57edf
| ← 56edf | 57edf | 58edf → |
57 equal divisions of the perfect fifth (abbreviated 57edf or 57ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 57 equal parts of about 12.3 ¢ each. Each step represents a frequency ratio of (3/2)1/57, or the 57th root of 3/2.
Theory
57edf corresponds to 97.4421edo. It is related to the regular temperament which tempers out [-32 33 0 -6 -1⟩ and [76 -8 0 -9 -11⟩ in the 11-limit, which is supported by 877-, 3313-, 4190-, 5067-, 5944-, 6821-, 7698-, and 11011edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.44 | -5.44 | +1.43 | -3.12 | +1.43 | +5.48 | -4.02 | +1.43 | +3.75 | -1.16 | -4.02 |
| Relative (%) | -44.2 | -44.2 | +11.6 | -25.4 | +11.6 | +44.5 | -32.6 | +11.6 | +30.4 | -9.4 | -32.6 | |
| Steps (reduced) |
97 (40) |
154 (40) |
195 (24) |
226 (55) |
252 (24) |
274 (46) |
292 (7) |
309 (24) |
324 (39) |
337 (52) |
349 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.19 | +0.04 | +3.75 | +2.85 | -3.59 | -4.02 | +0.90 | -1.70 | +0.04 | +5.71 | +2.64 |
| Relative (%) | +42.1 | +0.3 | +30.4 | +23.1 | -29.1 | -32.6 | +7.3 | -13.8 | +0.3 | +46.3 | +21.4 | |
| Steps (reduced) |
361 (19) |
371 (29) |
381 (39) |
390 (48) |
398 (56) |
406 (7) |
414 (15) |
421 (22) |
428 (29) |
435 (36) |
441 (42) | |
Related regular temperaments
2.3.7 subgroup 877&5067
Commas: [-428 371 0 -57⟩
POTE generator: ~1605632/1594323 = 12.3149
Mapping: [⟨1 1 -1], ⟨0 57 371]]
EDOs: 877, 4190, 5067, 5944, 6821, 7698, 8575
2.3.7.11 subgroup 877&5067
Commas: [-32 33 0 -6 -1⟩, [76 -8 0 -9 -11⟩
POTE generator: ~1605632/1594323 = 12.3150
Mapping: [⟨1 1 -1 7], ⟨0 57 371 -345]]
EDOs: 877, 3313, 4190, 5067, 5944, 6821, 7698, 11011
2.3.7.11.13 subgroup 877&5067
Commas: 257330216/257298363, 53722307808/53710650917, 1786706395136/1786568061663
POTE generator: ~1605632/1594323 = 12.3150
Mapping: [⟨1 1 -1 7 -10], ⟨0 57 371 -345 1335]]
EDOs: 877, 3313, 4190, 5067, 5944, 9257
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 12.3 | |
| 2 | 24.6 | |
| 3 | 36.9 | |
| 4 | 49.3 | 34/33 |
| 5 | 61.6 | 29/28 |
| 6 | 73.9 | |
| 7 | 86.2 | |
| 8 | 98.5 | |
| 9 | 110.8 | |
| 10 | 123.2 | |
| 11 | 135.5 | |
| 12 | 147.8 | |
| 13 | 160.1 | 23/21 |
| 14 | 172.4 | 21/19 |
| 15 | 184.7 | 10/9, 29/26 |
| 16 | 197 | 19/17, 28/25 |
| 17 | 209.4 | 26/23 |
| 18 | 221.7 | 25/22, 33/29 |
| 19 | 234 | |
| 20 | 246.3 | |
| 21 | 258.6 | 29/25 |
| 22 | 270.9 | 34/29 |
| 23 | 283.2 | 33/28 |
| 24 | 295.6 | |
| 25 | 307.9 | 31/26 |
| 26 | 320.2 | |
| 27 | 332.5 | 17/14, 23/19 |
| 28 | 344.8 | |
| 29 | 357.1 | |
| 30 | 369.5 | 21/17, 26/21 |
| 31 | 381.8 | |
| 32 | 394.1 | |
| 33 | 406.4 | |
| 34 | 418.7 | 14/11 |
| 35 | 431 | |
| 36 | 443.3 | 22/17 |
| 37 | 455.7 | |
| 38 | 468 | |
| 39 | 480.3 | 29/22, 33/25 |
| 40 | 492.6 | |
| 41 | 504.9 | |
| 42 | 517.2 | 27/20, 31/23 |
| 43 | 529.5 | 19/14, 34/25 |
| 44 | 541.9 | 26/19 |
| 45 | 554.2 | 29/21 |
| 46 | 566.5 | |
| 47 | 578.8 | |
| 48 | 591.1 | |
| 49 | 603.4 | |
| 50 | 615.8 | |
| 51 | 628.1 | |
| 52 | 640.4 | |
| 53 | 652.7 | 19/13 |
| 54 | 665 | 22/15, 25/17 |
| 55 | 677.3 | 31/21 |
| 56 | 689.6 | |
| 57 | 702 | 3/2 |