3edt

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← 2edt3edt4edt →
Prime factorization 3 (prime)
Step size 633.985¢ 
Octave 2\3edt (1267.97¢)
(convergent)
Consistency limit 4
Distinct consistency limit 2

3 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 3edt or 3ed3), is a nonoctave tuning system that divides the interval of 3/1 into 3 equal parts of about 634 ¢ each. Each step represents a frequency ratio of 31/3, or the 3rd root of 3.

Theory

3edt can be thought of as 2edo with the 3/1 made just, by stretching the octave by 67.97 cents.

Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63 cents flat of 13/1. One step of 3edt has two good 13-limit rational approximations, 13/9 and 75/52, both which are convergents. 3edt thus tempers out (13/9)3 / (3/1) = 2197/2187, the threedie, and (75/52)3 / (3/1) = 140625/140608, the catasma. The good approximation for 13/9 and 75/52 also implies a good approximation for 25/4, or (5/2)2.

Harmonics

Approximation of harmonics in 3edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +68 +0 +136 -250 +68 -199 +204 +0 -182 +287 +136
relative (%) +11 +0 +21 -39 +11 -31 +32 +0 -29 +45 +21
Steps
(reduced)
2
(2)
3
(0)
4
(1)
4
(1)
5
(2)
5
(2)
6
(0)
6
(0)
6
(0)
7
(1)
7
(1)

Relationship to octave temperaments

One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are:

Simple octave temperaments

Fractional-octave temperaments