57ed12
| ← 56ed12 | 57ed12 | 58ed12 → |
57 equal divisions of the 12th harmonic (abbreviated 57ed12) is a nonoctave tuning system that divides the interval of 12/1 into 57 equal parts of about 75.5 ¢ each. Each step represents a frequency ratio of 121/57, or the 57th root of 12.
Theory
57ed12 is similar to 16edo, but has the 12th harmonic tuned just instead of the octave, which stretches the octave by 7.6 ¢. It can be used as a tuning for mavila and has an antidiatonic (2L 5s) scale which approximates Pelog tunings in Indonesian gamelan music.
16edo's harmonics 3, 11 and 17 are all more than 25 cents away from just, making them unusable for most purposes. 57ed12 improves upon all of their tunings, bringing all those harmonics within 16 cents of just, and bringing 11 and 17 within an impressive 1 cent of just. This dramatically increases the number of consonant intervals and chords available in the tuning.
The trade-off is that 57ed12's harmonic 7 is significantly worse than 16edo. It has almost 28 cents of error, compared to 6, making it go from very consonant to completely unusable. So, if one intends to use harmonies involving 7's, 57ed12 is not a good choice. But if one is interested in the 5-limit, the 2.3.5.11 subgroup, or the no-7's 13-limit or 17-limit, then 57ed12 is by far the superior option, thanks to its dramatically improved 3/1 and 11/1.
Commas
As an equal temperament, 57ed12 tempers out 36/35 and 50/49 in the 7-limit; 33/32 and 45/44 in the 11-limit; 65/64, 66/65, and 78/77 in the 13-limit; 51/50 and 85/84 in the 17-limit; 39/38, 57/56, 77/76, and 96/95 in the 19-limit; 69/68 and 92/91 in the 23-limit; 58/57 and 87/85 in the 29-limit; 63/62, 93/92, and 93/91 in the 31-limit; and 75/74 in the 37-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.6 | -15.1 | +15.1 | +6.2 | -7.6 | +27.5 | +22.7 | -30.3 | +13.7 | -0.3 | +0.0 |
| Relative (%) | +10.0 | -20.1 | +20.1 | +8.2 | -10.0 | +36.4 | +30.1 | -40.1 | +18.2 | -0.4 | +0.0 | |
| Steps (reduced) |
16 (16) |
25 (25) |
32 (32) |
37 (37) |
41 (41) |
45 (45) |
48 (48) |
50 (50) |
53 (53) |
55 (55) |
57 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.4 | +35.0 | -8.9 | +30.3 | +0.8 | -22.7 | +34.6 | +21.3 | +12.3 | +7.3 | +5.8 | +7.6 |
| Relative (%) | +16.4 | +46.4 | -11.9 | +40.1 | +1.0 | -30.1 | +45.9 | +28.2 | +16.3 | +9.6 | +7.7 | +10.0 | |
| Steps (reduced) |
59 (2) |
61 (4) |
62 (5) |
64 (7) |
65 (8) |
66 (9) |
68 (11) |
69 (12) |
70 (13) |
71 (14) |
72 (15) |
73 (16) | |
Subsets and supersets
Since 57 factors into primes as 3 × 19, 57ed12 contains subset ed12's 3ed12 and 19ed12.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 75.5 | 23/22, 24/23 |
| 2 | 150.9 | 12/11 |
| 3 | 226.4 | 8/7, 33/29 |
| 4 | 301.9 | 19/16, 25/21, 31/26 |
| 5 | 377.4 | |
| 6 | 452.8 | 13/10 |
| 7 | 528.3 | 19/14, 34/25 |
| 8 | 603.8 | 17/12 |
| 9 | 679.3 | 34/23 |
| 10 | 754.7 | 17/11, 31/20 |
| 11 | 830.2 | 21/13, 29/18, 34/21 |
| 12 | 905.7 | 32/19 |
| 13 | 981.1 | 30/17 |
| 14 | 1056.6 | |
| 15 | 1132.1 | 25/13 |
| 16 | 1207.6 | |
| 17 | 1283 | 21/10 |
| 18 | 1358.5 | |
| 19 | 1434 | 16/7 |
| 20 | 1509.5 | |
| 21 | 1584.9 | 5/2 |
| 22 | 1660.4 | 34/13 |
| 23 | 1735.9 | 30/11 |
| 24 | 1811.3 | |
| 25 | 1886.8 | |
| 26 | 1962.3 | 31/10 |
| 27 | 2037.8 | 13/4 |
| 28 | 2113.2 | |
| 29 | 2188.7 | |
| 30 | 2264.2 | |
| 31 | 2339.7 | |
| 32 | 2415.1 | |
| 33 | 2490.6 | |
| 34 | 2566.1 | 22/5 |
| 35 | 2641.6 | 23/5 |
| 36 | 2717 | 24/5 |
| 37 | 2792.5 | |
| 38 | 2868 | 21/4 |
| 39 | 2943.4 | |
| 40 | 3018.9 | |
| 41 | 3094.4 | |
| 42 | 3169.9 | 25/4 |
| 43 | 3245.3 | |
| 44 | 3320.8 | 34/5 |
| 45 | 3396.3 | |
| 46 | 3471.8 | |
| 47 | 3547.2 | 31/4 |
| 48 | 3622.7 | |
| 49 | 3698.2 | |
| 50 | 3773.6 | |
| 51 | 3849.1 | |
| 52 | 3924.6 | 29/3 |
| 53 | 4000.1 | |
| 54 | 4075.5 | 21/2 |
| 55 | 4151 | 11/1 |
| 56 | 4226.5 | 23/2 |
| 57 | 4302 | 12/1 |