3684edo
| ← 3683edo | 3684edo | 3685edo → |
3684 equal divisions of the octave (abbreviated 3684edo or 3684ed2), also called 3684-tone equal temperament (3684tet) or 3684 equal temperament (3684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3684 equal parts of about 0.326 ¢ each. Each step represents a frequency ratio of 21/3684, or the 3684th root of 2.
3684edo is an extraordinarily strong 5-limit system, tempering out senior, [-17 62 -35⟩, gross, [144 -22 -47⟩; and the Kirnberger atom, [161 -84 -12⟩;. 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 the 7-limit atomic.
3684 = 12 * 307, 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 66 steps, 531441/524288, the Pythagorean comma, 72 steps, and 32805/32768, the schisma, 6 steps, making it exactly 1/12 of a Pythagorean comma and 1/11 of a syntonic comma, useful approximations when dealing with this problem. Raider, [71 -99 37⟩;, pirate, [-90 -15 49⟩; and the monzisma, [54 -37 2⟩;, are all one step of 3684et.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.0000 | -0.0006 | +0.0055 | -0.0963 | +0.1479 | -0.1368 | -0.0694 | -0.1189 | +0.0644 | +0.0645 |
| relative (%) | +0 | -0 | +2 | -30 | +45 | -42 | -21 | -36 | +20 | +20 | |
| Steps (reduced) |
3684 (0) |
5839 (2155) |
8554 (1186) |
10342 (2974) |
12745 (1693) |
13632 (2580) |
15058 (322) |
15649 (913) |
16665 (1929) |
17897 (3161) | |