# 339edo

 ← 338edo 339edo 340edo →
Prime factorization 3 × 113
Step size 3.53982¢
Fifth 198\339 (700.885¢) (→66\113)
Semitones (A1:m2) 30:27 (106.2¢ : 95.58¢)
Consistency limit 5
Distinct consistency limit 5

339 equal divisions of the octave (abbreviated 339edo or 339ed2), also called 339-tone equal temperament (339tet) or 339 equal temperament (339et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 339 equal parts of about 3.54 ¢ each. Each step represents a frequency ratio of 21/339, or the 339th root of 2.

## Theory

339edo is enfactored in the 3-limit with the same tuning as 113edo, and is only consistent to the 5-odd-limit. Using the 339d val (339 537 787 951]) in the 7-limit and the 339de val (339 537 787 951 1172]) in the 11-limit, it supports tritriple.

It can be used for the 2.3.5.13.19.23.31.41 subgroup, where it tempers out 621/620, 780/779, 1026/1025, 1426/1425, 2945/2944, 14391/14375 and 73853/73800.

### Odd harmonics

Approximation of odd harmonics in 339edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.07 -0.47 +1.09 +1.40 +0.89 -1.59 -1.54 +1.24 -0.17 +0.02 -1.73
Relative (%) -30.2 -13.4 +30.7 +39.5 +25.3 -44.9 -43.6 +35.0 -4.7 +0.4 -48.8
Steps
(reduced)
537
(198)
787
(109)
952
(274)
1075
(58)
1173
(156)
1254
(237)
1324
(307)
1386
(30)
1440
(84)
1489
(133)
1533
(177)

### Subsets and supersets

Since 339 factors into 3 × 113, 339edo has 3edo and 113edo as its subsets. 1017edo, which triples it, gives a good correction to the harmonics 7 and 11.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [-13 17 -6, [-44 -3 21 [339 537 787]] 0.2930 0.2828 7.99

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 32\339 113.27 16/15 Misneb
1 146\339 516.81 27/20 Gravity
1 158\339 559.29 864/625 Tritriple (339d)

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