111ed12: Difference between revisions

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~~{{Stub}}~~

111ed12 is nearly identical to [[31edo]] but with the 12/1 rather than the [[2/1]] being just. The octave is about 1.45 cents stretched compared to 31edo.

== Theory ==

111ed12 is nearly identical to [[31edo]], but with the [[12/1|12th]] [[harmonic]] rather than the [[2/1|octave]] being just. The octave is about 1.45 cents stretched compared to just. Like 31edo, 111ed12 is [[consistent]] through the [[integer limit|12-integer-limit]], and like [[80ed6]], it optimizes for the [[11-limit]] by trading the accuracy of the [[5/1|5th]] and [[7/1|7th harmonic]]s for improved [[3/1|3rd]] and [[11/1|11th harmonics]]. The stretch is quite mild, but still considerable: the [[11/1|11th harmonic]] is only 4.4 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the [[23/1|23rd harmonic]], which is now only 2.4 cents flat of just.

=== Harmonics ===

{{Harmonics in equal|111|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 111ed12 (continued)}}

=== Subsets and supersets ===

Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111ed12 contains [[3ed12]] and [[37ed12]] as subset ed12's.

== Intervals ==

== ~~Harmonics~~ ==

== See also ==

~~{{Harmonics in equal~~

* [[18edf]] – relative edf

~~| steps = 111~~

* [[31edo]] – relative edo

~~| num = 12~~

* [[49edt]] – relative edt

~~| denom = 1~~

* [[72ed5]] – relative ed5

}}

* [[80ed6]] – relative ed6

~~{{Harmonics in equal~~

* [[87ed7]] – relative ed7

~~| steps = 111~~

* [[107ed11]] – relative ed11

~~| num = 12~~

* [[138ed22]] – relative ed22

~~| denom = 1~~

* [[204ed96]] – close to the zeta-optimized tuning for 31edo

~~| start = 12~~

* [[39cET]]

~~| collapsed = 1~~

}}

[[Category:~~Edonoi~~]]

[[Category:31edo]]

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-{{Stub}}
 {{Infobox ET}}
 {{ED intro}}
-ed12 is nearly identical to [[31edo]] but with the 12/1 rather than the [[2/1]] being just. The octave is about 1.45 cents stretched compared to 31edo.
+== Theory ==
+ed12 is nearly identical to [[31edo]], but with the [[12/1|12th]] [[harmonic]] rather than the [[2/1|octave]] being just. The octave is about 1.45 cents stretched compared to just. Like 31edo, 111ed12 is [[consistent]] through the [[integer limit|12-integer-limit]], and like [[80ed6]], it optimizes for the [[11-limit]] by trading the accuracy of the [[5/1|5th]] and [[7/1|7th harmonic]]s for improved [[3/1|3rd]] and [[11/1|11th harmonics]]. The stretch is quite mild, but still considerable: the [[11/1|11th harmonic]] is only 4.4 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the [[23/1|23rd harmonic]], which is now only 2.4 cents flat of just.
+=== Harmonics ===
+{{Harmonics in equal|111|12|1|intervals=integer|columns=11}}
+{{Harmonics in equal|111|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 111ed12 (continued)}}
+=== Subsets and supersets ===
+Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111ed12 contains [[3ed12]] and [[37ed12]] as subset ed12's.
 == Intervals ==
 {{Interval table}}
-== Harmonics ==
+== See also ==
-{{Harmonics in equal
+* [[18edf]] – relative edf
-| steps = 111
+* [[31edo]] – relative edo
-| num = 12
+* [[49edt]] – relative edt
-| denom = 1
+* [[72ed5]] – relative ed5
-}}
+* [[80ed6]] – relative ed6
-{{Harmonics in equal
+* [[87ed7]] – relative ed7
-| steps = 111
+* [[107ed11]] – relative ed11
-| num = 12
+* [[138ed22]] – relative ed22
-| denom = 1
+* [[204ed96]] – close to the zeta-optimized tuning for 31edo
-| start = 12
+* [[39cET]]
-| collapsed = 1
-}}
-[[Category:Edonoi]]
+[[Category:31edo]]