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The '''59 equal division''' divides the octave into 59 equal steps of 20.339 cents each.
{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its major third 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_family|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]]. 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.
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&amp;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. The flat fifth also acts as a generator for [[flattone]] temperament in the 59bc val, a variant of meantone with 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.


59edo is the 17th [[prime_numbers|prime]] edo.
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}}
{{Harmonics in equal|59|columns=13}}
=== Subsets and supersets ===
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 ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4"
{{Interval table}}{{Todo|text=ADD 3|inline=1}}
|-
 
! rowspan="2"| Degrees
== Notation ==
! rowspan="2"| Cents
 
! colspan="2"| Approximate Ratios
=== Sagittal notation ===
|-
==== Best fifth notation ====
! 2.9.5.21.11.17 Subgroup
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].
! Full 11-limit in Patent Val
 
|-
===== Evo flavor =====
| 0
<imagemap>
| 0.000
File:59-EDO_Evo_Sagittal.svg
| [[1/1]]
desc none
| [[1/1]]
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 1
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
| 20.339
rect 190 80 320 106 [[144/143]]
| [[81/80]]
rect 320 80 430 106 [[81/80]]
| [[50/49]], [[99/98]]
rect 430 80 570 106 [[1053/1024]]
|-
default [[File:59-EDO_Evo_Sagittal.svg]]
| 2
</imagemap>
| 40.678
 
| [[128/125]]
===== Revo flavor =====
| [[49/48]]
<imagemap>
|-
File:59-EDO_Revo_Sagittal.svg
| 3
desc none
| 61.017
rect 80 0 300 50 [[Sagittal_notation]]
| [[648/625]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| [[25/24]], [[81/80]], [[36/35]], [[33/32]]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
|-
rect 190 80 320 106 [[144/143]]
| 4
rect 320 80 430 106 [[81/80]]
| 81.356
rect 430 80 570 106 [[1053/1024]]
| [[21/20]], [[22/21]]
default [[File:59-EDO_Revo_Sagittal.svg]]
|
</imagemap>
|-
 
| 5
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.
| 101.695
 
| [[17/16]], [[18/17]]
==== Second-best fifth notation ====
| [[16/15]]
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].
|-
 
| 6
===== Evo flavor =====
| 122.034
<imagemap>
|
File:59b_Evo_Sagittal.svg
| [[15/14]]
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 7
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 142.373
rect 20 80 130 106 [[36/35]]
|
default [[File:59b_Evo_Sagittal.svg]]
|
</imagemap>
|-
 
| 8
===== Revo flavor =====
| 162.712
<imagemap>
| [[11/10]]
File:59b_Revo_Sagittal.svg
| [[10/9]], [[11/10]], [[12/11]]
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 9
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 183.051
rect 20 80 130 106 [[36/35]]
| [[10/9]]
default [[File:59b_Revo_Sagittal.svg]]
|
</imagemap>
|-
 
| 10
===== Evo-SZ flavor =====
| 203.390
<imagemap>
| [[9/8]]
File:59b_Evo-SZ_Sagittal.svg
|
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 11
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 223.729
rect 20 80 130 106 [[36/35]]
|
default [[File:59b_Evo-SZ_Sagittal.svg]]
| [[9/8]], [[8/7]]
</imagemap>
|-
 
| 12
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
| 244.068
 
|
== 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.
|-
 
| 13
[[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.
| 264.407
 
|
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.
| [[7/6]]
 
|-
== Scales ==
| 14
; [[Porcupine]] scales
| 284.746
* Porcupine[7]: 8 8 8 11 8 8 8
| [[20/17]]
* 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'')
| 15
 
| 305.085
== Instruments ==
|
; Lumatone
|
 
|-
See [[Lumatone mapping for 59edo]].
| 16
 
| 325.424
== Music ==
|
; [[Bryan Deister]]
| [[6/5]], [[11/9]]
* [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)
| 17
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)
| 345.763
 
| [[11/9]]
; [[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]
| 18
| 366.102
| [[21/17]]
|
|-
| 19
| 386.441
| [[5/4]]
| [[5/4]]
|-
| 20
| 406.780
| [[81/64]]
|
|-
| 21
| 427.119
| [[32/25]]
| [[32/25]], [[14/11]]
|-
| 22
| 447.458
| [[22/17]]
| [[9/7]]
|-
| 23
| 467.797
| [[21/16]]
|
|-
| 24
| 488.136
|
| [[4/3]], [[21/16]]
|-
| 25
| 508.475
|
|
|-
| 26
| 528.814
|
|
|-
| 27
| 549.153
| [[11/8]]
| [[27/20]], [[11/8]], [[15/11]]
|-
| 28
| 569.492
| [[25/18]]
|
|-
| 29
| 589.831
| [[45/32]]
| [[7/5]]
|-
| 30
| 610.169
| [[64/45]]
| [[10/7]]
|-
| 31
| 630.508
| [[36/25]]
|
|-
| 32
| 650.847
| [[16/11]]
| [[40/27]], [[16/11]], [[22/15]]
|-
| 33
| 671.186
|
|
|-
| 34
| 691.525
|
|
|-
| 35
| 711.864
|
| [[3/2]], [[32/21]]
|-
| 36
| 732.203
| [[32/21]]
|
|-
| 37
| 752.542
| [[17/11]]
| [[14/9]]
|-
| 38
| 772.881
| [[25/16]]
| [[25/16]], [[11/7]]
|-
| 39
| 793.220
| [[128/81]]
|
|-
| 40
| 813.559
| [[8/5]]
| [[8/5]]
|-
| 41
| 833.898
| [[34/21]]
|
|-
| 42
| 854.237
| [[18/11]]
|
|-
| 43
| 874.576
|
| [[5/3]], [[18/11]]
|-
| 44
| 894.915
|
|
|-
| 45
| 915.254
| [[17/10]]
|
|-
| 46
| 935.593
|
| [[12/7]]
|-
| 47
| 955.932
|
|
|-
| 48
| 976.271
|
| [[16/9]], [[7/4]]
|-
| 49
| 996.610
| [[16/9]]
|
|-
| 50
| 1016.949
| [[9/5]]
|
|-
| 51
| 1037.288
| [[20/11]]
| [[9/5]], [[11/6]], [[20/11]]
|-
| 52
| 1057.627
|
|
|-
| 53
| 1077.966
|
| [[28/15]]
|-
| 54
| 1098.305
| [[17/9]], [[32/17]]
| [[15/8]]
|-
| 55
| 1118.644
| [[40/21]], [[21/11]]
|
|-
| 56
| 1138.983
| [[625/324]]
| [[48/25]], [[160/81]], [[35/18]], [[64/33]]
|-
| 57
| 1159.322
| [[125/64]]
| [[96/49]]
|-
| 58
| 1179.661
| [[160/81]]
| [[49/25]], [[196/99]]
|-
| 59
| 1200.000
| [[2/1]]
| [[2/1]]
|}


==Music==
; [[Budjarn Lambeth]]
; [[User:Francium|Francium]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)
* [https://www.youtube.com/watch?v=FyzN0P6icf0 too powerful if i had social skills]


; [[Ray Perlner]]
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI Chinchillian Fugue (First mode of Porcupine 7 scale, 59EDO)]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Prime EDO]]
[[Category:Porcupine]]
[[Category:Porcupine]]
[[Category:Subgroup]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/59edo"