450edo

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450edo

451edo →

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 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 2^1/450, 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