57ed6: Difference between revisions

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== 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}}. 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 ~~widely voiced harmonies~~. 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.

57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[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 [[Riemann zeta function #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 ===

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 == 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}}. 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 widely voiced harmonies. 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.
+ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[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 [[Riemann zeta function #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 ===