404edo

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← 403edo404edo405edo →
Prime factorization 22 × 101
Step size 2.9703¢
Fifth 236\404 (700.99¢) (→59\101)
Semitones (A1:m2) 36:32 (106.9¢ : 95.05¢)
Consistency limit 7
Distinct consistency limit 7

404 equal divisions of the octave (abbreviated 404edo or 404ed2), also called 404-tone equal temperament (404tet) or 404 equal temperament (404et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 404 equal parts of about 2.970 ¢ each. Each step represents a frequency ratio of 21/404, or the 404th root of 2.

Theory

404edo is enfactored in the 11-limit, with the same tuning as 202edo. The alternative 404e val, with lower overall error, tempers out 3025/3024 and 41503/41472 in the 11-limit; and 847/845, 1001/1000, and 1716/1715 in the 13-limit, whereas the patent val in the 13-limit tempers out 4225/4224.

Odd harmonics

Approximation of odd harmonics in 404edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.96 -0.18 -0.51 +1.04 +1.16 +0.07 -1.14 -1.00 -0.48 -1.47 +1.43
relative (%) -32 -6 -17 +35 +39 +2 -38 -33 -16 -50 +48
Steps
(reduced)
640
(236)
938
(130)
1134
(326)
1281
(69)
1398
(186)
1495
(283)
1578
(366)
1651
(35)
1716
(100)
1774
(158)
1828
(212)

Subsets and supersets

Since 404 factors into 22 × 101, 404edo has subset edos 2, 4, 101, and 202.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.7.11.13 243/242, 441/440, 4000/3993, 4225/4224, 65625/65536 [404 ​640​ 938​ 1134 ​1398 ​1495]] (404) +0.0854 0.2821 9.50

Trivia

A Scala file for this EDO is used as the 404 page of Sevish's website.

Music

Francium