57ed4
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Prime factorization
3 × 19
Step size
42.1053¢
Octave
29\57ed4 (1221.05¢)
Twelfth
45\57ed4 (1894.74¢) (→15\19ed4)
Consistency limit
1
Distinct consistency limit
1
 |
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| ← 55ed4 | 57ed4 | 59ed4 → |
57 equal divisions of the 4th harmonic (abbreviated 57ed4) is a nonoctave tuning system that divides the interval of 4/1 into 57 equal parts of about 42.1 ¢ each. Each step represents a frequency ratio of 41/57, or the 57th root of 4.
Intervals
| Steps | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 42.105 | |
| 2 | 84.211 | |
| 3 | 126.316 | 29/27 |
| 4 | 168.421 | 11/10 |
| 5 | 210.526 | 26/23 |
| 6 | 252.632 | 29/25 |
| 7 | 294.737 | |
| 8 | 336.842 | |
| 9 | 378.947 | |
| 10 | 421.053 | 14/11, 23/18 |
| 11 | 463.158 | 17/13, 30/23 |
| 12 | 505.263 | |
| 13 | 547.368 | 26/19 |
| 14 | 589.474 | |
| 15 | 631.579 | |
| 16 | 673.684 | 31/21, 34/23 |
| 17 | 715.789 | |
| 18 | 757.895 | |
| 19 | 800 | 27/17 |
| 20 | 842.105 | |
| 21 | 884.211 | 5/3 |
| 22 | 926.316 | 29/17 |
| 23 | 968.421 | |
| 24 | 1010.526 | 34/19 |
| 25 | 1052.632 | 11/6 |
| 26 | 1094.737 | |
| 27 | 1136.842 | 25/13, 29/15 |
| 28 | 1178.947 | |
| 29 | 1221.053 | |
| 30 | 1263.158 | 27/13 |
| 31 | 1305.263 | |
| 32 | 1347.368 | |
| 33 | 1389.474 | 29/13 |
| 34 | 1431.579 | |
| 35 | 1473.684 | |
| 36 | 1515.789 | |
| 37 | 1557.895 | |
| 38 | 1600 | |
| 39 | 1642.105 | |
| 40 | 1684.211 | |
| 41 | 1726.316 | 19/7 |
| 42 | 1768.421 | 25/9 |
| 43 | 1810.526 | |
| 44 | 1852.632 | |
| 45 | 1894.737 | |
| 46 | 1936.842 | |
| 47 | 1978.947 | |
| 48 | 2021.053 | 29/9 |
| 49 | 2063.158 | 23/7, 33/10 |
| 50 | 2105.263 | |
| 51 | 2147.368 | |
| 52 | 2189.474 | |
| 53 | 2231.579 | |
| 54 | 2273.684 | 26/7 |
| 55 | 2315.789 | 19/5 |
| 56 | 2357.895 | |
| 57 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +21.1 | -7.2 | +0.0 | -7.4 | +13.8 | -0.4 | +21.1 | -14.4 | +13.7 | +17.1 | -7.2 |
| Relative (%) | +50.0 | -17.1 | +0.0 | -17.5 | +32.9 | -1.0 | +50.0 | -34.3 | +32.5 | +40.6 | -17.1 | |
| Steps (reduced) |
29 (29) |
45 (45) |
57 (0) |
66 (9) |
74 (17) |
80 (23) |
86 (29) |
90 (33) |
95 (38) |
99 (42) |
102 (45) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -19.5 | +20.6 | -14.6 | +0.0 | -20.7 | +6.6 | -2.8 | -7.4 | -7.6 | -3.9 | +3.3 |
| Relative (%) | -46.3 | +49.0 | -34.6 | +0.0 | -49.3 | +15.7 | -6.6 | -17.5 | -18.1 | -9.4 | +7.8 | |
| Steps (reduced) |
105 (48) |
109 (52) |
111 (54) |
114 (0) |
116 (2) |
119 (5) |
121 (7) |
123 (9) |
125 (11) |
127 (13) |
129 (15) | |