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 tempers out the triwellisma and the varunisma in the 7-limit, supporting and providing the optimal patent val for 7-limit decile. 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.

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 (%)	-23.3	+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

Since 450 factors into 2 × 3² × 5², 450edo has 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	7\450	18.67	81/80	Quintile (5-limit)
10	7\450	18.67	81/80	Decile

* In minimal-generator form