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| {{Infobox ET}} | | {{Infobox ET}} |
| '''[[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]].
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| Lookalikes: [[53edo]], [[84edt]]
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| == Theory == | | == Theory == |
| 31edf provides excellent approximations for the classic [[5-limit]] just chords and scales, such as the Ptolemy-Zarlino "[[just major]]" scale. | | 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]]. Like 53edo, 31edf is consistent to the [[integer limit|10-integer-limit]]. |
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| {| class="wikitable"
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| |-
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| ! Interval
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| ! Ratio
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| ! Size
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| ! Difference
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| |-
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| | Perfect octave
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| | 2/1 | |
| | style="text-align: center;" | 31
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| | +0.12 cents
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| |-
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| | major third
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| | 5/4
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| | style="text-align: center;" | 17
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| | −1.37 cents
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| |-
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| | minor third
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| | 6/5
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| | style="text-align: center;" | 14
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| | +1.37 cents
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| |-
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| | major tone
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| | 9/8
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| | style="text-align: center;" | 9
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| | −0.12 cents
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| |-
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| | minor tone
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| | 10/9
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| | style="text-align: center;" | 8
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| | −1.25 cents
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| |-
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| | diat. semitone
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| | 16/15
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| | style="text-align: center;" | 5
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| | +1.49 cents
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| |}
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| One notable property of 31edf is that, like 53edo, it offers good approximations for both pure and [[Pythagorean tuning|Pythagorean]] major thirds.
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| 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.
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| 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.
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| === Harmonics === | | === Harmonics === |
| {{Harmonics in equal|31|3|2|intervals=prime}} | | {{Harmonics in equal|31|3|2|intervals=integer}} |
| | {{Harmonics in equal|31|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31edf (continued)}} |
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| [[Category:Edf]] | | == See also == |
| [[Category:Edonoi]] | | * [[53edo]] – relative edo |
| | * [[84edt]] – relative edt |
| | * [[137ed6]] – relative ed6 |