413edo

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← 412edo413edo414edo →
Prime factorization 7 × 59
Step size 2.90557¢ 
Fifth 242\413 (703.148¢)
Semitones (A1:m2) 42:29 (122¢ : 84.26¢)
Dual sharp fifth 242\413 (703.148¢)
Dual flat fifth 241\413 (700.242¢)
Dual major 2nd 70\413 (203.39¢) (→10\59)
Consistency limit 5
Distinct consistency limit 5

413 equal divisions of the octave (abbreviated 413edo or 413ed2), also called 413-tone equal temperament (413tet) or 413 equal temperament (413et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 413 equal parts of about 2.91 ¢ each. Each step represents a frequency ratio of 21/413, or the 413th root of 2.

Theory

413et is only consistent to the 5-limit. Omitting the harmonics 3 and 7, it can be used until the 31-limit.

Odd harmonics

Approximation of odd harmonics in 413edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.19 +0.13 -1.27 -0.52 +0.74 -0.82 +1.32 -0.35 -1.14 -0.08 -0.67
Relative (%) +41.0 +4.4 -43.8 -17.9 +25.5 -28.2 +45.4 -12.2 -39.4 -2.7 -23.1
Steps
(reduced)
655
(242)
959
(133)
1159
(333)
1309
(70)
1429
(190)
1528
(289)
1614
(375)
1688
(36)
1754
(102)
1814
(162)
1868
(216)

Subsets and supersets

413 factors into 7 × 59, with 7edo and 59edo as its subset edos. 826edo, which doubles it, gives a good correction to the harmonics 3 and 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [187 -59 [413 1309]] 0.0820 0.0821 2.83
2.9.5 32805/32768, [8 7 -13 [413 1309 959]] 0.0365 0.0930 3.20