31edf: Difference between revisions

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'''[[EDF|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo~~|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 [[~~9-odd-limit|10-~~integer-limit]].

'''[[EDF|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 [[Octave stretching|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]]

= ~~Just Approximation~~ =

== Theory ==

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.

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One notable property of ~~53EDO~~ is that it offers good approximations for both pure and Pythagorean major thirds.

One notable property of 31edf is that, like 53edo, it offers good approximations for both pure and [[Pythagorean tuning|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! 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 ===

[[Category:Edf]]

[[Category:Edonoi]]

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 {{Infobox ET}}
-'''[[EDF|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo|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 [[9-odd-limit|10-integer-limit]].
+'''[[EDF|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 [[Octave stretching|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]]
-= Just Approximation =
+== Theory ==
-edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
+edf provides excellent approximations for the classic [[5-limit]] just chords and scales, such as the Ptolemy-Zarlino "[[just major]]" scale.
 {| class="wikitable"
@@ Line 45: / Line 45: @@
 |}
-One notable property of 53EDO is that it offers good approximations for both pure and Pythagorean major thirds.
+One notable property of 31edf is that, like 53edo, it offers good approximations for both pure and [[Pythagorean tuning|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! 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 ===
+{{Harmonics in equal|31|3|2|intervals=prime}}
 [[Category:Edf]]
 [[Category:Edonoi]]