57edt
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Prime factorization
3 × 19
Step size
33.3676¢
Octave
36\57edt (1201.23¢) (→12\19edt)
Consistency limit
8
Distinct consistency limit
8
| ← 56edt | 57edt | 58edt → |
57 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 57edt or 57ed3), is a nonoctave tuning system that divides the interval of 3/1 into 57 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of 31/57, or the 57th root of 3.
57edt is related to 36edo (sixth-tone tuning), but with the 3/1 rather than the 2/1 being just. This stretches the octave by about 1.2347 cents. It is consistent to the 9-integer-limit. In comparison, 36edo is only consistent up to the 8-integer-limit.
Lookalikes: 21edf, 36edo, 93ed6, 101ed7, 129ed12
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 33.4 | 22.8 | |
| 2 | 66.7 | 45.6 | 27/26, 28/27 |
| 3 | 100.1 | 68.4 | 18/17 |
| 4 | 133.5 | 91.2 | |
| 5 | 166.8 | 114 | 32/29 |
| 6 | 200.2 | 136.8 | 9/8 |
| 7 | 233.6 | 159.6 | 8/7 |
| 8 | 266.9 | 182.5 | 7/6 |
| 9 | 300.3 | 205.3 | 19/16, 31/26 |
| 10 | 333.7 | 228.1 | 17/14, 23/19 |
| 11 | 367 | 250.9 | 21/17, 26/21 |
| 12 | 400.4 | 273.7 | 24/19, 29/23, 34/27 |
| 13 | 433.8 | 296.5 | 9/7 |
| 14 | 467.1 | 319.3 | 17/13, 21/16 |
| 15 | 500.5 | 342.1 | 4/3 |
| 16 | 533.9 | 364.9 | |
| 17 | 567.2 | 387.7 | 18/13, 32/23 |
| 18 | 600.6 | 410.5 | 17/12, 24/17 |
| 19 | 634 | 433.3 | 13/9 |
| 20 | 667.4 | 456.1 | 28/19 |
| 21 | 700.7 | 478.9 | 3/2 |
| 22 | 734.1 | 501.8 | 26/17, 29/19 |
| 23 | 767.5 | 524.6 | 14/9 |
| 24 | 800.8 | 547.4 | 27/17 |
| 25 | 834.2 | 570.2 | 21/13, 34/21 |
| 26 | 867.6 | 593 | 28/17 |
| 27 | 900.9 | 615.8 | 32/19 |
| 28 | 934.3 | 638.6 | 12/7 |
| 29 | 967.7 | 661.4 | 7/4 |
| 30 | 1001 | 684.2 | |
| 31 | 1034.4 | 707 | |
| 32 | 1067.8 | 729.8 | 13/7 |
| 33 | 1101.1 | 752.6 | 17/9 |
| 34 | 1134.5 | 775.4 | 27/14 |
| 35 | 1167.9 | 798.2 | |
| 36 | 1201.2 | 821.1 | 2/1 |
| 37 | 1234.6 | 843.9 | |
| 38 | 1268 | 866.7 | 27/13 |
| 39 | 1301.3 | 889.5 | 17/8 |
| 40 | 1334.7 | 912.3 | 13/6 |
| 41 | 1368.1 | 935.1 | |
| 42 | 1401.4 | 957.9 | 9/4 |
| 43 | 1434.8 | 980.7 | 16/7 |
| 44 | 1468.2 | 1003.5 | 7/3 |
| 45 | 1501.5 | 1026.3 | 19/8, 31/13 |
| 46 | 1534.9 | 1049.1 | 17/7 |
| 47 | 1568.3 | 1071.9 | |
| 48 | 1601.6 | 1094.7 | |
| 49 | 1635 | 1117.5 | 18/7 |
| 50 | 1668.4 | 1140.4 | 21/8, 34/13 |
| 51 | 1701.7 | 1163.2 | 8/3 |
| 52 | 1735.1 | 1186 | |
| 53 | 1768.5 | 1208.8 | |
| 54 | 1801.9 | 1231.6 | 17/6 |
| 55 | 1835.2 | 1254.4 | 26/9 |
| 56 | 1868.6 | 1277.2 | |
| 57 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.2 | +0.0 | +2.5 | +16.6 | +1.2 | +1.3 | +3.7 | +0.0 | -15.6 | -13.7 | +2.5 |
| Relative (%) | +3.7 | +0.0 | +7.4 | +49.7 | +3.7 | +3.9 | +11.1 | +0.0 | -46.6 | -41.2 | +7.4 | |
| Steps (reduced) |
36 (36) |
57 (0) |
72 (15) |
84 (27) |
93 (36) |
101 (44) |
108 (51) |
114 (0) |
119 (5) |
124 (10) |
129 (15) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | +2.5 | +16.6 | +4.9 | +0.1 | +1.2 | +7.7 | -14.3 | +1.3 | -12.5 | +10.6 |
| Relative (%) | -7.9 | +7.6 | +49.7 | +14.8 | +0.3 | +3.7 | +23.2 | -42.9 | +3.9 | -37.5 | +31.9 | |
| Steps (reduced) |
133 (19) |
137 (23) |
141 (27) |
144 (30) |
147 (33) |
150 (36) |
153 (39) |
155 (41) |
158 (44) |
160 (46) |
163 (49) | |