445edo

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← 444edo445edo446edo →
Prime factorization 5 × 89
Step size 2.69663¢
Fifth 260\445 (701.124¢) (→52\89)
Semitones (A1:m2) 40:35 (107.9¢ : 94.38¢)
Consistency limit 7
Distinct consistency limit 7

445 equal divisions of the octave (abbreviated 445edo), or 445-tone equal temperament (445tet), 445 equal temperament (445et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 445 equal parts of about 2.7 ¢ each. Each step of 445edo represents a frequency ratio of 21/445, or the 445th root of 2.

Theory

445edo is enfactored in the 3-limit with the same tuning as 89edo, but the approximation to some of the higher harmonics are improved. It is consistent to the 7-odd-limit with harmonics 3, 5, 7 all tuned flat, and it allows an extension to the 11-limit. The equal temperament tempers out 2401/2400, 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; 3025/3024, 8019/8000, 24057/24010, 35937/35840, 41503/41472, 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably supports neptune.

Odd harmonics

Approximation of odd harmonics in 445edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.83 -0.70 -0.74 +1.03 -1.21 +0.82 +1.17 +0.21 -0.88 +1.13 +0.04
relative (%) -31 -26 -27 +38 -45 +30 +43 +8 -33 +42 +1
Steps
(reduced)
705
(260)
1033
(143)
1249
(359)
1411
(76)
1539
(204)
1647
(312)
1739
(404)
1819
(39)
1890
(110)
1955
(175)
2013
(233)

Subsets and supersets

Since 445 factors into 5 × 89, 445edo has 5edo and 89edo as its subsets.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [-28 25 -5, [-29 -11 20 [445 705 1033]] +0.2748 0.2149 7.97
2.3.5.7 2401/2400, 7381125/7340032, 43046721/42875000 [445 705 1033 1249]] +0.2716 0.1862 6.90
2.3.5.7.11 2401/2400, 3025/3024, 8019/8000, 234375/234256 [445 705 1033 1249 1539]] +0.2870 0.1694 6.28

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 13\445 35.06 1990656/1953125 Gammic (5-limit)
1 42\445 113.26 16/15 Misneb
1 216\445 582.47 7/5 Neptune (7-limit)
5 185\445
(7\445)
498.88
(18.88)
4/3
(81/80)
Pental (5-limit)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct