4296edo

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← 4295edo4296edo4297edo →
Prime factorization 23 × 3 × 179
Step size 0.27933¢
Fifth 2513\4296 (701.955¢)
(semiconvergent)
Semitones (A1:m2) 407:323 (113.7¢ : 90.22¢)
Consistency limit 9
Distinct consistency limit 9

4296 equal divisions of the octave (abbreviated 4296edo), or 4296-tone equal temperament (4296tet), 4296 equal temperament (4296et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4296 equal parts of about 0.279 ¢ each. Each step of 4296edo represents a frequency ratio of 21/4296, or the 4296th root of 2.

4296edo is an extraordinarily strong 5-limit system, tempering out raider, [71 -99 37, pirate, [-90 -15 49 and the Kirnberger's atom, [161 -84 -12. Not until 73709 do we reach a division with a lower 5-limit relative error, and not until 6796263 do we find a lower logflat badness. It is uniquely consistent through the 9-odd-limit, and in the 7-limit, it tempers out the landscape comma, 250047/250000, and so supports septimal atomic, the 612 & 1848 temperament.

4296 = 12 × 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments, and which means that one cent is exactly 3.58 steps of 4296edo. From that point of view, one might note that 81/80 is 77 steps, 531441/524288, the Pythagorean comma, 84 steps, and 32805/32768, the schisma, 7 steps, making it exactly 1/12 of a Pythagorean comma and 1/11 of a syntonic comma, useful approximations when dealing with this problem. Senior, [-17 62 -35, fortune, [-107 47 14 and the monzisma, [54 -37 2, are all one step of 4296et.

Prime harmonics

Approximation of prime harmonics in 4296edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.0000 +0.0003 -0.0009 -0.1108 +0.0787 -0.0249 +0.0725 -0.0270 -0.0621 +0.0317 -0.0635
relative (%) +0 +0 -0 -40 +28 -9 +26 -10 -22 +11 -23
Steps
(reduced)
4296
(0)
6809
(2513)
9975
(1383)
12060
(3468)
14862
(1974)
15897
(3009)
17560
(376)
18249
(1065)
19433
(2249)
20870
(3686)
21283
(4099)