57ed6
| ← 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 ¢, corresponding to about 22.050610edo. 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 ¢), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-integer-limit. It corrects harmonics 3 and 7, but the 5 and 11 become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.
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 |