449edo

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← 448edo449edo450edo →
Prime factorization 449 (prime)
Step size 2.67261¢
Fifth 263\449 (702.895¢)
Semitones (A1:m2) 45:32 (120.3¢ : 85.52¢)
Dual sharp fifth 263\449 (702.895¢)
Dual flat fifth 262\449 (700.223¢)
Dual major 2nd 76\449 (203.118¢)
Consistency limit 7
Distinct consistency limit 7

449 equal divisions of the octave (abbreviated 449edo), or 449-tone equal temperament (449tet), 449 equal temperament (449et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 449 equal parts of about 2.67 ¢ each. Each step of 449edo represents a frequency ratio of 21/449, or the 449th root of 2.

Theory

449edo is consistent to the 7-odd-limit, but the errors of harmonics 3, 5, and 7 are all quite large, giving us the option of treating it as a full 7-limit temperament, or a 2.9.15.21.11.13 subgroup temperament.

Using the patent val, the equal temperament tempers out 4375/4374 and 26873856/26796875 in the 7-limit; 41503/41472, 160083/160000, 539055/537824, 805255/802816, and 825000/823543 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 449edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +0.94 +1.21 +1.33 -0.79 -0.76 -1.33 -0.52 -0.72 -0.85 -0.40 -0.21
relative (%) +35 +45 +50 -30 -28 -50 -19 -27 -32 -15 -8
Steps
(reduced)
712
(263)
1043
(145)
1261
(363)
1423
(76)
1553
(206)
1661
(314)
1754
(407)
1835
(39)
1907
(111)
1972
(176)
2031
(235)

Subsets and supersets

449edo is the 87th prime edo. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1423 449 [449 1423]] 0.1249 0.1249 4.67

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 127\449 339.421 243\200 Amity (7-limit)

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

Music

Francium