3edt

← 2edt

3edt

4edt →

Prime factorization	3 (prime)
Step size	633.985 ¢
Octave	2\3edt (1267.97 ¢) (convergent)
Consistency limit	4
Distinct consistency limit	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. This is reinforced by 3edt having 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.

Odd harmonics

Approximation of odd harmonics in 3edt

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0	-250	-199	+0	+287	-3	-250	+167	-26	-199	+278
	Relative (%)	+0.0	-39.5	-31.4	+0.0	+45.2	-0.4	-39.5	+26.3	-4.0	-31.4	+43.8
Steps (reduced)		3 (0)	4 (1)	5 (2)	6 (0)	7 (1)	7 (1)	7 (1)	8 (2)	8 (2)	8 (2)	9 (0)

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

• Liese
• Triton
• Tricot

Fractional-octave temperaments

• Augene, augmented, august - can be seen as a superset of 3edo and 3edt
• Soviet ferris wheel - 20edo and 3edt
• Akjayland - 21edo and 3edt
• Oganesson - 118edo and 3edt