# 31edf

**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.

# 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.