 7 and 11 are all more than 20 cents away from just, so they 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 is significantly worse than 23edo. It has almost 11 cents of error, compared to none. 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.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.9 | +10.9 | -21.8 | +5.4 | +0.0 | -8.4 | +18.9 | +21.8 | -5.5 | -15.4 | -10.9 |
| Relative (%) | -21.1 | +21.1 | -42.2 | +10.5 | +0.0 | -16.2 | +36.6 | +42.2 | -10.6 | -29.7 | -21.1 | |
| Steps (reduced) |
23 (23) |
37 (37) |
46 (46) |
54 (54) |
60 (0) |
65 (5) |
70 (10) |
74 (14) |
77 (17) |
80 (20) |
83 (23) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.6 | -19.3 | +16.4 | +8.0 | +6.5 | +10.9 | +20.7 | -16.4 | +2.5 | +25.4 | +0.1 | -21.8 |
| Relative (%) | +10.8 | -37.3 | +31.7 | +15.5 | +12.5 | +21.1 | +40.1 | -31.7 | +4.9 | +49.1 | +0.3 | -42.2 | |
| Steps (reduced) |
86 (26) |
88 (28) |
91 (31) |
93 (33) |
95 (35) |
97 (37) |
99 (39) |
100 (40) |
102 (42) |
104 (44) |
105 (45) |
106 (46) | |
Subsets and supersets
60ed6 is the 9th highly composite ed6, with subset ed6's 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.
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 |
See also
- 37edt – relative edt