680edo
| ← 679edo | 680edo | 681edo → |
680 equal divisions of the octave (abbreviated 680edo or 680ed2), also called 680-tone equal temperament (680tet) or 680 equal temperament (680et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 680 equal parts of about 1.76 ¢ each. Each step represents a frequency ratio of 21/680, or the 680th root of 2.
Odd harmonics
680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple 612edo; as a multiple of 34edo, 680edo borrows that edo's accurate representation of the interval 25/24, implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log2(7/4). It also has a very accurate 23rd harmonic.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.398 | +0.157 | -0.002 | -0.730 | -0.528 | -0.838 | +0.722 | -0.039 | -0.754 | +0.259 | -0.756 | -0.239 | +0.247 | -0.213 | +0.025 |
| Relative (%) | +0.0 | +22.5 | +8.9 | -0.1 | -41.4 | -29.9 | -47.5 | +40.9 | -2.2 | -42.7 | +14.7 | -42.8 | -13.5 | +14.0 | -12.0 | +1.4 | |
| Steps (reduced) |
680 (0) |
1078 (398) |
1579 (219) |
1909 (549) |
2352 (312) |
2516 (476) |
2779 (59) |
2889 (169) |
3076 (356) |
3303 (583) |
3369 (649) |
3542 (142) |
3643 (243) |
3690 (290) |
3777 (377) |
3895 (495) | |
Subsets and supersets
Since 680 factors into 23 × 5 × 17, 680edo has subset edos 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 170, and 340.
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