64edo: Difference between revisions

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== Theory ==

64edo is a [[~~The_Riemann_zeta_function_and_tuning~~#Valley_edos|zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.

64edo is a [[The Riemann zeta function and tuning #Valley_edos|zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.

Still, the patent val [[tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament.

Still, the patent val [[tempering out|tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament.

64be val is a tuning for the [[beatles]] temperament and for the [[rank-3]] temperaments [[heimlaug]] and [[vili]] in the 17-limit. 64bccc tunes [[dichotic]], although that is an [[exotemperament]]. 64cdf is a tuning for [[vibhu]].

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=== Octave stretch ===

~~64edo’s~~ approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as ~~180ed7’s~~.

[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.

If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

[[~~ed257~~/128#64ed257/128|64ed257/128]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 ~~isn’t~~ as accurate as ~~221ed11’s~~.

[[Ed257/128#64ed257/128|64ed257/128]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.

There are also some nearby [[~~Zeta~~ peak index]] (ZPI) tunings which can be used to improve ~~64edo’s~~ JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.

There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used to improve 64edo's JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.

=== Subsets and supersets ===

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 == Theory ==
-edo is a [[The_Riemann_zeta_function_and_tuning#Valley_edos|zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.
+edo is a [[The Riemann zeta function and tuning #Valley_edos|zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure unnamed 7c & 12c (or 19 & 64) temperament which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.
-Still, the patent val [[tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&amp;64 temperament.
+Still, the patent val [[tempering out|tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&amp;64 temperament.
 be val is a tuning for the [[beatles]] temperament and for the [[rank-3]] temperaments [[heimlaug]] and [[vili]] in the 17-limit. 64bccc tunes [[dichotic]], although that is an [[exotemperament]]. 64cdf is a tuning for [[vibhu]].
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 === Octave stretch ===
-edo’s approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
+edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
-[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7’s.
+[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
 If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
-[[ed257/128#64ed257/128|64ed257/128]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 221ed11’s.
+[[Ed257/128#64ed257/128|64ed257/128]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.
-There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 64edo’s JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.
+There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used to improve 64edo's JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.
 === Subsets and supersets ===