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 12 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 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.

=== Harmonics ===

{{Harmonics in equal|79|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ~~79ed12~~ (continued)}}

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

~~=== Subsets and supersets ===~~

~~79ed12 is the 22nd [[prime equal division|prime ed12]], so it does not contain any nontrivial subset ed12's.~~

== See also ==

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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 12 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 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.
 === Harmonics ===
-{{Harmonics in equal|79|12|1|intervals=integer|columns=11}}
+{{Harmonics in equal|204|96|1|intervals=integer|columns=11}}
-{{Harmonics in equal|79|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 79ed12 (continued)}}
+{{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}}
-=== Subsets and supersets ===
-ed12 is the 22nd [[prime equal division|prime ed12]], so it does not contain any nontrivial subset ed12's.
 == See also ==