57ed6
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| ← 56ed6 | 57ed6 | 58ed6 → |
57 equal divisions of the 6th harmonic (abbreviated 57ed6) is a nonoctave tuning system that divides the interval of 6/1 into 57 equal parts of about 54.4 ¢ each. Each step represents a frequency ratio of 61/57, or the 57th root of 6.
Theory
57ed6 is closely related to 22edo, but with the 6th harmonic rather than the octave being just, which results in octaves being compressed by about 2.75 ¢. The local zeta peak around 22 is located at 22.025147, which has a step size of 54.483 ¢ and an octave of 1198.63 ¢ (which is compressed by 1.37 ¢), making 57ed6 close to optimal for 22edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.8 | +2.8 | -5.5 | -10.9 | +0.0 | +5.2 | -8.3 | +5.5 | -13.6 | -15.4 | -2.8 |
| Relative (%) | -5.1 | +5.1 | -10.1 | -20.0 | +0.0 | +9.6 | -15.2 | +10.1 | -25.1 | -28.3 | -5.1 | |
| Steps (reduced) |
22 (22) |
35 (35) |
44 (44) |
51 (51) |
57 (0) |
62 (5) |
66 (9) |
70 (13) |
73 (16) |
76 (19) |
79 (22) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +21.9 | +2.5 | -8.1 | -11.0 | -7.1 | +2.8 | +18.0 | -16.4 | +8.0 | -18.1 | +13.8 | -5.5 |
| Relative (%) | +40.3 | +4.6 | -14.9 | -20.2 | -13.1 | +5.1 | +33.1 | -30.1 | +14.7 | -33.3 | +25.3 | -10.1 | |
| Steps (reduced) |
82 (25) |
84 (27) |
86 (29) |
88 (31) |
90 (33) |
92 (35) |
94 (37) |
95 (38) |
97 (40) |
98 (41) |
100 (43) |
101 (44) | |
Subsets and supersets
Since 57 factors into primes as 3 × 19, 57ed6 contains subset ed6's 3ed6 and 19ed6.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 54.4 | 30/29, 31/30, 32/31, 33/32, 34/33 |
| 2 | 108.8 | 16/15, 17/16, 33/31 |
| 3 | 163.3 | 11/10, 34/31 |
| 4 | 217.7 | 17/15, 25/22 |
| 5 | 272.1 | 34/29 |
| 6 | 326.5 | 23/19, 29/24 |
| 7 | 380.9 | |
| 8 | 435.4 | 9/7 |
| 9 | 489.8 | |
| 10 | 544.2 | 26/19 |
| 11 | 598.6 | 17/12, 24/17 |
| 12 | 653 | 16/11, 19/13 |
| 13 | 707.5 | |
| 14 | 761.9 | 14/9, 31/20 |
| 15 | 816.3 | 8/5 |
| 16 | 870.7 | 33/20 |
| 17 | 925.1 | 29/17 |
| 18 | 979.6 | 30/17 |
| 19 | 1034 | 20/11, 29/16 |
| 20 | 1088.4 | 15/8 |
| 21 | 1142.8 | 29/15, 31/16 |
| 22 | 1197.2 | 2/1 |
| 23 | 1251.7 | 33/16 |
| 24 | 1306.1 | 17/8 |
| 25 | 1360.5 | 11/5 |
| 26 | 1414.9 | 34/15 |
| 27 | 1469.3 | 7/3 |
| 28 | 1523.8 | 29/12 |
| 29 | 1578.2 | |
| 30 | 1632.6 | 18/7 |
| 31 | 1687 | |
| 32 | 1741.4 | 30/11 |
| 33 | 1795.9 | 31/11 |
| 34 | 1850.3 | 32/11 |
| 35 | 1904.7 | 3/1 |
| 36 | 1959.1 | 31/10 |
| 37 | 2013.5 | 16/5 |
| 38 | 2068 | 33/10 |
| 39 | 2122.4 | 17/5 |
| 40 | 2176.8 | |
| 41 | 2231.2 | 29/8 |
| 42 | 2285.7 | 15/4 |
| 43 | 2340.1 | 27/7 |
| 44 | 2394.5 | |
| 45 | 2448.9 | 33/8 |
| 46 | 2503.3 | 17/4 |
| 47 | 2557.8 | |
| 48 | 2612.2 | |
| 49 | 2666.6 | 14/3 |
| 50 | 2721 | |
| 51 | 2775.4 | |
| 52 | 2829.9 | |
| 53 | 2884.3 | |
| 54 | 2938.7 | |
| 55 | 2993.1 | |
| 56 | 3047.5 | 29/5 |
| 57 | 3102 | 6/1 |