421edo: 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 36: | Line 37: | ||
| {{monzo| -667 421 }} | | {{monzo| -667 421 }} | ||
| {{mapping| 421 667 }} | | {{mapping| 421 667 }} | ||
| 0.2421 | | +0.2421 | ||
| 0.2421 | | 0.2421 | ||
| 8.49 | | 8.49 | ||
| Line 43: | Line 44: | ||
| {{monzo| -44 26 1 }}, {{monzo| 37 5 -16 }} | | {{monzo| -44 26 1 }}, {{monzo| 37 5 -16 }} | ||
| {{mapping| 421 667 1182 }} | | {{mapping| 421 667 1182 }} | ||
| 0.1263 | | +0.1263 | ||
| 0.2567 | | 0.2567 | ||
| 9.01 | | 9.01 | ||
| Line 50: | Line 51: | ||
| 88209/87808, 2893401/2883584, 208971104256/208422380089 | | 88209/87808, 2893401/2883584, 208971104256/208422380089 | ||
| {{mapping| 421 667 1182 1456 }} | | {{mapping| 421 667 1182 1456 }} | ||
| 0.1814 | | +0.1814 | ||
| 0.2419 | | 0.2419 | ||
| 8.49 | | 8.49 | ||
| Line 57: | Line 58: | ||
| 24192/24167, 53361/53248, 85293/85184, 88209/87808 | | 24192/24167, 53361/53248, 85293/85184, 88209/87808 | ||
| {{mapping| 421 667 1182 1456 1558 }} | | {{mapping| 421 667 1182 1456 1558 }} | ||
| 0.1274 | | +0.1274 | ||
| 0.2418 | | 0.2418 | ||
| 8.48 | | 8.48 | ||
Latest revision as of 12:47, 21 February 2025
| ← 420edo | 421edo | 422edo → |
421 equal divisions of the octave (abbreviated 421edo or 421ed2), also called 421-tone equal temperament (421tet) or 421 equal temperament (421et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 421 equal parts of about 2.85 ¢ each. Each step represents a frequency ratio of 21/421, or the 421st root of 2.
Theory
421edo is inconsistent to the 5-odd-limit, with its harmonic 5 being way too sharp. To start with, consider the following breeds:
- ⟨421 667 977 1182] (421c)
- ⟨421 667 977 1181] (421cd)
- ⟨421 667 978 1182] (patent val)
The 421c val tempers out 4375/4374 and 2100875/2097152, supporting mitonic.
The 421cd val tempers out 1029/1024 and 823543/820125.
The 421 val tempers out 2401/2400 and 3136/3125, supporting hemiwürschmidt.
Omitting harmonic 5, it is suitable for the 2.3.7.11.13.29.37 subgroup, where it tempers out 638/637, 5292/5291, 24192/24167, 53361/53248, 88209/87808 and 85293/85184.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.77 | +1.33 | +0.30 | +1.32 | -1.20 | +0.33 | +0.57 | +0.51 | -1.08 | -0.47 | -1.20 |
| Relative (%) | -26.9 | +46.8 | +10.4 | +46.2 | -42.1 | +11.5 | +19.9 | +17.8 | -37.7 | -16.6 | -42.0 | |
| Steps (reduced) |
667 (246) |
978 (136) |
1182 (340) |
1335 (72) |
1456 (193) |
1558 (295) |
1645 (382) |
1721 (37) |
1788 (104) |
1849 (165) |
1904 (220) | |
Subsets and supersets
421edo is the 82nd prime edo. 1263edo, 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 | [-667 421⟩ | [⟨421 667]] | +0.2421 | 0.2421 | 8.49 |
| 2.3.7 | [-44 26 1⟩, [37 5 -16⟩ | [⟨421 667 1182]] | +0.1263 | 0.2567 | 9.01 |
| 2.3.7.11 | 88209/87808, 2893401/2883584, 208971104256/208422380089 | [⟨421 667 1182 1456]] | +0.1814 | 0.2419 | 8.49 |
| 2.3.7.11.13 | 24192/24167, 53361/53248, 85293/85184, 88209/87808 | [⟨421 667 1182 1456 1558]] | +0.1274 | 0.2418 | 8.48 |