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 {{nowrap|7c & 12c}} (or {{nowrap|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 [[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.

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== Notation ==

===Sagittal notation===

=== Ups and downs notation ===

====Best fifth notation====

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

Using [[Helmholtz–Ellis]] accidentals, 64edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

=== Sagittal notation ===

==== Best fifth notation ====

This notation uses the same sagittal sequence as EDOs [[50edo#Sagittal notation|50]], [[57edo#Sagittal notation|57]], and [[71edo#Second-best fifth notation|71b]].

~~=====Evo flavor=====~~

===== Evo flavor =====

File:64-EDO_Evo_Sagittal.svg

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</imagemap>

=====Revo flavor=====

===== Revo flavor =====

File:64-EDO_Revo_Sagittal.svg

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In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

====Second-best fifth notation====

==== Second-best fifth notation ====

This notation is a superset of the notation for [[32edo#Sagittal notation|32-EDO]].

~~=====Evo flavor=====~~

===== Evo flavor =====

File:64b_Evo_Sagittal.svg

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</imagemap>

=====Revo flavor=====

===== Revo flavor =====

File:64b_Revo_Sagittal.svg

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</imagemap>

=====Evo-SZ flavor=====

===== Evo-SZ flavor =====

File:64b_Evo-SZ_Sagittal.svg

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|64}}
+{{ED intro}}
 == 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 {{nowrap|7c &amp; 12c}} (or {{nowrap|19 &amp; 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 [[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.
@@ Line 30: / Line 30: @@
 == Notation ==
-===Sagittal notation===
+=== Ups and downs notation ===
-====Best fifth notation====
+Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
+{{sharpness-sharp3a}}
+Using [[Helmholtz–Ellis]] accidentals, 64edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:
+{{Sharpness-sharp3}}
+Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
+=== Sagittal notation ===
+==== Best fifth notation ====
 This notation uses the same sagittal sequence as EDOs [[50edo#Sagittal notation|50]], [[57edo#Sagittal notation|57]], and [[71edo#Second-best fifth notation|71b]].
-=====Evo flavor=====
+===== Evo flavor =====
 <imagemap>
 File:64-EDO_Evo_Sagittal.svg
@@ Line 44: / Line 52: @@
 </imagemap>
-=====Revo flavor=====
+===== Revo flavor =====
 <imagemap>
 File:64-EDO_Revo_Sagittal.svg
@@ Line 57: / Line 64: @@
 In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
-====Second-best fifth notation====
+==== Second-best fifth notation ====
 This notation is a superset of the notation for [[32edo#Sagittal notation|32-EDO]].
-=====Evo flavor=====
+===== Evo flavor =====
 <imagemap>
 File:64b_Evo_Sagittal.svg
@@ Line 70: / Line 77: @@
 </imagemap>
-=====Revo flavor=====
+===== Revo flavor =====
 <imagemap>
 File:64b_Revo_Sagittal.svg
@@ Line 81: / Line 87: @@
 </imagemap>
-=====Evo-SZ flavor=====
+===== Evo-SZ flavor =====
 <imagemap>
 File:64b_Evo-SZ_Sagittal.svg