450edo
← 449edo | 450edo | 451edo → |
450 equal divisions of the octave (abbreviated 450edo or 450ed2), also called 450-tone equal temperament (450tet) or 450 equal temperament (450et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 450 equal parts of about 2.67 ¢ each. Each step represents a frequency ratio of 21/450, or the 450th root of 2.
Theory
450edo is consistent to the 7-odd-limit. It can be considered for the 2.3.5.7.13.17.29.31.37 subgroup, where it tempers out 651/650, 1666/1665, 1887/1885, 2016/2015, 2295/2294, 5916/5915, 4901/4900 and 14229/14210. It supports decal and varuna.
Odd harmonics
Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -0.62 | +0.35 | -0.83 | -1.24 | +0.68 | -0.53 | -0.27 | -0.96 | +1.15 | +1.22 | +1.06 |
Relative (%) | -23.3 | +13.2 | -31.0 | -46.6 | +25.6 | -19.8 | -10.1 | -35.8 | +43.3 | +45.7 | +39.7 | |
Steps (reduced) |
713 (263) |
1045 (145) |
1263 (363) |
1426 (76) |
1557 (207) |
1665 (315) |
1758 (408) |
1839 (39) |
1912 (112) |
1977 (177) |
2036 (236) |
Subsets and supersets
450 factors into 2 × 32 × 52, with subset edos 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, and 225.
Regular temperament properties
Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.3 | [-713 450⟩ | [⟨450 713]] | +0.1961 | 0.1961 | 7.35 |
2.3.5 | [-28 25 -5⟩, [25 15 -21⟩ | [⟨450 713 1045]] | +0.0800 | 0.2293 | 8.60 |
2.3.5.7 | 235298/234375, 321489/320000, 26873856/26796875 | [⟨450 713 1045 1263]] | +0.1336 | 0.2192 | 8.22 |
Rank-2 temperaments
Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
---|---|---|---|---|
2 | 61\450 | 162.67 | 1125/1024 | Kwazy |
5 | 187\450 (7\450) |
498.67 (18.67) |
4/3 (81/80) |
Pental (5-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct