59edo: Difference between revisions

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== Theory ==

59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127 ~~cents~~), as the denominator of a convergent to log<sub>2</sub>5. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log<sub>2</sub>5. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the 12 & 35 temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50 & 59 temperament with a subminor third generator provides an interesting temperament.

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== Notation ==

===Sagittal notation===

=== Sagittal notation ===

====Best fifth notation====

==== Best fifth notation ====

This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

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

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

File:59-EDO_Evo_Sagittal.svg

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</imagemap>

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

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

File:59-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 uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

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

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

File:59b_Evo_Sagittal.svg

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</imagemap>

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

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

File:59b_Revo_Sagittal.svg

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</imagemap>

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

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

File:59b_Evo-SZ_Sagittal.svg

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</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a ~~Stein-Zimmerman~~ notation.

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Instruments ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|59}}
+{{ED intro}}
 == Theory ==
-edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127 cents), as the denominator of a convergent to log<sub>2</sub>5. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].
+edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log<sub>2</sub>5. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].
-Using the flat fifth instead of the sharp one allows for the 12 &amp; 35 temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.
+Using the flat fifth instead of the sharp one allows for the {{nowrap|12 &amp; 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.
 As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50 &amp; 59 temperament with a subminor third generator provides an interesting temperament.
@@ Line 27: / Line 27: @@
 == Notation ==
-===Sagittal notation===
+=== Sagittal notation ===
-====Best fifth notation====
+==== Best fifth notation ====
 This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].
-=====Evo flavor=====
+===== Evo flavor =====
 <imagemap>
 File:59-EDO_Evo_Sagittal.svg
@@ Line 44: / Line 44: @@
 </imagemap>
-=====Revo flavor=====
+===== Revo flavor =====
 <imagemap>
 File:59-EDO_Revo_Sagittal.svg
@@ Line 60: / Line 59: @@
 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 uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].
-=====Evo flavor=====
+===== Evo flavor =====
 <imagemap>
 File:59b_Evo_Sagittal.svg
@@ Line 73: / Line 72: @@
 </imagemap>
-=====Revo flavor=====
+===== Revo flavor =====
 <imagemap>
 File:59b_Revo_Sagittal.svg
@@ Line 84: / Line 82: @@
 </imagemap>
-=====Evo-SZ flavor=====
+===== Evo-SZ flavor =====
 <imagemap>
 File:59b_Evo-SZ_Sagittal.svg
@@ Line 95: / Line 92: @@
 </imagemap>
-Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
+Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
 == Instruments ==