31edf: Difference between revisions
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Lookalikes: [[53edo]], [[84edt]] | Lookalikes: [[53edo]], [[84edt]] | ||
=Just Approximation= | |||
= Just Approximation = | |||
31edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale. | 31edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Interval | ||
! | ! Ratio | ||
! | ! Size | ||
! | ! Difference | ||
|- | |- | ||
| | | Perfect octave | ||
| 2/1 | |||
| style="text-align:center;" |31 | | style="text-align: center;" | 31 | ||
| +0.12 cents | |||
|- | |- | ||
| major third | |||
| 5/4 | |||
| style="text-align:center;" |17 | | style="text-align: center;" | 17 | ||
| | | −1.37 cents | ||
|- | |- | ||
| minor third | |||
| 6/5 | |||
| style="text-align:center;" |14 | | style="text-align: center;" | 14 | ||
| +1.37 cents | |||
|- | |- | ||
| major tone | |||
| 9/8 | |||
| style="text-align:center;" |9 | | style="text-align: center;" | 9 | ||
| | | −0.12 cents | ||
|- | |- | ||
| minor tone | |||
| 10/9 | |||
| style="text-align:center;" |8 | | style="text-align: center;" | 8 | ||
| | | −1.25 cents | ||
|- | |- | ||
| diat. semitone | |||
| 16/15 | |||
| style="text-align:center;" |5 | | style="text-align: center;" | 5 | ||
| +1.49 cents | |||
|}One notable property of 53EDO is that it offers good approximations for both pure and | |} | ||
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. | 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. | ||
[[Category:Edf]] | [[Category:Edf]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
Revision as of 04:52, 16 November 2024
| ← 30edf | 31edf | 32edf → |
(convergent)
(convergent)
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.