441edo

← 440edo

441edo

442edo →

Prime factorization	32 × 72
Step size	2.72109 ¢
Fifth	258\441 (702.041 ¢) (→ 86\147)
Semitones (A1:m2)	42:33 (114.3 ¢ : 89.8 ¢)
Consistency limit	17
Distinct consistency limit	17

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

Approximation of prime harmonics in 441edo

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 divisors 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

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

Template:Comma basis begin |- | 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 Template:Comma basis end

• 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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

Music

Eliora

• Etude in G Akjayland for Piano and Tribal Pan (2022)

Gene Ward Smith

• Bodacious Breed (archived 2010) – details | play – breed in 441edo tuning