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← 30edf31edf32edf →
Prime factorization 31 (prime)
Step size 22.6437¢ 
Octave 53\31edf (1200.12¢)
Twelfth 84\31edf (1902.07¢)
Consistency limit 10
Distinct consistency limit 7

Division of the just perfect fifth into 31 equal parts (31EDF) is almost identical to 53 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 0.1166 cents stretched and the step size is about 22.6437 cents. It is consistent to the 10-integer-limit.

Lookalikes: 53edo, 84edt

Just Approximation

31edf provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy-Zarlino "just major" scale.

interval ratio size difference
perfect octave 2/1 31 +0.12 cents
major third 5/4 17 −1.37 cents
minor third 6/5 14 +1.37 cents
major tone 9/8 9 −0.12 cents
minor tone 10/9 8 −1.25 cents
diat. semitone 16/15 5 +1.49 cents

One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.