413edo
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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
| ← 412edo | 413edo | 414edo → |
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
| 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 |