204ed96: Difference between revisions

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== Theory ==

The 96th harmonic ~~would~~ be ~~extremely wide for an~~ equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is ~~compressed~~ by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.

The 96th harmonic is far too wide to be a useful equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.

=== Harmonics ===

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{{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}}

~~== See also ==~~

* [[18edf]] – relative edf

* [[31edo]] – relative edo

* [[49edt]] – relative edt

* [[72ed5]] – relative ed5

* [[80ed6]] – relative ed6

* [[87ed7]] – relative ed7

* [[107ed11]] – relative ed11

* [[111ed12]] – relative ed12

* [[138ed22]] – relative ed22

* [[39cET]]

[[Category:31edo]]

[[Category:Zeta-optimized tunings]]

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+{{Mathematical interest}}
 {{Infobox ET}}
 {{ED intro}}
 == Theory ==
-The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is compressed by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.
+The 96th harmonic is far too wide to be a useful equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.
 === Harmonics ===
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 {{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}}
-== See also ==
+* [[18edf]] – relative edf
 * [[31edo]] – relative edo
 * [[49edt]] – relative edt
 * [[72ed5]] – relative ed5
 * [[80ed6]] – relative ed6
+* [[87ed7]] – relative ed7
+* [[107ed11]] – relative ed11
+* [[111ed12]] – relative ed12
+* [[138ed22]] – relative ed22
+* [[39cET]]
+[[Category:31edo]]
+[[Category:Zeta-optimized tunings]]