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The ''59 equal division'' divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is very (9.9 cents) sharp, and yet its [[major_third|major third]] is nearly pure. It is a good [[Porcupine_family|porcupine]] tuning, giving in fact the [[Optimal_patent_val|optimal patent val]] for [[11-limit|11-limit]] porcupine. This patent val tempers out 250/243 in the [[5-limit|5-limit]], 64/63 and 16875/16807 in the [[7-limit|7-limit]], and 55/54, 100/99 and 176/175 in the [[11-limit|11-limit]]. 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 as 118et. 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.
{{Infobox ET}}
{{ED intro}}


Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to [[Schismatic_family|garibaldi temperament]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth.
== 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{{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]].


59edo is the 17th [[prime_numbers|prime]] edo.
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.


{| class="wikitable"
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.
|-
 
| | Degrees
=== Odd harmonics ===
| | Cents Value
{{Harmonics in equal|59|columns=13}}
|-
 
| | 1
=== Subsets and supersets ===
| | 20.339
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.
|-
 
| | 2
== Intervals ==
| | 40.678
{{Interval table}}{{Todo|text=ADD 3|inline=1}}
|-
 
| | 3
== Notation ==
| | 61.017
 
|-
=== Sagittal notation ===
| | 4
==== Best fifth notation ====
| | 81.356
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].
|-
 
| | 5
===== Evo flavor =====
| | 101.695
<imagemap>
|-
File:59-EDO_Evo_Sagittal.svg
| | 6
desc none
| | 122.034
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 7
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
| | 142.373
rect 190 80 320 106 [[144/143]]
|-
rect 320 80 430 106 [[81/80]]
| | 8
rect 430 80 570 106 [[1053/1024]]
| | 162.712
default [[File:59-EDO_Evo_Sagittal.svg]]
|-
</imagemap>
| | 9
 
| | 183.051
===== Revo flavor =====
|-
<imagemap>
| | 10
File:59-EDO_Revo_Sagittal.svg
| | 203.390
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| | 11
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 223.729
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
|-
rect 190 80 320 106 [[144/143]]
| | 12
rect 320 80 430 106 [[81/80]]
| | 244.068
rect 430 80 570 106 [[1053/1024]]
|-
default [[File:59-EDO_Revo_Sagittal.svg]]
| | 13
</imagemap>
| | 264.407
 
|-
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.
| | 14
 
| | 284.746
==== Second-best fifth notation ====
|-
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].
| | 15
 
| | 305.085
===== Evo flavor =====
|-
<imagemap>
| | 16
File:59b_Evo_Sagittal.svg
| | 325.424
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| | 17
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 345.763
rect 20 80 130 106 [[36/35]]
|-
default [[File:59b_Evo_Sagittal.svg]]
| | 18
</imagemap>
| | 366.102
 
|-
===== Revo flavor =====
| | 19
<imagemap>
| | 386.441
File:59b_Revo_Sagittal.svg
|-
desc none
| | 20
rect 80 0 300 50 [[Sagittal_notation]]
| | 406.780
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 130 106 [[36/35]]
| | 21
default [[File:59b_Revo_Sagittal.svg]]
| | 427.119
</imagemap>
|-
 
| | 22
===== Evo-SZ flavor =====
| | 447.458
<imagemap>
|-
File:59b_Evo-SZ_Sagittal.svg
| | 23
desc none
| | 467.797
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 24
rect 20 80 130 106 [[36/35]]
| | 488.136
default [[File:59b_Evo-SZ_Sagittal.svg]]
|-
</imagemap>
| | 25
 
| | 508.475
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
|-
 
| | 26
== Octave stretch or compression ==
| | 528.814
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.
|-
 
| | 27
[[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.
| | 549.153
 
|-
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.
| | 28
 
| | 569.492
== Scales ==
|-
; [[Porcupine]] scales
| | 29
* Porcupine[7]: 8 8 8 11 8 8 8
| | 589.831
* 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
| | 30
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')
| | 610.169
 
|-
== Instruments ==
| | 31
; Lumatone
| | 630.508
 
|-
See [[Lumatone mapping for 59edo]].
| | 32
 
| | 650.847
== Music ==
|-
; [[Bryan Deister]]
| | 33
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
| | 671.186
* [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)
| | 34
 
| | 691.525
; [[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]
| | 35
* "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]
| | 711.864
 
|-
; [[Budjarn Lambeth]]
| | 36
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)
| | 732.203
 
|-
; [[Ray Perlner]]
| | 37
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo
| | 752.542
 
|-
[[Category:Porcupine]]
| | 38
[[Category:Listen]]
| | 772.881
[[Category:Todo:add rank 2 temperaments table]]
|-
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| | 793.220
|-
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| | 813.559
|-
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| | 833.898
|-
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| | 854.237
|-
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| | 874.576
|-
| | 44
| | 894.915
|-
| | 45
| | 915.254
|-
| | 46
| | 935.593
|-
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| | 955.932
|-
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| | 976.271
|-
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| | 996.610
|-
| | 50
| | 1016.949
|-
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| | 1037.288
|-
| | 52
| | 1057.627
|-
| | 53
| | 1077.966
|-
| | 54
| | 1098.305
|-
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| | 1118.644
|-
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| | 1138.983
|-
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| | 1159.322
|-
| | 58
| | 1179.661
|}
[[Category:edo]]
[[Category:porcupine]]
[[Category:prime_edo]]
[[Category:subgroup]]
Retrieved from "https://en.xen.wiki/w/59edo"