405edo
| ← 404edo | 405edo | 406edo → |
405 equal divisions of the octave (abbreviated 405edo or 405ed2), also called 405-tone equal temperament (405tet) or 405 equal temperament (405et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 405 equal parts of about 2.96 ¢ each. Each step represents a frequency ratio of 21/405, or the 405th root of 2.
Theory
405edo is enfactored in the 3-limit, with the same tuning as 135edo. Like 135edo, it is consistent to the 7-odd-limit with a poor approximation to the harmonic 5. Using the patent val, the equal temperament tempers out 15625/15552 in the 5-limit; 2100875/2097152, and 2460375/2458624 in the 7-limit; 1375/1372, 4000/3993, 19712/19683, and 41503/41472 in the 11-limit. It supports marthirds, novemkleismic and kleirtismic. It provides the optimal patent val for 7- and 11-limit novemkleismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.27 | -1.13 | +0.06 | -0.21 | +0.95 | -1.25 | -1.22 | -0.13 | -1.43 | -1.33 |
| Relative (%) | +0.0 | +9.0 | -38.1 | +2.1 | -7.0 | +32.2 | -42.2 | -41.1 | -4.3 | -48.2 | -45.0 | |
| Steps (reduced) |
405 (0) |
642 (237) |
940 (130) |
1137 (327) |
1401 (186) |
1499 (284) |
1655 (35) |
1720 (100) |
1832 (212) |
1967 (347) |
2006 (386) | |
Subsets and supersets
Since 405 factors into 34 × 5, 405edo has subset edos 3, 5, 9, 15, 27, 45, 81, and 135.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, [110 -65 -3⟩ | [⟨405 642 940]] | +0.1058 | 0.2776 | 9.37 |
| 2.3.5.7 | 15625/15552, 2100875/2097152, 2460375/2458624 | [⟨405 642 940 1137]] | +0.0737 | 0.2467 | 8.33 |
| 2.3.5.7.11 | 1375/1372, 4000/3993, 19712/19683, 41503/41472 | [⟨405 642 940 1137 1401]] | +0.0709 | 0.2207 | 7.45 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 56\405 | 165.93 | 11/10 | Satin |
| 1 | 107\405 | 317.04 | 6/5 | Hanson |
| 9 | 107\405 (17\405) |
317.04 (50.37) |
6/5 (36/35) |
Novemkleismic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct