# 362edo

← 361edo | 362edo | 363edo → |

**362 equal divisions of the octave** (abbreviated **362edo** or **362ed2**), also called **362-tone equal temperament** (**362tet**) or **362 equal temperament** (**362et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 362 equal parts of about 3.31 ¢ each. Each step represents a frequency ratio of 2^{1/362}, or the 362nd root of 2.

## Theory

362edo is enfactored in the 3-limit and is only consistent to the 5-odd-limit, with two mappings possible for the 7-limit:

- ⟨362 574 841 1016] (patent val),
- ⟨362 574 841
**1017**] (362d).

Using the patent val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 4375/4374, 458752/455625 and 11529602/11390625 in the 7-limit, supporting barbados.

Using the 362d val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 5120/5103, 118098/117649 and 1959552/1953125 in the 7-limit.

### Odd harmonics

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

Error | Absolute (¢) | +0.81 | +1.53 | -0.87 | +1.61 | -1.04 | +1.46 | -0.98 | +1.12 | +0.83 | -0.06 | +1.56 |

Relative (%) | +24.4 | +46.2 | -26.2 | +48.7 | -31.4 | +44.1 | -29.4 | +33.8 | +25.0 | -1.9 | +47.1 | |

Steps (reduced) |
574 (212) |
841 (117) |
1016 (292) |
1148 (62) |
1252 (166) |
1340 (254) |
1414 (328) |
1480 (32) |
1538 (90) |
1590 (142) |
1638 (190) |

### Subsets and supersets

Since 362 factors into 2 × 181, 372edo has 2edo and 181edo as its subsets. 1448edo, which quadruples it, is a strong full 13-limit system.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.3.5 | 393216/390625, [25 -48 22⟩ | [⟨362 574 841]] | -0.3896 | 0.2822 | 8.51 |

### Rank-2 temperaments

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

1 | 117\362 | 387.85 | 5/4 | Würschmidt |

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