59edf
59 equal divisions of the perfect fifth (abbreviated 59edf or 59ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 59 equal parts of about 11.9 ¢ each. Each step represents a frequency ratio of (3/2)1/59, or the 59th root of 3/2.
Theory
59edf corresponds is 101edo but with the perfect fifth rather than the octave being just. The octave is stretched by about 1.65 cents. 58edf is consistent to the 7-integer-limit. In comparison, 101edo is only consistent up to the 3-integer-limit.
Where each of 101edo's primes 5, 7 and 11 have two about equally good mappings with 40-60% relative error, 59edf instead has one mapping for each with 5-30% relative error, at the cost of only minimal damage to the 2 and 3.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.65 | +1.65 | +3.30 | -2.29 | +3.30 | -1.82 | +4.96 | +3.30 | -0.64 | +0.92 | +4.96 |
| Relative (%) | +13.9 | +13.9 | +27.8 | -19.2 | +27.8 | -15.3 | +41.7 | +27.8 | -5.4 | +7.8 | +41.7 | |
| Steps (reduced) |
101 (42) |
160 (42) |
202 (25) |
234 (57) |
261 (25) |
283 (47) |
303 (8) |
320 (25) |
335 (40) |
349 (54) |
362 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.74 | -0.17 | -0.64 | -5.29 | -3.17 | +4.96 | -5.36 | +1.01 | -0.17 | +2.58 | -3.00 | -5.29 |
| Relative (%) | -23.1 | -1.4 | -5.4 | -44.5 | -26.6 | +41.7 | -45.1 | +8.5 | -1.4 | +21.7 | -25.2 | -44.5 | |
| Steps (reduced) |
373 (19) |
384 (30) |
394 (40) |
403 (49) |
412 (58) |
421 (8) |
428 (15) |
436 (23) |
443 (30) |
450 (37) |
456 (43) |
462 (49) | |
Subsets and supersets
59 is a prime number, so 59edf has no subset edfs. Its smallest supersets are 118edf and 177edf.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 11.9 | |
| 2 | 23.8 | |
| 3 | 35.7 | |
| 4 | 47.6 | 34/33 |
| 5 | 59.5 | 28/27, 29/28, 30/29, 31/30, 32/31 |
| 6 | 71.4 | 26/25 |
| 7 | 83.3 | 21/20, 22/21 |
| 8 | 95.2 | |
| 9 | 107.1 | 33/31 |
| 10 | 119 | 15/14, 31/29 |
| 11 | 130.9 | 14/13 |
| 12 | 142.8 | 25/23 |
| 13 | 154.7 | 12/11, 23/21 |
| 14 | 166.6 | 11/10 |
| 15 | 178.5 | 10/9, 31/28 |
| 16 | 190.4 | 19/17, 29/26 |
| 17 | 202.3 | 9/8 |
| 18 | 214.2 | 17/15, 26/23 |
| 19 | 226.1 | 33/29 |
| 20 | 238 | 23/20, 31/27 |
| 21 | 249.8 | 15/13 |
| 22 | 261.7 | |
| 23 | 273.6 | 34/29 |
| 24 | 285.5 | 13/11, 20/17, 33/28 |
| 25 | 297.4 | 25/21, 32/27 |
| 26 | 309.3 | |
| 27 | 321.2 | |
| 28 | 333.1 | 17/14, 23/19 |
| 29 | 345 | 11/9, 28/23 |
| 30 | 356.9 | 27/22 |
| 31 | 368.8 | 21/17, 26/21 |
| 32 | 380.7 | |
| 33 | 392.6 | |
| 34 | 404.5 | 29/23 |
| 35 | 416.4 | 14/11, 33/26 |
| 36 | 428.3 | |
| 37 | 440.2 | 31/24 |
| 38 | 452.1 | 13/10 |
| 39 | 464 | 17/13, 30/23 |
| 40 | 475.9 | 25/19, 29/22 |
| 41 | 487.8 | |
| 42 | 499.7 | 4/3 |
| 43 | 511.6 | |
| 44 | 523.5 | 23/17, 27/20 |
| 45 | 535.4 | 15/11, 34/25 |
| 46 | 547.3 | 11/8, 26/19 |
| 47 | 559.2 | 29/21 |
| 48 | 571.1 | |
| 49 | 583 | 7/5 |
| 50 | 594.9 | 31/22 |
| 51 | 606.8 | |
| 52 | 618.7 | 10/7 |
| 53 | 630.6 | |
| 54 | 642.5 | 29/20 |
| 55 | 654.4 | 19/13 |
| 56 | 666.3 | 22/15, 25/17 |
| 57 | 678.2 | 31/21, 34/23 |
| 58 | 690.1 | |
| 59 | 702 | 3/2 |
See also
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