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.

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 [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===

=== ~~Octave stretch~~ ===

=== Subsets and supersets ===

~~59edo’s approximations~~ of 3/~~1, 7~~/~~1 and 11~~/~~1 are improved by~~ [[~~ed257~~/~~128~~#~~59ed257~~/~~128|59ed257~~/~~128~~]], a [[~~Octave stretch~~|~~stretched-octave~~]] version of ~~59edo~~. ~~The trade~~-~~off is a slightly worse 2~~/~~1 and 5~~/1.

59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==

== Notation ==

=== Sagittal notation ===

==== Best fifth notation ====

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

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

File:59-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

rect 190 80 320 106 [[144/143]]

rect 320 80 430 106 [[81/80]]

rect 430 80 570 106 [[1053/1024]]

default [[File:59-EDO_Evo_Sagittal.svg]]

</imagemap>

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

File:59-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

rect 190 80 320 106 [[144/143]]

rect 320 80 430 106 [[81/80]]

rect 430 80 570 106 [[1053/1024]]

default [[File:59-EDO_Revo_Sagittal.svg]]

</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.

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

This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

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

File:59b_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 130 106 [[36/35]]

default [[File:59b_Evo_Sagittal.svg]]

</imagemap>

~~There are also some nearby~~ [[~~Zeta peak index~~]] ~~(ZPI) tunings which can be used for this same purpose~~: ~~293zpi, 294zpi, 295zpi, 296zpi and 297zpi~~. ~~The main Zeta peak index page details all five tunings~~.

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

File:59b_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 130 106 [[36/35]]

default [[File:59b_Revo_Sagittal.svg]]

</imagemap>

=== ~~Subsets and supersets~~ ===

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

59edo is ~~the 17th~~ [[~~prime edo~~]], ~~following~~ [[~~53edo~~]] ~~and before~~ [[~~61edo~~]].

File:59b_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 130 106 [[36/35]]

default [[File:59b_Evo-SZ_Sagittal.svg]]

</imagemap>

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

== Octave stretch or compression ==

59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== ~~Intervals~~ ==

== Scales ==

~~{{Interval table}}~~

; [[Porcupine]] scales

* Porcupine[7]: 8 8 8 11 8 8 8

* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3

* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3

* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==

; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)

* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)

* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]

* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]

* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]

* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]

@@ 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.
+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 [[50edo|50]] &amp; 59 temperament with a subminor third generator provides an interesting temperament.
 === Odd harmonics ===
 {{Harmonics in equal|59|columns=13}}
-=== Octave stretch ===
+=== Subsets and supersets ===
-edo’s approximations of 3/1,  7/1 and 11/1 are improved by [[ed257/128#59ed257/128|59ed257/128]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.
+edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.
+== Intervals ==
+{{Interval table}}{{Todo|text=ADD 3|inline=1}}
+== Notation ==
+=== Sagittal notation ===
+==== Best fifth notation ====
+This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].
+===== Evo flavor =====
+<imagemap>
+File:59-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
+rect 190 80 320 106 [[144/143]]
+rect 320 80 430 106 [[81/80]]
+rect 430 80 570 106 [[1053/1024]]
+default [[File:59-EDO_Evo_Sagittal.svg]]
+</imagemap>
+===== Revo flavor =====
+<imagemap>
+File:59-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
+rect 190 80 320 106 [[144/143]]
+rect 320 80 430 106 [[81/80]]
+rect 430 80 570 106 [[1053/1024]]
+default [[File:59-EDO_Revo_Sagittal.svg]]
+</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.
+==== Second-best fifth notation ====
+This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].
+===== Evo flavor =====
+<imagemap>
+File:59b_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 130 106 [[36/35]]
+default [[File:59b_Evo_Sagittal.svg]]
+</imagemap>
-There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 293zpi, 294zpi, 295zpi, 296zpi and 297zpi. The main Zeta peak index page details all five tunings.
+===== Revo flavor =====
+<imagemap>
+File:59b_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 130 106 [[36/35]]
+default [[File:59b_Revo_Sagittal.svg]]
+</imagemap>
-=== Subsets and supersets ===
+===== Evo-SZ flavor =====
-edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]].
+<imagemap>
+File:59b_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 130 106 [[36/35]]
+default [[File:59b_Evo-SZ_Sagittal.svg]]
+</imagemap>
+Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
+== Octave stretch or compression ==
+edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.
+[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.
+If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.
-== Intervals ==
+== Scales ==
-{{Interval table}}
+; [[Porcupine]] scales
+* Porcupine[7]: 8 8 8 11 8 8 8
+* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
+* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
+* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')
 == Instruments ==
 ; Lumatone
 See [[Lumatone mapping for 59edo]].
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
+* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
+* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)
 ; [[Francium]]
 * "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
+* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]
+; [[Budjarn Lambeth]]
+* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)
 ; [[Ray Perlner]]