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 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [38 -2 -15⟩, [1 -27 18⟩ | [⟨441 699 1024]] | -0.0297 | 0.0224 | 0.82 |
| 2.3.5.7 | 2401/2400, 4375/4374, [38 -2 -15⟩ | [⟨441 699 1024 1238]] | -0.0117 | 0.0367 | 1.35 |
| 2.3.5.7.11 | 2401/2400, 4000/3993, 4375/4374, 131072/130977 | [⟨441 699 1024 1238 1526]] | -0.0708 | 0.1227 | 4.51 |
| 2.3.5.7.11.13 | 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | [⟨441 699 1024 1238 1526 1632]] | -0.0720 | 0.1120 | 4.12 |
| 2.3.5.7.11.13.17 | 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095 | [⟨441 699 1024 1238 1526 1632 1803]] | -0.1025 | 0.1278 | 4.70 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 71\441 | 193.20 | 262144/234375 | Luna / lunatic |
| 1 | 95\441 | 258.50 | [-32 13 5⟩ | Lafa |
| 1 | 116\441 | 315.65 | 6/5 | Egads |
| 1 | 128\441 | 348.30 | 57344/46875 | Subneutral |
| 1 | 206\441 | 560.54 | 864/625 | Whoosh |
| 1 | 208\441 | 565.99 | 104/75 | Tricot / trillium |
| 7 | 191\441 (2\441) |
519.73 (5.44) |
27/20 (325/324) |
Brahmagupta |
| 9 | 92\441 (6\441) |
250.34 (16.33) |
140/121 (100/99) |
Semiennealimmal |
| 9 | 116\441 (18\441) |
315.65 (48.98) |
6/5 (36/35) |
Ennealimmal / ennealimmia |
| 21 | 215\441 (5\441) |
585.03 (13.61) |
91875/65536 (126/125) |
Akjayland |
Scales
Scales used in Etude in G Akjayland, in order of size:
- Balzano-200[9]: 77 41 41 41 77 41 41 41 41 (2L 7s, generator = 200\441)
- OEIS-A163205[11]: 9 12 24 4 44 12 84 28 8 156 60 (rank 10)
- Lafa[14]: 34 34 27 34 34 27 34 34 27 34 34 27 34 27 - 9L 5s (m-chro semiquartal)
- Ennealimmal[27]: 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 (18L 9s)
- Akjayland[84]: 6 5 5 5, repeated 21 times