441edo
| ← 440edo | 441edo | 442edo → |
The 441 equal divisions of the octave (441edo), or the 441(-tone) equal temperament (441tet, 441et) when viewed from a regular temperament perspective, is the equal division of the octave into 441 parts of about 2.72 cents each, a size close to 625/624, the tunbarsma.
Theory
441edo is a very strong 7-limit system; strong enough to qualify as a zeta peak edo. It is also very strong simply considered as a 5-limit system; it is the first division past 118 with a lower 5-limit relative error. In the 5-limit It tempers out the hemithirds comma, [38 -2 -15⟩, the ennealimma, [1 -27 18⟩, whoosh, [37 25 -33⟩, and egads, [-36 -52 51⟩. In the 7-limit it tempers out 2401/2400, 4375/4374, 420175/419904 and 250047/250000, so that it supports ennealimmal temperament. In the 11-limit it tempers out 4000/3993, and in the 13-limit, 1575/1573, 2080/2079 and 4096/4095. It provides the optimal patent val for 11- and 13-limit semiennealimmal temperament, and the 7-limit 41&359 temperament. Since it tempers out 1575/1573, the nicola, it allows the nicolic tetrad.
The steps of 441 are only 1/30 of a cent sharp of 1/8 syntonic comma. Lowering the fifth, which is only 1/12 of a cent sharp, by two steps gives a generator, 256\441, close to 1/4 comma meantone. Like 205edo but even more accurately, 441 can be used as a basis for a Vicentino style "adaptive JI" system.
441 factors into primes as 32 × 72, and has divisors 3, 7, 9, 21, 49, 63 and 147.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.086 | +0.081 | -0.118 | +1.063 | +0.289 | +1.167 | -0.914 | +0.297 | -1.006 | +0.543 |
| Relative (%) | +0.0 | +3.2 | +3.0 | -4.4 | +39.1 | +10.6 | +42.9 | -33.6 | +10.9 | -37.0 | +19.9 | |
| Steps (reduced) |
441 (0) |
699 (258) |
1024 (142) |
1238 (356) |
1526 (203) |
1632 (309) |
1803 (39) |
1873 (109) |
1995 (231) |
2142 (378) |
2185 (421) | |
Selected intervals
| Step | Eliora's Naming System | Asosociated Ratio |
|---|---|---|
| 0 | Prime | 1/1 |
| 8 | Syntonic comma | 81/80 |
| 9 | Pythagorean comma | 531441/524288 |
| 10 | Septimal comma | 64/63 |
| 75 | Whole tone | 9/8 |
| 85 | Septimal supermajor second | 8/7 |
| 98 | Septimal subminor third | 7/6 |
| 142 | Classical major 3rd | 5/4 |
| 150 | Pythagorean major 3rd | 81/64 |
| 258 | Perfect 5th | 3/2 |
| 356 | Harmonic 7th | 7/4 |
| 441 | Octave | 2/1 |