287ed6
287 equal divisions of the 6th harmonic (abbreviated 287ed6) is a nonoctave tuning system that divides the interval of 6/1 into 287 equal parts of about 10.8 ¢ each. Each step represents a frequency ratio of 61/287, or the 287th root of 6.
Theory
287ed6 is closely related to 111edo, but with the 6th harmonic tuned just instead of the octave. The octave is compressed by about 0.289 cents. Like 111edo, 287ed6 is consistent to the 22-integer-limit. While it tunes 2 and 11 flat, the 3, 5, 7, 13, 17, and 19 remain sharp as in 111edo but less so. The 23, which is flat to begin with, becomes slightly worse.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.29 | +0.29 | -0.58 | +2.20 | +0.00 | +3.33 | -0.87 | +0.58 | +1.91 | -0.97 | -0.29 |
| Relative (%) | -2.7 | +2.7 | -5.4 | +20.4 | +0.0 | +30.8 | -8.0 | +5.4 | +17.7 | -8.9 | -2.7 | |
| Steps (reduced) |
111 (111) |
176 (176) |
222 (222) |
258 (258) |
287 (0) |
312 (25) |
333 (46) |
352 (65) |
369 (82) |
384 (97) |
398 (111) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.64 | +3.05 | +2.49 | -1.16 | +1.97 | +0.29 | +3.96 | +1.62 | +3.62 | -1.26 | -2.56 | -0.58 |
| Relative (%) | +15.2 | +28.2 | +23.1 | -10.7 | +18.2 | +2.7 | +36.6 | +15.0 | +33.5 | -11.6 | -23.6 | -5.4 | |
| Steps (reduced) |
411 (124) |
423 (136) |
434 (147) |
444 (157) |
454 (167) |
463 (176) |
472 (185) |
480 (193) |
488 (201) |
495 (208) |
502 (215) |
509 (222) | |
Subsets and supersets
Since 287 factors into primes as 7 × 41, 287ed6 contains 7ed6 and 41ed6 as subset ed6's.