692edo
692 equal divisions of the octave (abbreviated 692edo or 692ed2), also called 692-tone equal temperament (692tet) or 692 equal temperament (692et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 692 equal parts of about 1.73 ¢ each. Each step represents a frequency ratio of 21/692, or the 692nd root of 2.
Theory
692edo is consistent to the 19-odd-limit and almost the 21-odd-limit; the only inconsistently mapped intervals in the 21-odd-limit are 21/16 and its octave complement. It has a sharp tendency, with odd harmonics 3 through 19 all tuned sharp.
As an equal temperament, it tempers out [23 6 -14⟩ (vishnuzma) in the 5-limit; 4375/4374 (ragisma) and 29360128/29296875 (quasiorwellisma) in the 7-limit; 3025/3024, 5632/5625, 9801/9800, and 19712/19683 in the 11-limit; 1716/1715, 2080/2079, and 6656/6655 in the 13-limit; 2431/2430, 2500/2499, and 4914/4913 in the 17-limit; 1216/1215, 1331/1330, 1540/1539, 2432/2431, 2926/2925, 3136/3135, and 3250/3249 in the 19-limit. It supports and provides a good tuning for vishnu and its 13-limit extension acyuta.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.357 | +0.391 | +0.538 | +0.127 | +0.513 | +0.825 | +0.753 | -0.529 | +0.481 | -0.527 |
| Relative (%) | +0.0 | +20.6 | +22.6 | +31.0 | +7.3 | +29.6 | +47.6 | +43.4 | -30.5 | +27.7 | -30.4 | |
| Steps (reduced) |
692 (0) |
1097 (405) |
1607 (223) |
1943 (559) |
2394 (318) |
2561 (485) |
2829 (61) |
2940 (172) |
3130 (362) |
3362 (594) |
3428 (660) | |
Subsets and supersets
Since 692 factors into primes as 22 × 173, 692edo has subset edos 2, 4, 173, and 346.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1097 -692⟩ | [⟨692 1097]] | −0.1127 | 0.1127 | 6.50 |
| 2.3.5 | [23 6 -14⟩, [80 -49 1⟩ | [⟨692 1097 1607]] | −0.1313 | 0.0957 | 5.52 |
| 2.3.5.7 | 4375/4374, 29360128/29296875, 283115520/282475249 | [⟨692 1097 1607 1943]] | −0.1464 | 0.0869 | 5.01 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 5632/5625, 283115520/282475249 | [⟨692 1097 1607 1943 2394]] | −0.1245 | 0.0892 | 5.15 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 5632/5625, 200000/199927 | [⟨692 1097 1607 1943 2394 2561]] | −0.1268 | 0.0816 | 4.71 |
| 2.3.5.7.11.13.17 | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4914/4913, 5632/5625 | [⟨692 1097 1607 1943 2394 2561 2829]] | −0.1375 | 0.0800 | 4.61 |
| 2.3.5.7.11.13.17.19 | 1216/1215, 1331/1330, 1540/1539, 1716/1715, 2431/2430, 2500/2499, 4914/4913 | [⟨692 1097 1607 1943 2394 2561 2829 2940]] | −0.1425 | 0.0760 | 4.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 259\692 | 449.13 | 35/27 | Semidimi |
| 2 | 41\692 | 71.10 | 25/24 | Vishnu / acyuta |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct