312edo

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← 311edo312edo313edo →
Prime factorization 23 × 3 × 13
Step size 3.84615¢
Fifth 183\312 (703.846¢) (→61\104)
Semitones (A1:m2) 33:21 (126.9¢ : 80.77¢)
Dual sharp fifth 183\312 (703.846¢) (→61\104)
Dual flat fifth 182\312 (700¢) (→7\12)
Dual major 2nd 53\312 (203.846¢)
Consistency limit 3
Distinct consistency limit 3

312 equal divisions of the octave (abbreviated 312edo or 312ed2), also called 312-tone equal temperament (312tet) or 312 equal temperament (312et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 312 equal parts of about 3.846 ¢ each. Each step represents a frequency ratio of 21/312, or the 312th root of 2.

This edo is the first multiple of 12 to have a patent val fifth that does not correspond to the 12edo fifth of 700 cents. It is strong in the 2.9.15.7 subgroup. Beyond that, it is harmonic quality is quite poor for its size.

Odd harmonics

Approximation of odd harmonics in 312edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25
Error absolute (¢) +1.89 -1.70 +0.40 -0.06 -1.32 +1.78 +0.19 -1.11 -1.36 -1.55 -1.35 +0.45
relative (%) +49 -44 +11 -2 -34 +46 +5 -29 -35 -40 -35 +12
Steps
(reduced)
495
(183)
724
(100)
876
(252)
989
(53)
1079
(143)
1155
(219)
1219
(283)
1275
(27)
1325
(77)
1370
(122)
1411
(163)
1449
(201)

Subsets and supersets

Since 312 factors into 23 × 3 × 13, 312edo has subset edos 2, 3, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, and 156. 624edo, which doubles it, provides the much needed correction to many of the lower harmonics.