3684edo: Difference between revisions
Created page with "The 3684 equal division divides the octave into 3684 steps of 0.325733 cents each, which means that one cent is exactly 3.07 steps of 3684 edo. It is an extraordinarily strong..." Tags: Mobile edit Mobile web edit |
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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 | 3684edo is an extraordinarily strong 5-limit system, tempering out senior, {{monzo|-17 62 -35}}, gross, {{monzo|144 -22 -47}}; and the Kirnberger atom, {{monzo|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 [[support]]s 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, {{monzo|71 -99 37}};, pirate, {{monzo|-90 -15 49}}; and the monzisma, {{monzo|54 -37 2}};, are all one step of 3684et. | |||
{{Harmonics in equal|3684|columns=10|prec=4}} | |||
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number --> | |||
Latest revision as of 16:38, 20 February 2025
| ← 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 | -0.2 | +1.7 | -29.6 | +45.4 | -42.0 | -21.3 | -36.5 | +19.8 | +19.8 | |
| Steps (reduced) |
3684 (0) |
5839 (2155) |
8554 (1186) |
10342 (2974) |
12745 (1693) |
13632 (2580) |
15058 (322) |
15649 (913) |
16665 (1929) |
17897 (3161) | |