76ed80: Difference between revisions

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== Theory ==

The 80th harmonic ~~would~~ be ~~extremely wide for an~~ equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}} ~~(which is compressed by 2.31{{c}})~~, making 76ed80 extremely close to optimal for 12edo.

The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}, making 76ed80 extremely close to optimal for 12edo.

=== Harmonics ===

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* [[31ed6]] – relative ed6

* [[40ed10]] – relative ed10

* [[43ed12]] – relative ed12

[[Category:12edo]]

[[Category:Zeta-optimized tunings]]

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-{{mathematical interest}}
+{{Mathematical interest}}
 {{Infobox ET}}
 {{ED intro}}
 == Theory ==
-The 80th harmonic would be extremely wide for an equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}} (which is compressed by 2.31{{c}}), making 76ed80 extremely close to optimal for 12edo.
+The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}, making 76ed80 extremely close to optimal for 12edo.
 === Harmonics ===
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 * [[31ed6]] – relative ed6
 * [[40ed10]] – relative ed10
+* [[43ed12]] – relative ed12
+[[Category:12edo]]
+[[Category:Zeta-optimized tunings]]