362edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 361edo362edo363edo →
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 362-tone equal temperament (362tet), 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 of 362edo 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 +46 -26 +49 -31 +44 -29 +34 +25 -2 +47
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