292edo
| ← 291edo | 292edo | 293edo → |
292 equal divisions of the octave (abbreviated 292edo or 292ed2), also called 292-tone equal temperament (292tet) or 292 equal temperament (292et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 292 equal parts of about 4.11 ¢ each. Each step represents a frequency ratio of 21/292, or the 292nd root of 2.
Theory
292edo is closely related to 146edo, but the patent vals differ on the mapping for 3. The equal temperament tempers out [3 -18 11⟩ (quartonic comma) and [38 -2 -15⟩ (luna/hemithirds comma) in the 5-limit; 5120/5103 (hemifamity), 390625/388962 (dimcomp), 420175/419904 (wizma), and 4802000/4782969 (canousma) in the 7-limit; 1375/1372, 5632/5625, 6250/6237, 9801/9800 and 14641/14580 in the 11-limit; 352/351, 625/624, 847/845, 1716/1715, and 2080/2079 in the 13-limit.
It provides the optimal patent val for the undim temperament in the 7-, 11-, and 13-limit, and notably supports semiseptiquarter and semiluna.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.78 | -0.01 | +1.04 | -0.63 | +1.94 | +1.89 | -1.62 | +0.49 | +1.93 | +1.54 |
| relative (%) | +0 | +19 | -0 | +25 | -15 | +47 | +46 | -39 | +12 | +47 | +37 | |
| Steps (reduced) |
292 (0) |
463 (171) |
678 (94) |
820 (236) |
1010 (134) |
1081 (205) |
1194 (26) |
1240 (72) |
1321 (153) |
1419 (251) |
1447 (279) | |
Subsets and supersets
Since 292 factors into 22 × 73, 292edo has subset edos 2, 4, 73, and 146.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [463 -292⟩ | [⟨292 463]] | -0.2476 | 0.2475 | 6.02 |
| 2.3.5 | [3 -18 11⟩, [38 -2 -15⟩ | [⟨292 463 678]] | -0.1633 | 0.2346 | 5.71 |
| 2.3.5.7 | 5120/5103, 390625/388962, 420175/419904 | [⟨292 463 678 820]] | -0.2148 | 0.2219 | 5.40 |
| 2.3.5.7.11 | 1375/1372, 5120/5103, 5632/5625, 14641/14580 | [⟨292 463 678 820 1010]] | -0.1353 | 0.2544 | 6.19 |
| 2.3.5.7.11.13 | 352/351, 625/624, 847/845, 1716/1715, 14641/14580 | [⟨292 463 678 820 1010 1081]] | -0.3480 | 0.2736 | 6.66 |
| 2.3.5.7.11.13.17 | 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023 | [⟨292 463 678 820 1010 1081 1194]] | -0.2376 | 0.2696 | 6.56 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\292 | 45.21 | 250/243 | Quartonic (5-limit) |
| 1 | 47\292 | 193.15 | 262144/234375 | Luna |
| 1 | 59\292 | 242.47 | 147/128 | Septiquarter |
| 1 | 111\292 | 456.16 | 125/96 | Qak |
| 2 | 47\292 | 193.15 | 121/108 | Semiluna |
| 2 | 59\292 | 242.47 | 121/105 | Semiseptiquarter |
| 4 | 121\292 | 497.26 | 4/3 | Undim |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct