1312edo

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1312edo

1313edo →

Prime factorization	25 × 41
Step size	0.914634 ¢
Fifth	767\1312 (701.524 ¢)
Semitones (A1:m2)	121:101 (110.7 ¢ : 92.38 ¢)
Dual sharp fifth	768\1312 (702.439 ¢) (→ 24\41)
Dual flat fifth	767\1312 (701.524 ¢)
Dual major 2nd	223\1312 (203.963 ¢)
Consistency limit	7
Distinct consistency limit	7

1312 equal divisions of the octave (abbreviated 1312edo or 1312ed2), also called 1312-tone equal temperament (1312tet) or 1312 equal temperament (1312et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1312 equal parts of about 0.915 ¢ each. Each step represents a frequency ratio of 2^1/1312, or the 1312th root of 2.

1312edo is consistent in the 7-odd-limit and is a satisfactory 2.9.13.23 subgroup tuning, but otherwise it represents low harmonics poorly. It also has a very strong approximation to 399/256.

Nonetheless, 1312edo provides the optimal patent val for the bezique temperament in the 7, 11, and 13-limit, despite being inconsistent.

Odd harmonics

Approximation of odd harmonics in 1312edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.431	-0.338	-0.228	+0.053	+0.206	+0.021	+0.146	+0.228	-0.257	+0.256	+0.079
	Relative (%)	-47.1	-37.0	-25.0	+5.8	+22.6	+2.3	+16.0	+24.9	-28.1	+28.0	+8.7
Steps (reduced)		2079 (767)	3046 (422)	3683 (1059)	4159 (223)	4539 (603)	4855 (919)	5126 (1190)	5363 (115)	5573 (325)	5763 (515)	5935 (687)

Subsets and supersets

1312edo notably contains 32edo and 41edo.