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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.
 
=== Odd harmonics ===
{{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 ==
{| class="wikitable center-1 right-2"
|-
|-
| | Degrees
! Steps
| | Cents Value
! Cents
! Approximate ratios<br>(2.9.5.21.11.39.17-subgroup)
! Ratios of 3, 7, 13<br>(tending sharp)
! Ratios of 3, 7, 13<br>(tending flat)
|-
|-
| | 1
| 0
| | 20.339
| 0.0
| [[1/1]]
|
|
|-
|-
| | 2
| 1
| | 40.678
| 20.3
| [[81/80]]
|
|
|-
|-
| | 3
| 2
| | 61.017
| 40.7
| [[40/39]], [[45/44]]
|
|
|-
|-
| | 4
| 3
| | 81.356
| 61.0
| [[27/26]], [[28/27]]
|
|
|-
|-
| | 5
| 4
| | 101.695
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
|-
| | 6
| 5
| | 122.034
| 101.7
| [[17/16]], [[18/17]], [[35/33]]
|
|
|-
|-
| | 7
| 6
| | 142.373
| 122.0
| [[15/14]], [[14/13]]
|
|
|-
|-
| | 8
| 7
| | 162.712
| 142.4
| [[13/12]]
|
|
|-
|-
| | 9
| 8
| | 183.051
| 162.7
| [[11/10]]
|
|
|-
|-
| | 10
| 9
| | 203.390
| 183.1
| [[10/9]]
|
|
|-
|-
| | 11
| 10
| | 223.729
| 203.4
| [[9/8]], [[44/39]]
|
|
|-
|-
| | 12
| 11
| | 244.068
| 223.7
| [[25/22]]
| [[8/7]]
|  
|-
|-
| | 13
| 12
| | 264.407
| 244.1
| [[15/13]], [[39/34]]
|
| [[8/7]]
|-
|-
| | 14
| 13
| | 284.746
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
|-
| | 15
| 14
| | 305.085
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
|-
| | 16
| 15
| | 325.424
| 305.1
| [[25/21]]
|
|
|-
|-
| | 17
| 16
| | 345.763
| 325.4
|  
|
|
|-
|-
| | 18
| 17
| | 366.102
| 345.8
| [[11/9]], [[39/32]], [[128/105]]
| [[16/13]]
|
|-
|-
| | 19
| 18
| | 386.441
| 366.1
| [[21/17]]
|
| [[16/13]]
|-
|-
| | 20
| 19
| | 406.780
| 386.4
| [[5/4]]
|
|
|-
|-
| | 21
| 20
| | 427.119
| 406.8
| [[81/64]]
|
|
|-
|-
| | 22
| 21
| | 447.458
| 427.1
| [[32/25]], [[50/39]]
|
|
|-
|-
| | 23
| 22
| | 467.797
| 447.5
| [[22/17]], [[35/27]], [[128/99]]
|
|
|-
|-
| | 24
| 23
| | 488.136
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
|-
| | 25
| 24
| | 508.475
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
|-
| | 26
| 25
| | 528.814
| 508.5
| [[35/26]]
|
| [[4/3]]
|-
|-
| | 27
| 26
| | 549.153
| 528.8
| [[34/25]]
|
|
|-
|-
| | 28
| 27
| | 569.492
| 549.2
| [[11/8]], [[48/35]]
|
|
|-
|-
| | 29
| 28
| | 589.831
| 569.5
| [[25/18]]
|
|
|-
|-
| | 30
| 29
| | 610.169
| 589.8
| [[45/32]], [[128/91]]
|
|
|-
|-
| | 31
| 30
| | 630.508
| 610.2
| [[64/45]], [[91/64]]
|
|
|-
|-
| | 32
| 31
| | 650.847
| 630.5
| [[36/25]]
|
|
|-
|-
| | 33
| 32
| | 671.186
| 650.8
| [[16/11]], [[35/24]]
|
|
|-
|-
| | 34
| 33
| | 691.525
| 671.2
| [[25/17]]
|
|
|-
|-
| | 35
| 34
| | 711.864
| 691.5
| [[52/35]]
|
| [[3/2]]
|-
|-
| | 36
| 35
| | 732.203
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
|-
| | 37
| 36
| | 752.542
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
|-
| | 38
| 37
| | 772.881
| 752.5
| [[17/11]], [[54/35]], [[99/64]]
|
|
|-
|-
| | 39
| 38
| | 793.220
| 772.9
| [[25/16]], [[39/25]]
|
|
|-
|-
| | 40
| 39
| | 813.559
| 793.2
| [[128/81]]
|
|
|-
|-
| | 41
| 40
| | 833.898
| 813.6
| [[8/5]]
|
|
|-
|-
| | 42
| 41
| | 854.237
| 833.9
| [[34/21]]
|
| [[13/8]]
|-
|-
| | 43
| 42
| | 874.576
| 854.2
| [[18/11]], [[64/39]], [[105/64]]
| [[13/8]]
|
|-
|-
| | 44
| 43
| | 894.915
| 874.6
|  
|
|
|-
|-
| | 45
| 44
| | 915.254
| 894.9
| [[42/25]]
|
|
|-
|-
| | 46
| 45
| | 935.593
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
|-
| | 47
| 46
| | 955.932
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
|-
| | 48
| 47
| | 976.271
| 955.9
| [[26/15]], [[68/39]]
|
| [[7/4]]
|-
|-
| | 49
| 48
| | 996.610
| 976.3
| [[44/25]]
| [[7/4]]
|  
|-
|-
| | 50
| 49
| | 1016.949
| 996.6
| [[16/9]], [[39/22]]
|
|
|-
|-
| | 51
| 50
| | 1037.288
| 1016.9
| [[9/5]]
|
|
|-
|-
| | 52
| 51
| | 1057.627
| 1037.3
| [[20/11]]
|
|
|-
|-
| | 53
| 52
| | 1077.966
| 1057.6
| [[24/13]]
|
|
|-
|-
| | 54
| 53
| | 1098.305
| 1078.0
| [[13/7]], [[28/15]]
|
|
|-
|-
| | 55
| 54
| | 1118.644
| 1098.3
| [[17/9]], [[32/17]], [[66/35]]
|
|
|-
|-
| | 56
| 55
| | 1138.983
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
|-
| | 57
| 56
| | 1159.322
| 1139.0
| [[27/14]], [[52/27]]
|
|
|-
|-
| | 58
| 57
| | 1179.661
| 1159.3
|}
| [[39/20]], [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}
 
== 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>
 
===== 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>
 
===== Evo-SZ flavor =====
<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 ==
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.
 
== Scales ==
; [[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]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo


[[Category:Edo]]
[[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"