# 467edo

← 466edo | 467edo | 468edo → |

**467 equal divisions of the octave** (abbreviated **467edo** or **467ed2**), also called **467-tone equal temperament** (**467tet**) or **467 equal temperament** (**467et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 467 equal parts of about 2.570 ¢ each. Each step represents a frequency ratio of 2^{1/467}, or the 467th root of 2.

## Theory

467edo is consistent to the 9-odd-limit with harmonics 3, 5, and 7 all tuned flat. Using the patent val, the equal temperament tempers out 4375/4374, 2100875/2097152, 5250987/5242880, and [-16 4 9 -4⟩ in the 7-limit. It supports mitonic and counterkleismic, supplying the optimal patent val for the latter.

In the 11-limit, the 467e val scores much better than the patent val. The 467e val tempers out 1375/1372, 24057/24010, 35937/35840, and 41503/41472, and in the 13-limit, 625/624, 729/728, 1716/1715, and 2200/2197. The patent val tempers out 540/539, 6250/6237, 12005/11979, and 14700/14641, and in the 13-limit, 625/624, 729/728, and 2080/2079.

In the 17-limit, it supplies the optimal patent val for the rank-6 temperament tempering out 375/374.

### Odd harmonics

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

Error | absolute (¢) | -0.46 | -0.87 | -0.09 | -0.91 | +1.14 | -0.27 | +1.24 | +0.40 | +0.56 | -0.55 | +1.28 |

relative (%) | -18 | -34 | -3 | -35 | +45 | -11 | +48 | +15 | +22 | -21 | +50 | |

Steps (reduced) |
740 (273) |
1084 (150) |
1311 (377) |
1480 (79) |
1616 (215) |
1728 (327) |
1825 (424) |
1909 (41) |
1984 (116) |
2051 (183) |
2113 (245) |

### Subsets and supersets

467edo is the 91st prime edo.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.3 | [-740 467⟩ | [⟨467 740]] | 0.1439 | 0.1439 | 5.38 |

2.3.5 | [-36 11 8⟩, [-16 35 -17⟩ | [⟨467 740 1084]] | 0.2215 | 0.1608 | 6.02 |

2.3.5.7 | 4375/4374, 2100875/2097152, [-16 4 9 -4⟩ | [⟨467 740 1084 1311]] | 0.1741 | 0.1617 | 6.05 |

### Rank-2 temperaments

Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
---|---|---|---|---|

1 | 71\467 | 182.441 | 10/9 | Mitonic |

1 | 123\467 | 316.060 | 6/5 | Counterhanson |

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct