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]].
+{{Infobox ET}}
+{{ED intro}}
-Lookalikes: [[53edo]], [[84edt]]
+== Theory ==
-=Just Approximation=
+edf is almost identical to [[53edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1|7]], the [[5/1|5]], [[11/1|11]], [[13/1|13]], and [[19/1|19]] remain flat but significantly less so than in 53edo.
-edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
-{| class="wikitable"
-|-
-! |interval
-! |ratio
-! |size
-! |difference
-|-
-| |perfect octave
-| |2/1
-| style="text-align:center;" |31
-| | +0.12 cents
-|-
-| |major third
-| |5/4
-| style="text-align:center;" |17
-| |−1.37 cents
-|-
-| |minor third
-| |6/5
-| style="text-align:center;" |14
-| | +1.37 cents
-|-
-| |major tone
-| |9/8
-| style="text-align:center;" |9
-| |−0.12 cents
-|-
-| |minor tone
-| |10/9
-| style="text-align:center;" |8
-| |−1.25 cents
-|-
-| |diat. semitone
-| |16/15
-| style="text-align:center;" |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.
+The [[The Riemann zeta function and tuning|local zeta peak]] around 53 is located at 52.996829, which has the octave stretched by 0.0718{{c}}; the octave of 31edf comes extremely close (differing by only {{sfrac|1|22}}{{c}}), thus minimizing relative error as much as possible.
-[[Category:Edf]]
-[[Category:Edonoi]]
+=== Harmonics ===
+{{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)}}
+=== Subsets and supersets ===
+edf is the 11th [[prime equal division|prime edf]], following [[29edf]] and coming before [[37edf]]. It does not contain any nontrivial subset edfs.
+== See also ==
+* [[9ed9/8]] – relative ed9/8
+* [[53edo]] – relative edo
+* [[84edt]] – relative edt
+* [[137ed6]] – relative ed6
+[[Category:53edo]]
+[[Category:Zeta-optimized tunings]]