441edo
| ← 440edo | 441edo | 442edo → |
441 equal divisions of the octave (abbreviated 441edo or 441ed2), also called 441-tone equal temperament (441tet) or 441 equal temperament (441et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 441 equal parts of about 2.72 ¢ each. Each step represents a frequency ratio of 21/441, or the 441st root of 2.
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. 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, the 72 & 369f temperament, and for the 7-limit 41 & 400 temperament. Since it tempers out 1575/1573, the nicola, it allows the nicolic chords in the 15-odd-limit.
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.
One step of 441edo is also of a size close to 625/624, the tunbarsma.
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) | |
Subsets and supersets
441 factors into primes as 32 × 72, and 441edo has subset edos 3, 7, 9, 21, 49, 63 and 147.
882edo, which doubles it, gives an alternative mapping for harmonics 11 and 17. 1323edo, which divides the edostep into three, is the smallest distinctly consistent edo in the 29-odd-limit and thus provides good correction for prime harmonics from 11 to 29.
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 |
- 441et has a lower relative error than any previous equal temperaments in the 5-limit, past 118 and before 559.
- 441et is also notable in the 7-limit, where it has a lower absolute error than any previous equal temperaments, past 171 and before 612.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
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