157edt
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| ← 156edt | 157edt | 158edt → |
157 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 157edt or 157ed3), is a nonoctave tuning system that divides the interval of 3/1 into 157 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of 31/157, or the 157th root of 3.
Theory
157edt is related to 99edo, but with the 3/1 rather than the 2/1 being just. The octave is compressed by about 0.678 cents. 157edt is consistent to the 12-integer-limit. In comparison, 99edo is only consistent up to the 10-integer-limit. 157edt is notable for its excellent 5/3, as a convergent to log3(5), and can be used effectively both with and without twos.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.68 | +0.00 | -1.36 | -0.01 | -0.68 | -1.03 | -2.03 | +0.00 | -0.69 | +3.91 | -1.36 |
| Relative (%) | -5.6 | +0.0 | -11.2 | -0.1 | -5.6 | -8.5 | -16.8 | +0.0 | -5.7 | +32.3 | -11.2 | |
| Steps (reduced) |
99 (99) |
157 (0) |
198 (41) |
230 (73) |
256 (99) |
278 (121) |
297 (140) |
314 (0) |
329 (15) |
343 (29) |
355 (41) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.44 | -1.71 | -0.01 | -2.71 | +1.36 | -0.68 | +2.63 | -1.37 | -1.03 | +3.23 | -1.04 | -2.03 |
| Relative (%) | +44.9 | -14.1 | -0.1 | -22.4 | +11.2 | -5.6 | +21.7 | -11.3 | -8.5 | +26.7 | -8.6 | -16.8 | |
| Steps (reduced) |
367 (53) |
377 (63) |
387 (73) |
396 (82) |
405 (91) |
413 (99) |
421 (107) |
428 (114) |
435 (121) |
442 (128) |
448 (134) |
454 (140) | |
Subsets and supersets
157edt is the 37th prime edt. It does not contain any nontrivial edts as subsets.