339edo

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← 338edo339edo340edo →
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 339-tone equal temperament (339tet), 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 of 339edo 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 -13 +31 +40 +25 -45 -44 +35 -5 +0 -49
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