409edo
| ← 408edo | 409edo | 410edo → |
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
| 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 |