680edo
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.
Theory
680edo contains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior superset of 34edo – 612edo; it borrows 34edo's accurate representation of the interval 25/24, implying that the error on the 3rd harmonic is approximately twice that on the 5th harmonic. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log2(7/4).
Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high prime limits, with great representation of prime 23 (inherited from 170edo) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as 37/29, of which 680edo is a weak circle.
As an equal temperament, it tempers out the quintosec comma in the 5-limit and the breedsma in the 7-limit, supporting decoid. In the 11-limit, the 680e val, better in overall accuracy, tempers out 4000/3993, 12005/11979 and 19712/19683; the patent val tempers out 5632/5625 and 9801/9800. In the 13-limit, the 680ef val tempers out 1575/1573, 2080/2079, and 2200/2197; the patent val tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, and 4096/4095 among others.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.398 | +0.157 | -0.002 | -0.730 | -0.528 | -0.838 | +0.722 | -0.039 | -0.754 | +0.259 |
| Relative (%) | +0.0 | +22.5 | +8.9 | -0.1 | -41.4 | -29.9 | -47.5 | +40.9 | -2.2 | -42.7 | +14.7 | |
| Steps (reduced) |
680 (0) |
1078 (398) |
1579 (219) |
1909 (549) |
2352 (312) |
2516 (476) |
2779 (59) |
2889 (169) |
3076 (356) |
3303 (583) |
3369 (649) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.756 | -0.239 | +0.247 | -0.213 | +0.025 | -0.348 | +0.174 | +0.105 | +0.303 | -0.142 | +0.757 |
| Relative (%) | -42.8 | -13.5 | +14.0 | -12.0 | +1.4 | -19.7 | +9.9 | +5.9 | +17.2 | -8.1 | +42.9 | |
| Steps (reduced) |
3542 (142) |
3643 (243) |
3690 (290) |
3777 (377) |
3895 (495) |
4000 (600) |
4033 (633) |
4125 (45) |
4182 (102) |
4209 (129) |
4287 (207) | |
Subsets and supersets
Since 680 factors into primes as 23 × 5 × 17, 680edo has subset edos 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [47 -15 -10⟩, [-5 -32 24⟩ | [⟨680 1078 1579]] | -0.1062 | 0.1061 | 6.01 |
| 2.3.5.7 | 2401/2400, 67108864/66976875, [8 -20 9 1⟩ | [⟨680 1078 1579 1909]] | -0.0795 | 0.1029 | 5.83 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 5 | 63\612 | 111.18 | 16/15 | Quintosec (680) |
| 10 | 5\612 | 8.82 | 225/224 | Decoid (680) |