409edo

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← 408edo409edo410edo →
Prime factorization 409 (prime)
Step size 2.93399¢
Fifth 239\409 (701.222¢)
Semitones (A1:m2) 37:32 (108.6¢ : 93.89¢)
Consistency limit 3
Distinct consistency limit 3

409 equal divisions of the octave (abbreviated 409edo), or 409-tone equal temperament (409tet), 409 equal temperament (409et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 409 equal parts of about 2.93 ¢ each. Each step of 409edo represents a frequency ratio of 21/409, or the 409th root of 2.

Theory

409et is inconsistent to the 5-odd-limit. In the 7-limit, the 409c val 409 648 949 1148] is about as viable as the patent val 409 648 950 1148]. The 409c val tempers out 15625/15552 and 16875/16807, supporting sqrtphi. The patent val tempers out 3136/3125 and 19683/19600, supporting subpental.

Odd harmonics

Approximation of prime harmonics in 409edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.73 +0.97 -0.61 +0.27 -1.41 +0.67 -1.18 -0.40 +0.25 -0.78
relative (%) +0 -25 +33 -21 +9 -48 +23 -40 -14 +9 -27
Steps
(reduced)
409
(0)
648
(239)
950
(132)
1148
(330)
1415
(188)
1513
(286)
1672
(36)
1737
(101)
1850
(214)
1987
(351)
2026
(390)

Subsets and supersets

409edo is the 80th prime edo. 1227edo, which triples it, gives a good correction to the harmonic 5.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-648 409 [409 648]] 0.2311 0.2311 7.88
2.3.7 [-44 26 1, [12 19 -15 [409 648 1148]] 0.2266 0.1888 6.43
2.3.7.11 117649/117612, 5038848/5021863, 134775333/134217728 [409 648 1148 1415]] 0.1503 0.2102 7.16
2.3.7.11.13 729/728, 19773/19712, 50421/50336, 718848/717409 [409 648 1148 1415 1513]] 0.1963 0.2093 7.13

Music

Francium