111ed12: Difference between revisions

(8 intermediate revisions by 4 users not shown)

Line 3:

== Theory ==

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.

111ed12 is nearly identical to [[31edo]], but with the [[12/1|12th]] [[harmonic]] rather than the [[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 ==

== See also ==

* [[18edf]] – relative edf

* [[31edo]] – relative edo

* [[49edt]] – relative edt

* [[72ed5]] – relative ed5

* [[80ed6]] – relative ed6

* [[87ed7]] – relative ed7

* [[107ed11]] – relative ed11

* [[138ed22]] – relative ed22

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

* [[39cET]]

[[Category:31edo]]

@@ Line 3: / Line 3: @@
 == Theory ==
-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.
+ed12 is nearly identical to [[31edo]], but with the [[12/1|12th]] [[harmonic]] rather than the [[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}}
+== See also ==
+* [[18edf]] – relative edf
+* [[31edo]] – relative edo
+* [[49edt]] – relative edt
+* [[72ed5]] – relative ed5
+* [[80ed6]] – relative ed6
+* [[87ed7]] – relative ed7
+* [[107ed11]] – relative ed11
+* [[138ed22]] – relative ed22
+* [[204ed96]] – close to the zeta-optimized tuning for 31edo
+* [[39cET]]
+[[Category:31edo]]