424edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
424edo is [[consistent]] to the [[9-odd-limit]], but the [[harmonic]] [[5/1|5]] is about halfway between its steps. It is [[enfactoring|enfactored]] in the 7-limit, with the same tuning as [[212edo]]. The approximation to [[11/1|11]], although closer to just than 212edo's, tends sharp, so its improvement is debatable. All things considered, a 2.3.13.17.19.23 [[subgroup]] interpretation with optional additions of 7, 11, or both, seems most reasonable. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
424 factors into 2<sup>3</sup> × 53, | Since 424 factors into 2<sup>3</sup> × 53, 424edo has subset edos {{EDOs| 2, 4, 8, 53, 106, and 212 }}. [[848edo]], which doubles it, gives a good correction to the harmonic 5. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|- | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|424 672}} | ! rowspan="2" | [[Mapping]] | ||
| 0. | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
| 0. | ! colspan="2" | Tuning error | ||
| | |- | ||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.7.11 | |||
| 41503/41472, 117649/117128, {{monzo| -26 19 1 -2 }} | |||
| {{mapping| 424 672 1190 1467 }} | |||
| +0.0499 | |||
| 0.1747 | |||
| 6.17 | |||
|} | |} | ||
Latest revision as of 12:39, 21 February 2025
| ← 423edo | 424edo | 425edo → |
424 equal divisions of the octave (abbreviated 424edo or 424ed2), also called 424-tone equal temperament (424tet) or 424 equal temperament (424et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 424 equal parts of about 2.83 ¢ each. Each step represents a frequency ratio of 21/424, or the 424th root of 2.
Theory
424edo is consistent to the 9-odd-limit, but the harmonic 5 is about halfway between its steps. It is enfactored in the 7-limit, with the same tuning as 212edo. The approximation to 11, although closer to just than 212edo's, tends sharp, so its improvement is debatable. All things considered, a 2.3.13.17.19.23 subgroup interpretation with optional additions of 7, 11, or both, seems most reasonable.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.07 | -1.41 | -0.90 | -0.14 | +0.57 | +0.04 | +1.35 | -0.24 | -0.34 | -0.97 | +0.03 |
| Relative (%) | -2.4 | -49.8 | -31.8 | -4.8 | +20.1 | +1.4 | +47.8 | -8.4 | -12.1 | -34.3 | +1.0 | |
| Steps (reduced) |
672 (248) |
984 (136) |
1190 (342) |
1344 (72) |
1467 (195) |
1569 (297) |
1657 (385) |
1733 (37) |
1801 (105) |
1862 (166) |
1918 (222) | |
Subsets and supersets
Since 424 factors into 23 × 53, 424edo has subset edos 2, 4, 8, 53, 106, and 212. 848edo, which doubles it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.7.11 | 41503/41472, 117649/117128, [-26 19 1 -2⟩ | [⟨424 672 1190 1467]] | +0.0499 | 0.1747 | 6.17 |