404edo
| ← 403edo | 404edo | 405edo → |
404 equal divisions of the octave (abbreviated 404edo or 404ed2), also called 404-tone equal temperament (404tet) or 404 equal temperament (404et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 404 equal parts of about 2.97 ¢ each. Each step represents a frequency ratio of 21/404, or the 404th root of 2.
Theory
404edo is enfactored in the 11-limit, with the same tuning as 202edo. The alternative 404e val, with lower overall error, tempers out 3025/3024 and 41503/41472 in the 11-limit; and 847/845, 1001/1000, and 1716/1715 in the 13-limit, whereas the patent val in the 13-limit tempers out 4225/4224.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.96 | -0.18 | -0.51 | +1.04 | +1.16 | +0.07 | -1.14 | -1.00 | -0.48 | -1.47 | +1.43 |
| relative (%) | -32 | -6 | -17 | +35 | +39 | +2 | -38 | -33 | -16 | -50 | +48 | |
| Steps (reduced) |
640 (236) |
938 (130) |
1134 (326) |
1281 (69) |
1398 (186) |
1495 (283) |
1578 (366) |
1651 (35) |
1716 (100) |
1774 (158) |
1828 (212) | |
Subsets and supersets
Since 404 factors into 22 × 101, 404edo has subset edos 2, 4, 101, and 202.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11.13 | 243/242, 441/440, 4000/3993, 4225/4224, 65625/65536 | [⟨404 640 938 1134 1398 1495]] (404) | +0.0854 | 0.2821 | 9.50 |
Trivia
A Scala file for this EDO is used as the 404 page of Sevish's website.
Music
- Error 404: Tuning Not Found (2023)