# 362edo

 ← 361edo 362edo 363edo →
Prime factorization 2 × 181
Step size 3.31492¢
Fifth 212\362 (702.762¢) (→106\181)
Semitones (A1:m2) 36:26 (119.3¢ : 86.19¢)
Consistency limit 5
Distinct consistency limit 5

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 21/362, 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

Approximation of odd harmonics in 362edo
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

Table of rank-2 temperaments by generator
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