59ed6
| ← 58ed6 | 59ed6 | 60ed6 → |
59 equal divisions of the 6th harmonic (abbreviated 59ed6) is a nonoctave tuning system that divides the interval of 6/1 into 59 equal parts of about 52.6 ¢ each. Each step represents a frequency ratio of 61/59, or the 59th root of 6.
Theory
59ed6 corresponds to 22.8243…edo. It can be viewed as a stretched version of 23edo or a compressed version of 36edt.
23edo's harmonics 3, 5, 7 and 11 are all more than 20 cents away from just, so they exhibit very little consonance. 59ed6 improves upon all of their tunings, bringing all of them within 10 cents of just. This dramatically increases the number of consonant intervals and chords available in the tuning.
The trade-off is that 59ed6's octave is significantly worse than 23edo. It has just over 9 cents of error, compared to none. For some composers, 9 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 (¢) | +9.2 | -9.2 | +18.5 | +0.2 | +0.0 | -4.0 | -24.9 | -18.5 | +9.4 | +2.1 | +9.2 |
| Relative (%) | +17.6 | -17.6 | +35.1 | +0.4 | +0.0 | -7.6 | -47.3 | -35.1 | +17.9 | +4.1 | +17.6 | |
| Steps (reduced) |
23 (23) |
36 (36) |
46 (46) |
53 (53) |
59 (0) |
64 (5) |
68 (9) |
72 (13) |
76 (17) |
79 (20) |
82 (23) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -24.2 | +5.2 | -9.0 | -15.6 | -15.4 | -9.2 | +2.3 | +18.7 | -13.2 | +11.4 | -13.0 | +18.5 |
| Relative (%) | -46.0 | +10.0 | -17.2 | -29.7 | -29.4 | -17.6 | +4.4 | +35.5 | -25.2 | +21.7 | -24.7 | +35.1 | |
| Steps (reduced) |
84 (25) |
87 (28) |
89 (30) |
91 (32) |
93 (34) |
95 (36) |
97 (38) |
99 (40) |
100 (41) |
102 (43) |
103 (44) |
105 (46) | |
Subsets and supersets
59ed6 is the 17th prime ed6, so it does not contain any nontrivial subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 52.6 | 31/30, 34/33 |
| 2 | 105.2 | 33/31 |
| 3 | 157.7 | 23/21, 34/31 |
| 4 | 210.3 | 26/23 |
| 5 | 262.9 | 7/6 |
| 6 | 315.5 | 6/5 |
| 7 | 368 | 21/17, 26/21, 31/25 |
| 8 | 420.6 | 14/11, 23/18 |
| 9 | 473.2 | 25/19 |
| 10 | 525.8 | 19/14, 23/17 |
| 11 | 578.3 | 7/5 |
| 12 | 630.9 | |
| 13 | 683.5 | |
| 14 | 736.1 | 23/15, 26/17, 29/19 |
| 15 | 788.6 | 30/19 |
| 16 | 841.2 | |
| 17 | 893.8 | |
| 18 | 946.4 | 19/11 |
| 19 | 998.9 | |
| 20 | 1051.5 | 11/6 |
| 21 | 1104.1 | 17/9 |
| 22 | 1156.7 | |
| 23 | 1209.2 | |
| 24 | 1261.8 | 27/13, 29/14 |
| 25 | 1314.4 | |
| 26 | 1367 | 11/5 |
| 27 | 1419.5 | 25/11, 34/15 |
| 28 | 1472.1 | |
| 29 | 1524.7 | 29/12 |
| 30 | 1577.3 | |
| 31 | 1629.8 | |
| 32 | 1682.4 | 29/11 |
| 33 | 1735 | 30/11 |
| 34 | 1787.6 | |
| 35 | 1840.1 | 26/9, 29/10 |
| 36 | 1892.7 | |
| 37 | 1945.3 | |
| 38 | 1997.9 | 19/6 |
| 39 | 2050.4 | |
| 40 | 2103 | |
| 41 | 2155.6 | |
| 42 | 2208.2 | 25/7 |
| 43 | 2260.7 | |
| 44 | 2313.3 | 19/5 |
| 45 | 2365.9 | |
| 46 | 2418.5 | |
| 47 | 2471 | 25/6 |
| 48 | 2523.6 | 30/7 |
| 49 | 2576.2 | 31/7 |
| 50 | 2628.8 | |
| 51 | 2681.4 | 33/7 |
| 52 | 2733.9 | 34/7 |
| 53 | 2786.5 | 5/1 |
| 54 | 2839.1 | 31/6 |
| 55 | 2891.7 | |
| 56 | 2944.2 | |
| 57 | 2996.8 | |
| 58 | 3049.4 | |
| 59 | 3102 | 6/1 |
See also
- 36edt – relative edt