31edf
| ← 30edf | 31edf | 32edf → |
(convergent)
(convergent)
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.
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
| 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) | |