57ed6: Difference between revisions

(15 intermediate revisions by 4 users not shown)

Line 1:

~~'''[[Ed6|Division of the sixth harmonic]] into 70 equal parts''' (70ED6) 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~~{{c}} [[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 27 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.

== ~~Harmonics~~ ==

== Theory ==

{{~~Harmonics in equal~~|57|6|1|~~prec=~~2~~|columns=15}}~~

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.

[[~~Category:Edonoi~~]]

=== 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 ==

== See also ==

* [[22edo]] – relative edo

* [[35edt]] – relative edt

* [[79ed12]] – relative ed12

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''[[Ed6|Division of the sixth harmonic]] into 70 equal parts''' (70ED6) 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{{c}} [[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 27 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.
+{{ED intro}}
-== Harmonics ==
+== Theory ==
-{{Harmonics in equal|57|6|1|prec=2|columns=15}}
+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.
-[[Category:Edonoi]]
+=== 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}}
+== See also ==
+* [[22edo]] – relative edo
+* [[35edt]] – relative edt
+* [[79ed12]] – relative ed12