# 312edo

← 311edo | 312edo | 313edo → |

^{3}× 3 × 13**312 equal divisions of the octave** (abbreviated **312edo** or **312ed2**), also called **312-tone equal temperament** (**312tet**) or **312 equal temperament** (**312et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 312 equal parts of about 3.85 ¢ each. Each step represents a frequency ratio of 2^{1/312}, or the 312th root of 2.

This edo is the first multiple of 12 to have a patent val fifth that does not correspond to the 12edo fifth of 700 cents. It is strong in the 2.9.15.7 subgroup. Beyond that, it is harmonic quality is quite poor for its size.

### Odd harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +1.89 | -1.70 | +0.40 | -0.06 | -1.32 | +1.78 | +0.19 | -1.11 | -1.36 | -1.55 | -1.35 | +0.45 |

Relative (%) | +49.2 | -44.2 | +10.5 | -1.7 | -34.3 | +46.3 | +5.0 | -28.8 | -35.3 | -40.3 | -35.1 | +11.7 | |

Steps (reduced) |
495 (183) |
724 (100) |
876 (252) |
989 (53) |
1079 (143) |
1155 (219) |
1219 (283) |
1275 (27) |
1325 (77) |
1370 (122) |
1411 (163) |
1449 (201) |

### Subsets and supersets

Since 312 factors into 2^{3} × 3 × 13, 312edo has subset edos 2, 3, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, and 156. 624edo, which doubles it, provides the much needed correction to many of the lower harmonics.