407edo
| ← 406edo | 407edo | 408edo → |
407 equal divisions of the octave (abbreviated 407edo or 407ed2), also called 407-tone equal temperament (407tet) or 407 equal temperament (407et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 407 equal parts of about 2.95 ¢ each. Each step represents a frequency ratio of 21/407, or the 407th root of 2.
Theory
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 subgroup system. The equal temperament tempers out 32805/32768 in the 5-limit; using the patent val, 16875/16807, 4096000/4084101, and 26873856/26796875 in the 7-limit. It supports and provides the optimal patent val for the subsemifourth temperament in the 7- and 11-limit. Essentially tempered chords available in 407et include pinkanberry chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.24 | -0.07 | +1.20 | +0.03 | -0.23 | +1.19 | +0.28 | -0.26 | -0.58 | -1.06 |
| Relative (%) | +0.0 | -8.0 | -2.5 | +40.7 | +1.1 | -7.9 | +40.3 | +9.4 | -9.0 | -19.8 | -35.8 | |
| Steps (reduced) |
407 (0) |
645 (238) |
945 (131) |
1143 (329) |
1408 (187) |
1506 (285) |
1664 (36) |
1729 (101) |
1841 (213) |
1977 (349) |
2016 (388) | |
Subsets and supersets
407 factors into 11 × 37, with 11edo and 37edo as its subset edos. 814edo, which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-645 407⟩ | [⟨407 645]] | +0.0742 | 0.0742 | 2.52 |
| 2.3.5 | 32805/32768, [30 47 -45⟩ | [⟨407 645 945]] | +0.0599 | 0.0638 | 2.16 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 63\407 | 185.75 | [24 4 -13⟩ | Pirate |
| 1 | 83\407 | 244.72 | 15/13 | Subsemifourth (407f) |
| 1 | 169\407 | 498.28 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct