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