# 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 2^{1/407}, 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