# 409edo

 ← 408edo 409edo 410edo →
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 409ed2), also called 409-tone equal temperament (409tet) or 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 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 (%) +0.0 -25.0 +33.1 -20.8 +9.2 -48.0 +22.8 -40.2 -13.7 +8.6 -26.6
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

Francium