# 433edo

 ← 432edo 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 21/433, 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 (%) -28.9 -39.5 +41.5 +42.2 +6.6 -29.0 +31.6 +12.9 -35.3 +12.7 +29.8
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

Francium