69edf
| ← 68edf | 69edf | 70edf → |
69 equal divisions of the perfect fifth (abbreviated 69edf or 69ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 69 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of (3/2)1/69, or the 69th root of 3/2.
Theory
69edf is closely related to 118edo, but with the perfect fifth rather than the octave being just. The octave is stretched by about 0.445 cents. Like 118edo, 69edf is consistent to the 12-integer-limit. While the 5-limit microtempering quality of 118edo is sort of compromised here due to the prime 5 being sharp by more than a cent, the approximated prime harmonics 7, 11, 17, and 19 are much improved, as befits the purpose of no-13 19-limit harmony.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.44 | +0.44 | +0.89 | +1.16 | +0.89 | -1.48 | +1.33 | +0.89 | +1.60 | -0.63 | +1.33 |
| Relative (%) | +4.4 | +4.4 | +8.7 | +11.4 | +8.7 | -14.5 | +13.1 | +8.7 | +15.8 | -6.2 | +13.1 | |
| Steps (reduced) |
118 (49) |
187 (49) |
236 (29) |
274 (67) |
305 (29) |
331 (55) |
354 (9) |
374 (29) |
392 (47) |
408 (63) |
423 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.99 | -1.03 | +1.60 | +1.78 | -1.44 | +1.33 | -0.71 | +2.05 | -1.03 | -0.18 | +4.25 | +1.78 |
| Relative (%) | -49.0 | -10.1 | +15.8 | +17.5 | -14.2 | +13.1 | -7.0 | +20.1 | -10.1 | -1.8 | +41.7 | +17.5 | |
| Steps (reduced) |
436 (22) |
449 (35) |
461 (47) |
472 (58) |
482 (68) |
492 (9) |
501 (18) |
510 (27) |
518 (35) |
526 (43) |
534 (51) |
541 (58) | |
Subsets and supersets
Since 69 factors into primes as 3 × 23, 69edf contains 3edf and 23edf as subset edfs.