4296edo: Difference between revisions
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4296edo is an extraordinarily strong 5-limit system, tempering out raider, {{monzo| 71 -99 37 }}, pirate, {{monzo| -90 -15 49 }} and the [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}. Not until [[73709edo|73709]] do we reach a division with a lower 5-limit relative error, and not until [[6796263edo|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 [[support]]s 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, {{monzo| -17 62 -35 }}, fortune, {{monzo| -107 47 14 }} and the [[monzisma]], {{monzo| 54 -37 2 }}, are all one step of 4296et. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|4296|prec=4}} | |||
Latest revision as of 23:34, 6 March 2026
| ← 4295edo | 4296edo | 4297edo → |
(semiconvergent)
4296 equal divisions of the octave (abbreviated 4296edo or 4296ed2), also called 4296-tone equal temperament (4296tet) or 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 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
| 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) | |