416edo
← 415edo | 416edo | 417edo → |
416 equal divisions of the octave (abbreviated 416edo or 416ed2), also called 416-tone equal temperament (416tet) or 416 equal temperament (416et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 416 equal parts of about 2.88 ¢ each. Each step represents a frequency ratio of 21/416, or the 416th root of 2.
Theory
416et is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. Nonetheless, it gives reasonable approximations to harmonics 5, 7, 9, 11, 19, and 23, making it suitable for a 2.9.5.7.11.19.23 subgroup interpretation, where it notably tempers out 1331/1330, 1540/1539, 5632/5625, 9801/9800, and 10241/10240.
Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, supporting tridecatonic and fermionic.
Odd harmonics
Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -0.99 | +0.22 | +0.40 | +0.90 | -0.36 | -1.10 | -0.77 | -1.11 | -0.40 | -0.59 | +0.57 |
Relative (%) | -34.4 | +7.8 | +14.0 | +31.1 | -12.4 | -38.3 | -26.6 | -38.5 | -13.8 | -20.4 | +19.8 | |
Steps (reduced) |
659 (243) |
966 (134) |
1168 (336) |
1319 (71) |
1439 (191) |
1539 (291) |
1625 (377) |
1700 (36) |
1767 (103) |
1827 (163) |
1882 (218) |
Subsets and supersets
Since 416 factors into 25 × 13, 416edo subset edos 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208. 832edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.9 | [1319 -416⟩ | [⟨416 1319]] | -0.1416 | 0.1416 | 4.91 |
2.9.5 | [56 -14 -5⟩, [-5 -16 24⟩ | [⟨416 1319 966]] | -0.1267 | 0.1175 | 4.07 |
2.9.5.7 | 420175/419904, 102760448/102515625, [14 -6 7 -4⟩ | [⟨416 1319 966 1168]] | -0.1310 | 0.1021 | 3.54 |
2.9.5.7.11 | 5632/5625, 9801/9800, 41503/41472, 774400000/771895089 | [⟨416 1319 966 1168 1439]] | -0.0842 | 0.1308 | 4.53 |
Music
- palate cleanser (2024) – tetracot[13] in 416edo tuning