31edf: Difference between revisions

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== Theory ==

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]].

31edf 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.

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.

=== Harmonics ===

{{Harmonics in equal|31|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31edf (continued)}}

=== Subsets and supersets ===

31edf 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]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edf 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]].
+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.
+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.
 === 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]]