433edo

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433edo

434edo →

Prime factorization	433 (prime)
Step size	2.77136¢
Fifth	253\433 (701.155¢)
Semitones (A1:m2)	39:34 (108.1¢ : 94.23¢)
Consistency limit	5
Distinct consistency limit	5

433 equal divisions of the octave (abbreviated 433edo or 433ed2), also called 433-tone equal temperament (433tet) or 433 equal temperament (433et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 433 equal parts of about 2.77 ¢ each. Each step represents a frequency ratio of 2^1/433, or the 433rd root of 2.

Theory

443edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. To start with, the patent val ⟨433 686 1005 1216] as well as the 433d val ⟨433 686 1005 1215] are worth considering.

Using the patent val, the equal temperament tempers out 19683/19600 and 4096000/4084101 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 161280/161051, and 180224/180075 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 433edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-0.80	-1.09	+1.15	+1.17	+0.18	-0.80	+0.88	+0.36	-0.98	+0.35	+0.82
	relative (%)	-29	-39	+42	+42	+7	-29	+32	+13	-35	+13	+30
Steps (reduced)		686 (253)	1005 (139)	1216 (350)	1373 (74)	1498 (199)	1602 (303)	1692 (393)	1770 (38)	1839 (107)	1902 (170)	1959 (227)

Subsets and supersets

433edo is the 84th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-686 433⟩	[⟨433 686]]	0.2525	0.2525	9.11
2.3.5	2109375/2097152, [-29 52 -23⟩	[⟨433 686 1005]]	0.3254	0.2306	8.32
2.3.5.7	19683/19600, 4096000/4084101, 2109375/2097152	[⟨433 686 1005 1216]]	0.1414	0.3759	13.56
2.3.5.7.11	3025/3024, 6250/6237, 30375/30184, 180224/180075	[⟨433 686 1005 1216 1498]]	0.1026	0.3451	12.45
2.3.5.7.11.13	2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405	[⟨433 686 1005 1216 1498 1602]]	0.1217	0.3179	11.47
2.3.5.7.11.13.17	2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803	[⟨433 686 1005 1216 1498 1602 1770]]	0.0919	0.3033	10.94

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	98\433	271.594	75/64	Orson

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Francium

• sleeping as we don't know (2023)