450edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 449edo450edo451edo →
Prime factorization 2 × 32 × 52
Step size 2.66667¢
Fifth 263\450 (701.333¢)
Semitones (A1:m2) 41:35 (109.3¢ : 93.33¢)
Consistency limit 7
Distinct consistency limit 7

450 equal divisions of the octave (abbreviated 450edo), or 450-tone equal temperament (450tet), 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 of 450edo 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

Approximation of odd harmonics in 450edo
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 +13 -31 -47 +26 -20 -10 -36 +43 +46 +40
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

Table of rank-2 temperaments by generator
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