392edo
| ← 391edo | 392edo | 393edo → |
392 equal divisions of the octave (abbreviated 392edo or 392ed2), also called 392-tone equal temperament (392tet) or 392 equal temperament (392et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 392 equal parts of about 3.06 ¢ each. Each step represents a frequency ratio of 21/392, or the 392nd root of 2.
Theory
392et is consistent to the 7-odd-limit with a flat tendency in the prime harmonics. The equal temperament tempers out the parakleisma in the 5-limit; 321489/320000 (varunisma), 420175/419904 (wizma), 703125/702464 (meter), and 823543/819200 (quince comma) in the 7-limit; and 441/440, 8019/8000, 9801/9800, and 41503/41472 in the 11-limit. It supports qak and octowerck.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.93 | -0.60 | -1.48 | +1.19 | -0.30 | +1.31 | +1.53 | -0.87 | -0.57 | +0.65 | -0.72 |
| relative (%) | -31 | -20 | -48 | +39 | -10 | +43 | +50 | -29 | -19 | +21 | -24 | |
| Steps (reduced) |
621 (229) |
910 (126) |
1100 (316) |
1243 (67) |
1356 (180) |
1451 (275) |
1532 (356) |
1602 (34) |
1665 (97) |
1722 (154) |
1773 (205) | |
Subsets and supersets
392 factors into 23 × 72, with subset edos 2, 4, 7, 8, 14, 28, 49, 56, 98, and 196.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-621 392⟩ | [⟨392 621]] | +0.2948 | 0.2949 | 9.63 |
| 2.3.5 | [8 14 -13⟩, [-49 28 2⟩ | [⟨392 621 910]] | +0.2826 | 0.2414 | 7.89 |
| 2.3.5.7 | 321489/320000, 420175/419904, 703125/702464 | [⟨392 621 910 1100]] | +0.3437 | 0.2343 | 7.65 |
| 2.3.5.7.11 | 441/440, 8019/8000, 41503/41472, 703125/702464 | [⟨392 621 910 1100 1356]] | +0.2922 | 0.2335 | 7.63 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 37\392 | 113.27 | 16/15 | Misneb (5-limit) |
| 1 | 103\392 | 315.31 | 6/5 | Parakleismic (5-limit) |
| 1 | 149\392 | 456.12 | 125/96 | Qak |
| 8 | 185\392 (11\392) |
566.33 (33.67) |
104/75 (55/54) |
Octowerck (392f) |
| 28 | 163\392 (5\392) |
498.98 (15.31) |
4/3 (105/104) |
Oquatonic (5-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct