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 {{nowrap|7c & 12c}} (or {{nowrap|19 & 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) 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 [[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 {{nowrap|7c & 12c}} (or {{nowrap|19 & 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) 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 ==

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould notation ===

64edo can be notated with [[Stein–Zimmermann–Gould notation]]:

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.

=== Kite's ups and downs 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]]:~~

~~{{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]].

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 =====

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</imagemap>

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~~.

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 ====

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

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

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

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* [https://www.youtube.com/shorts/ag1FbgAuR6I ''microtonal improvisation in 64edo''] (2025)

* [https://www.youtube.com/shorts/cwEY-eXjMFw ''Servant of Evil - Mothy (microtonal cover in 64edo)''] (2026)

* [https://www.youtube.com/shorts/X9JjH40xCnU ''Waltz in 64edo''] (2026)

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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 {{nowrap|7c &amp; 12c}} (or {{nowrap|19 &amp; 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) 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 [[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 {{nowrap|7c &amp; 12c}} (or {{nowrap|19 &amp; 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) 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.
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 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
+edo can be notated with [[Stein–Zimmermann–Gould notation]]:
+{{Sharpness-sharp3-szg}}
+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.
+=== Kite's ups and downs 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]].
+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 =====
@@ Line 53: / Line 55: @@
 </imagemap>
-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.
+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 ====
-This notation is a superset of the notation for [[32edo#Sagittal notation|32-EDO]].
+This notation is a superset of the notation for [[32edo #Sagittal notation|32edo]].
 ===== Evo flavor =====
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 * [https://www.youtube.com/shorts/ag1FbgAuR6I ''microtonal improvisation in 64edo''] (2025)
 * [https://www.youtube.com/shorts/cwEY-eXjMFw ''Servant of Evil - Mothy (microtonal cover in 64edo)''] (2026)
+* [https://www.youtube.com/shorts/X9JjH40xCnU ''Waltz in 64edo''] (2026)