129ed12: Difference between revisions

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~~129ED12~~ is very nearly identical to [[36edo]] (sixth-tone tuning), but with the [[12/1]] ~~rather than the 2/1~~ being just. ~~The~~ octave ~~is stretched~~ by 0.55[[~~cent~~]]s.

== Theory ==

129ed12 is very nearly identical to [[36edo]] (sixth-tone tuning), but with the 12th harmonic rather than the [[2/1|octave]] being just. This stretches the octave by about 0.546{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 36 is located at 35.982388, which has a step size of 33.3496{{c}} and has octaves stretched by 0.587{{c}}; 129ed12's octave is extremely close to optimal, being only 0.0418{{c}} ({{sfrac|1|23}} of a cent) off from the zeta peak.

~~Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]], [[101ed7]]~~

=== Harmonics ===

{{Harmonics in equal|129|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 129ed12 (continued)}}

== ~~Harmonics~~ ==

=== Subsets and supersets ===

Since 129 factors into primes as {{nowrap| 3 × 43 }}, 129ed12 contains subset ed12's [[3ed12]] and [[43ed12]].

[[Category:~~Edonoi~~]]

== See also ==

* [[21edf]] – relative edf

* [[36edo]] – relative edo

* [[57edt]] – relative edt

* [[93ed6]] – relative ed6

* [[101ed7]] – relative ed7

[[Category:36edo]]

[[Category:Zeta-optimized tunings]]

@@ Line 2: / Line 2: @@
 {{ED intro}}
-ED12 is very nearly identical to [[36edo]] (sixth-tone tuning), but with the [[12/1]] rather than the 2/1 being just. The octave is stretched by 0.55[[cent]]s.
+== Theory ==
+ed12 is very nearly identical to [[36edo]] (sixth-tone tuning), but with the 12th harmonic rather than the [[2/1|octave]] being just. This stretches the octave by about 0.546{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 36 is located at 35.982388, which has a step size of 33.3496{{c}} and has octaves stretched by 0.587{{c}}; 129ed12's octave is extremely close to optimal, being only 0.0418{{c}} ({{sfrac|1|23}} of a cent) off from the zeta peak.
-Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]], [[101ed7]]
+=== Harmonics ===
+{{Harmonics in equal|129|12|1|intervals=integer|columns=11}}
+{{Harmonics in equal|129|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 129ed12 (continued)}}
-== Harmonics ==
+=== Subsets and supersets ===
-{{Harmonics in equal|129|12|1|prec=2|columns=15}}
+Since 129 factors into primes as {{nowrap| 3 × 43 }}, 129ed12 contains subset ed12's [[3ed12]] and [[43ed12]].
-[[Category:Edonoi]]
+== See also ==
+* [[21edf]] – relative edf
+* [[36edo]] – relative edo
+* [[57edt]] – relative edt
+* [[93ed6]] – relative ed6
+* [[101ed7]] – relative ed7
+[[Category:36edo]]
+[[Category:Zeta-optimized tunings]]