669edo
| ← 668edo | 669edo | 670edo → |
669 equal divisions of the octave (abbreviated 669edo or 669ed2), also called 669-tone equal temperament (669tet) or 669 equal temperament (669et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 669 equal parts of about 1.79 ¢ each. Each step represents a frequency ratio of 21/669, or the 669th root of 2.
669edo is consistent in the 7-odd-limit, although it has significant errors on the 3rd and the 5th harmonics. Besides that, 669c val is a tuning for the sensipent temperament in the 5-limit.
669edo appears better at approximating higher harmonics, with harmonics 37 through 53 all having an error of 20% or less, with a comma basis for the 2.37.41.43.47.53 subgroup being {75809/75776, 1874161/1873232, 151124317/151101728, 9033613312/9032089499, 9795995841727/9788230467584}. Overall, the subgroup which provides satisfactory results for 669edo is 2.7.19.29.37.41.43.47.53.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.610 | -0.663 | -0.216 | +0.574 | -0.645 | +0.728 | +0.521 | +0.874 | +0.245 | -0.826 | -0.472 |
| Relative (%) | -34.0 | -37.0 | -12.0 | +32.0 | -36.0 | +40.6 | +29.0 | +48.7 | +13.6 | -46.0 | -26.3 | |
| Steps (reduced) |
1060 (391) |
1553 (215) |
1878 (540) |
2121 (114) |
2314 (307) |
2476 (469) |
2614 (607) |
2735 (59) |
2842 (166) |
2938 (262) |
3026 (350) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.467 | -0.035 | +0.019 | -0.641 | +0.539 | -0.880 | -0.223 | +0.118 | -0.363 | -0.307 | -0.089 |
| Relative (%) | +26.0 | -2.0 | +1.1 | -35.7 | +30.0 | -49.0 | -12.4 | +6.6 | -20.2 | -17.1 | -5.0 | |
| Steps (reduced) |
3107 (431) |
3181 (505) |
3250 (574) |
3314 (638) |
3375 (30) |
3431 (86) |
3485 (140) |
3536 (191) |
3584 (239) |
3630 (285) |
3674 (329) | |
| Harmonic | 47 | 49 | 51 | 53 | 55 | 57 | 59 | 61 | 63 | 65 | 67 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.036 | -0.432 | +0.264 | +0.038 | +0.485 | -0.365 | -0.876 | +0.604 | +0.358 | +0.064 | -0.383 |
| Relative (%) | -2.0 | -24.1 | +14.7 | +2.1 | +27.0 | -20.3 | -48.8 | +33.7 | +20.0 | +3.6 | -21.4 | |
| Steps (reduced) |
3716 (371) |
3756 (411) |
3795 (450) |
3832 (487) |
3868 (523) |
3902 (557) |
3935 (590) |
3968 (623) |
3999 (654) |
4029 (15) |
4058 (44) | |
Subsets and supersets
Since 669 factors into 3 × 223, 669edo contains 3edo and 223edo as subsets.