476edo
| ← 475edo | 476edo | 477edo → |
476 equal divisions of the octave (abbreviated 476edo or 476ed2), also called 476-tone equal temperament (476tet) or 476 equal temperament (476et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 476 equal parts of about 2.52 ¢ each. Each step represents a frequency ratio of 21/476, or the 476th root of 2.
Theory
476edo is consistent to the 7-odd-limit, but the harmonic 3 is about halfway its steps, while its 5 and 7 are both tuned flat. To start with, consider the 2.3.5.7 patent val, as well as 2.9.15.21 and 2.9.5.7 subgroups.
Using the patent val, the equal temperament tempers out 2401/2400 and 19683/19600 in the 7-limit, supporting harry. The 11-limit 476e val tempers out 3025/3024 and 41503/41472, whereas the patent val tempers out 243/242, 441/440, 540/539, 4000/3993, 8019/8000, and 9801/9800, supporting 11-limit harry.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.11 | -0.60 | -0.76 | +0.29 | +0.78 | -1.03 | +0.81 | +0.93 | -0.03 | +0.65 | -0.54 |
| Relative (%) | -44.2 | -23.8 | -30.1 | +11.6 | +31.1 | -40.9 | +32.0 | +36.8 | -1.3 | +25.7 | -21.5 | |
| Steps (reduced) |
754 (278) |
1105 (153) |
1336 (384) |
1509 (81) |
1647 (219) |
1761 (333) |
1860 (432) |
1946 (42) |
2022 (118) |
2091 (187) |
2153 (249) | |
Subsets and supersets
476 factors into 22 × 7 × 17, with subset edos 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238. 952edo, which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the 5-odd-limit.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [1509 -476⟩ | [⟨476 1509]] | -0.0460 | 0.0460 | 1.82 |
| 2.9.5 | [33 -17 9⟩, [-65 0 28⟩ | [⟨476 1509 1105]] | +0.0554 | 0.1482 | 5.88 |
| 2.9.5.7 | 703125/702464, 4802000/4782969, [25 3 -3 8⟩ | [⟨476 1509 1105 1336]] | +0.1091 | 0.1586 | 6.29 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 205\476 | 516.81 | 27/20 | Larry (476) |
| 2 | 205\476 (33\476) |
516.81 (83.19) |
27/20 (21/20) |
Harry (11-limit, 476) |
| 28 | 197\476 (6\476) |
496.64 (15.13) |
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