269ed6
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269 equal divisions of the 6th harmonic (abbreviated 269ed6) is a nonoctave tuning system that divides the interval of 6/1 into 269 equal parts of about 11.5 ¢ each. Each step represents a frequency ratio of 61/269, or the 269th root of 6.
Theory
269ed6 is closely related to 104edo, but with the 6th harmonic instead of the octave tuned just. The octave is compressed by about 0.731 cents. Unlike 104edo, which is only consistent to the 4-integer-limit, 269ed6 is consistent to the 6-integer-limit. It tunes prime harmonics 3 and 5 sharp, 2, 7 and 13 flat, and 11 practically pure.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.73 | +0.73 | -1.46 | +4.29 | +0.00 | -1.65 | -2.19 | +1.46 | +3.56 | -0.00 | -0.73 |
| Relative (%) | -6.3 | +6.3 | -12.7 | +37.2 | +0.0 | -14.3 | -19.0 | +12.7 | +30.9 | -0.0 | -6.3 | |
| Steps (reduced) |
104 (104) |
165 (165) |
208 (208) |
242 (242) |
269 (0) |
292 (23) |
312 (43) |
330 (61) |
346 (77) |
360 (91) |
373 (104) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.93 | -2.38 | +5.02 | -2.92 | -4.10 | +0.73 | -0.62 | +2.83 | -0.92 | -0.73 | +3.03 | -1.46 |
| Relative (%) | -8.0 | -20.6 | +43.6 | -25.4 | -35.5 | +6.3 | -5.4 | +24.5 | -8.0 | -6.4 | +26.3 | -12.7 | |
| Steps (reduced) |
385 (116) |
396 (127) |
407 (138) |
416 (147) |
425 (156) |
434 (165) |
442 (173) |
450 (181) |
457 (188) |
464 (195) |
471 (202) |
477 (208) | |
Subsets and supersets
269ed6 is the 57th prime ed6. It does not contain any nontrivial subset ed6's.