31edf

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← 30edf 31edf 32edf →
Prime factorization 31 (prime)
Step size 22.6437¢ 
Octave 53\31edf (1200.12¢)
(convergent)
Twelfth 84\31edf (1902.07¢)
(convergent)
Consistency limit 10
Distinct consistency limit 7

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

Lookalikes: 53edo, 84edt

Theory

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 31edf is that, like 53edo, 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! Like 53edo, 31edf 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.

Harmonics

Approximation of prime harmonics in 31edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.12 +0.12 -1.14 +5.09 -7.52 -2.36 +8.73 -2.68 +6.22 -10.14 +10.26
Relative (%) +0.5 +0.5 -5.0 +22.5 -33.2 -10.4 +38.6 -11.8 +27.5 -44.8 +45.3
Steps
(reduced)
53
(22)
84
(22)
123
(30)
149
(25)
183
(28)
196
(10)
217
(0)
225
(8)
240
(23)
257
(9)
263
(15)