# 424edo

← 423edo | 424edo | 425edo → |

^{3}× 53**424 equal divisions of the octave** (abbreviated **424edo** or **424ed2**), also called **424-tone equal temperament** (**424tet**) or **424 equal temperament** (**424et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 424 equal parts of about 2.83 ¢ each. Each step represents a frequency ratio of 2^{1/424}, or the 424th root of 2.

## Theory

424edo is consistent to the 9-odd-limit, but the harmonic 5 is about halfway between its steps. It is enfactored in the 7-limit, with the same tuning as 212edo. The approximation to 11, although closer to just than 212edo's, tends sharp, so its improvement is debatable. All things considered, a 2.3.13.17.19.23 subgroup interpretation with optional additions of 7, 11, or both, seems most reasonable.

### Odd harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | -0.07 | -1.41 | -0.90 | -0.14 | +0.57 | +0.04 | +1.35 | -0.24 | -0.34 | -0.97 | +0.03 |

Relative (%) | -2.4 | -49.8 | -31.8 | -4.8 | +20.1 | +1.4 | +47.8 | -8.4 | -12.1 | -34.3 | +1.0 | |

Steps (reduced) |
672 (248) |
984 (136) |
1190 (342) |
1344 (72) |
1467 (195) |
1569 (297) |
1657 (385) |
1733 (37) |
1801 (105) |
1862 (166) |
1918 (222) |

### Subsets and supersets

Since 424 factors into 2^{3} × 53, 424edo has subset edos 2, 4, 8, 53, 106, and 212. 848edo, which doubles it, gives a good correction to the harmonic 5.

## Regular temperament properties

Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|

Absolute (¢) | Relative (%) | ||||

2.3.7.11 | 41503/41472, 117649/117128, [-26 19 1 -2⟩ | [⟨424 672 1190 1467]] | +0.0499 | 0.1747 | 6.17 |