409edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| -648 409 }} | | {{monzo| -648 409 }} | ||
| {{mapping| 409 648 }} | | {{mapping| 409 648 }} | ||
| 0.2311 | | +0.2311 | ||
| 0.2311 | | 0.2311 | ||
| 7.88 | | 7.88 | ||
| Line 32: | Line 33: | ||
| {{monzo| -44 26 1 }}, {{monzo| 12 19 -15 }} | | {{monzo| -44 26 1 }}, {{monzo| 12 19 -15 }} | ||
| {{mapping| 409 648 1148 }} | | {{mapping| 409 648 1148 }} | ||
| 0.2266 | | +0.2266 | ||
| 0.1888 | | 0.1888 | ||
| 6.43 | | 6.43 | ||
| Line 39: | Line 40: | ||
| 117649/117612, 5038848/5021863, 134775333/134217728 | | 117649/117612, 5038848/5021863, 134775333/134217728 | ||
| {{mapping| 409 648 1148 1415 }} | | {{mapping| 409 648 1148 1415 }} | ||
| 0.1503 | | +0.1503 | ||
| 0.2102 | | 0.2102 | ||
| 7.16 | | 7.16 | ||
| Line 46: | Line 47: | ||
| 729/728, 19773/19712, 50421/50336, 718848/717409 | | 729/728, 19773/19712, 50421/50336, 718848/717409 | ||
| {{mapping| 409 648 1148 1415 1513 }} | | {{mapping| 409 648 1148 1415 1513 }} | ||
| 0.1963 | | +0.1963 | ||
| 0.2093 | | 0.2093 | ||
| 7.13 | | 7.13 | ||
Latest revision as of 12:58, 21 February 2025
| ← 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 |