# 404edo

← 403edo | 404edo | 405edo → |

^{2}× 101**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 2^{1/404}, 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 (%) | -32.5 | -5.9 | -17.1 | +35.0 | +39.0 | +2.2 | -38.4 | -33.5 | -16.3 | -49.6 | +48.1 | |

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 2^{2} × 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)