421edo
| ← 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 (%) | -27 | +47 | +10 | +46 | -42 | +11 | +20 | +18 | -38 | -17 | -42 | |
| 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 |