4296edo
| ← 4295edo | 4296edo | 4297edo → |
(semiconvergent)
The 4296 equal divisions of the octave (4296edo), or 4296-tone equal temperament (4296tet), 4296 equal temperament (4296et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 4296 equal parts of about of 0.2793 cents each, which means that one cent is exactly 3.58 steps of 4296edo. It 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. 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
| 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 | +0.1 | -0.3 | -39.7 | +28.2 | -8.9 | +26.0 | -9.7 | -22.2 | +11.4 | -22.7 | |
| Steps (reduced) |
4296 (0) |
6809 (2513) |
9975 (1383) |
12060 (3468) |
14862 (1974) |
15897 (3009) |
17560 (376) |
18249 (1065) |
19433 (2249) |
20870 (3686) |
21283 (4099) | |