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Created page with "{{Infobox ET}} {{EDO intro|413}} == 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 =..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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" | ||
|- | |- | ||
|2.9 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|187 -59}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|413 1309}} | ! rowspan="2" | [[Mapping]] | ||
| 0.0820 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.9 | |||
| {{monzo|187 -59}} | |||
| {{mapping|413 1309}} | |||
| +0.0820 | |||
| 0.0821 | | 0.0821 | ||
| 2.83 | | 2.83 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|32805/32768, {{monzo|8 7 -13}} | | 32805/32768, {{monzo|8 7 -13}} | ||
|{{mapping|413 1309 959}} | | {{mapping|413 1309 959}} | ||
| 0.0365 | | +0.0365 | ||
| 0.0930 | | 0.0930 | ||
| 3.20 | | 3.20 | ||
|} | |} | ||
Latest revision as of 12:55, 21 February 2025
| ← 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 |