57ed6: Difference between revisions

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57ed6 is ~~very nearly identical~~ to [[22edo~~|22 EDO~~]], but with the [[6/1]] ~~rather than the 2/1~~ being just~~. The octave is about 2.754{{cent}}~~ [[stretched and compressed tuning|compressed]] ~~and the step size is~~ about 54.~~4203~~{{c}}. The local [[~~The_Riemann_zeta_function_and_tuning~~#~~Optimal_octave_stretch~~|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), ~~making 57ed6 very close~~ to ~~optimal for~~ 22edo.

== Theory ==

57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1|octave]] being just, which results in octaves being [[stretched and compressed tuning|compressed]] by about 2.75{{c}}, corresponding to about 22.050610edo. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1|3]] and [[7/1|7]], but the [[5/1|5]] and [[11/1|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.

=== Harmonics ===

{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}

=== Subsets and supersets ===

Since 57 factors into primes as {{nowrap| 3 × 19 }}, 57ed6 contains subset ed6's [[3ed6]] and [[19ed6]].

== Intervals ==

== ~~Harmonics~~ ==

== See also ==

~~{{Harmonics in equal~~

* [[22edo]] – relative edo

~~| steps = 57~~

* [[35edt]] – relative edt

~~| num = 6~~

* [[79ed12]] – relative ed12

~~| denom = 1~~

}}

~~{{Harmonics in equal~~

~~| steps = 57~~

~~| num = 6~~

~~| denom = 1~~

~~| start = 12~~

~~| collapsed = 1~~

}}

[[~~Category:~~22edo]]

[[~~Category:Edonoi~~]]

~~{{stub}}~~

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 {{ED intro}}
-ed6 is very nearly identical to [[22edo|22 EDO]], but with the [[6/1]] rather than the 2/1 being just. The octave is about 2.754{{cent}} [[stretched and compressed tuning|compressed]] and the step size is about 54.4203{{c}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.
+== Theory ==
+ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1|octave]] being just, which results in octaves being [[stretched and compressed tuning|compressed]] by about 2.75{{c}}, corresponding to about 22.050610edo. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1|3]] and [[7/1|7]], but the [[5/1|5]] and [[11/1|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.
+=== Harmonics ===
+{{Harmonics in equal|57|6|1|intervals=integer|columns=11}}
+{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}
+=== Subsets and supersets ===
+Since 57 factors into primes as {{nowrap| 3 × 19 }}, 57ed6 contains subset ed6's [[3ed6]] and [[19ed6]].
 == Intervals ==
 {{Interval table}}
-== Harmonics ==
+== See also ==
-{{Harmonics in equal
+* [[22edo]] – relative edo
-| steps = 57
+* [[35edt]] – relative edt
-| num = 6
+* [[79ed12]] – relative ed12
-| denom = 1
-}}
-{{Harmonics in equal
-| steps = 57
-| num = 6
-| denom = 1
-| start = 12
-| collapsed = 1
-}}
-[[Category:22edo]]
-[[Category:Edonoi]]
-{{stub}}