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The '''59 equal division''' divides the octave into 59 equal steps of 20.339 cents each.
{{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]]. 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&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.
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}}
 
=== 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"
{| class="wikitable center-1 right-2"
|-
! rowspan="2"| Degrees
! rowspan="2"| Cents
! colspan="2"| Approximate Ratios
|-
|-
! 2.9.5.21.11.17 Subgroup
! Steps
! Full 11-limit in Patent Val
! 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)
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| [[1/1]]
|
|
|-
|-
| 1
| 1
| 20.339
| 20.3
| [[81/80]]
| [[81/80]]
| [[50/49]], [[99/98]]
|
|
|-
|-
| 2
| 2
| 40.678
| 40.7
| [[128/125]]
| [[40/39]], [[45/44]]
| [[49/48]]
|
|
|-
|-
| 3
| 3
| 61.017
| 61.0
| [[648/625]]
| [[27/26]], [[28/27]]
| [[25/24]], [[81/80]], [[36/35]], [[33/32]]
|
|
|-
|-
| 4
| 4
| 81.356
| 81.4
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
|
|
|
|-
|-
| 5
| 5
| 101.695
| 101.7
| [[17/16]], [[18/17]]
| [[17/16]], [[18/17]], [[35/33]]
| [[16/15]]
|
|
|-
|-
| 6
| 6
| 122.034
| 122.0
| [[15/14]], [[14/13]]
|
|
|
| [[15/14]]
|-
|-
| 7
| 7
| 142.373
| 142.4
| [[13/12]]
|
|
|
|
|-
|-
| 8
| 8
| 162.712
| 162.7
| [[11/10]]
| [[11/10]]
| [[10/9]], [[11/10]], [[12/11]]
|
|
|-
|-
| 9
| 9
| 183.051
| 183.1
| [[10/9]]
| [[10/9]]
|
|
|
|-
|-
| 10
| 10
| 203.390
| 203.4
| [[9/8]]
| [[9/8]], [[44/39]]
|
|
|
|-
|-
| 11
| 11
| 223.729
| 223.7
|
| [[25/22]]
| [[9/8]], [[8/7]]
| [[8/7]]
|
|-
|-
| 12
| 12
| 244.068
| 244.1
|
| [[15/13]], [[39/34]]
|
|
| [[8/7]]
|-
|-
| 13
| 13
| 264.407
| 264.4
| [[7/6]], [[64/55]]
|
|
|
| [[7/6]]
|-
|-
| 14
| 14
| 284.746
| 284.7
| [[20/17]]
| [[20/17]], [[33/28]]
|
|
|
|-
|-
| 15
| 15
| 305.085
| 305.1
| [[25/21]]
|
|
|
|
|-
|-
| 16
| 16
| 325.424
| 325.4
|
|
|
|
| [[6/5]], [[11/9]]
|-
|-
| 17
| 17
| 345.763
| 345.8
| [[11/9]]
| [[11/9]], [[39/32]], [[128/105]]
| [[16/13]]
|
|
|-
|-
| 18
| 18
| 366.102
| 366.1
| [[21/17]]
| [[21/17]]
|
|
| [[16/13]]
|-
|-
| 19
| 19
| 386.441
| 386.4
| [[5/4]]
| [[5/4]]
| [[5/4]]
|
|
|-
|-
| 20
| 20
| 406.780
| 406.8
| [[81/64]]
| [[81/64]]
|
|
|
|-
|-
| 21
| 21
| 427.119
| 427.1
| [[32/25]]
| [[32/25]], [[50/39]]
| [[32/25]], [[14/11]]
|
|
|-
|-
| 22
| 22
| 447.458
| 447.5
| [[22/17]]
| [[22/17]], [[35/27]], [[128/99]]
| [[9/7]]
|
|
|-
|-
| 23
| 23
| 467.797
| 467.8
| [[21/16]]
| [[21/16]], [[64/49]]
|
|
|
|-
|-
| 24
| 24
| 488.136
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|
| [[4/3]], [[21/16]]
|-
|-
| 25
| 25
| 508.475
| 508.5
|
| [[35/26]]
|
|
| [[4/3]]
|-
|-
| 26
| 26
| 528.814
| 528.8
| [[34/25]]
|
|
|
|
|-
|-
| 27
| 27
| 549.153
| 549.2
| [[11/8]]
| [[11/8]], [[48/35]]
| [[27/20]], [[11/8]], [[15/11]]
|
|
|-
|-
| 28
| 28
| 569.492
| 569.5
| [[25/18]]
| [[25/18]]
|
|
|
|-
|-
| 29
| 29
| 589.831
| 589.8
| [[45/32]]
| [[45/32]], [[128/91]]
| [[7/5]]
|
|
|-
|-
| 30
| 30
| 610.169
| 610.2
| [[64/45]]
| [[64/45]], [[91/64]]
| [[10/7]]
|
|
|-
|-
| 31
| 31
| 630.508
| 630.5
| [[36/25]]
| [[36/25]]
|
|
|
|-
|-
| 32
| 32
| 650.847
| 650.8
| [[16/11]]
| [[16/11]], [[35/24]]
| [[40/27]], [[16/11]], [[22/15]]
|
|
|-
|-
| 33
| 33
| 671.186
| 671.2
| [[25/17]]
|
|
|
|
|-
|-
| 34
| 34
| 691.525
| 691.5
|
| [[52/35]]
|
|
| [[3/2]]
|-
|-
| 35
| 35
| 711.864
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|
| [[3/2]], [[32/21]]
|-
|-
| 36
| 36
| 732.203
| 732.2
| [[32/21]]
| [[32/21]], [[49/32]]
|
|
|
|-
|-
| 37
| 37
| 752.542
| 752.5
| [[17/11]]
| [[17/11]], [[54/35]], [[99/64]]
| [[14/9]]
|
|
|-
|-
| 38
| 38
| 772.881
| 772.9
| [[25/16]]
| [[25/16]], [[39/25]]
| [[25/16]], [[11/7]]
|
|
|-
|-
| 39
| 39
| 793.220
| 793.2
| [[128/81]]
| [[128/81]]
|
|
|
|-
|-
| 40
| 40
| 813.559
| 813.6
| [[8/5]]
| [[8/5]]
| [[8/5]]
|
|
|-
|-
| 41
| 41
| 833.898
| 833.9
| [[34/21]]
| [[34/21]]
|
|
| [[13/8]]
|-
|-
| 42
| 42
| 854.237
| 854.2
| [[18/11]]
| [[18/11]], [[64/39]], [[105/64]]
| [[13/8]]
|
|
|-
|-
| 43
| 43
| 874.576
| 874.6
|
|
|
|
| [[5/3]], [[18/11]]
|-
|-
| 44
| 44
| 894.915
| 894.9
| [[42/25]]
|
|
|
|
|-
|-
| 45
| 45
| 915.254
| 915.3
| [[17/10]]
| [[17/10]], [[56/33]]
|
|
|
|-
|-
| 46
| 46
| 935.593
| 935.6
| [[12/7]], [[55/32]]
|
|
|
| [[12/7]]
|-
|-
| 47
| 47
| 955.932
| 955.9
|
| [[26/15]], [[68/39]]
|
|
| [[7/4]]
|-
|-
| 48
| 48
| 976.271
| 976.3
|
| [[44/25]]
| [[16/9]], [[7/4]]
| [[7/4]]
|
|-
|-
| 49
| 49
| 996.610
| 996.6
| [[16/9]]
| [[16/9]], [[39/22]]
|
|
|
|-
|-
| 50
| 50
| 1016.949
| 1016.9
| [[9/5]]
| [[9/5]]
|
|
|
|-
|-
| 51
| 51
| 1037.288
| 1037.3
| [[20/11]]
| [[20/11]]
| [[9/5]], [[11/6]], [[20/11]]
|
|
|-
|-
| 52
| 52
| 1057.627
| 1057.6
| [[24/13]]
|
|
|
|
|-
|-
| 53
| 53
| 1077.966
| 1078.0
| [[13/7]], [[28/15]]
|
|
|
| [[28/15]]
|-
|-
| 54
| 54
| 1098.305
| 1098.3
| [[17/9]], [[32/17]]
| [[17/9]], [[32/17]], [[66/35]]
| [[15/8]]
|
|
|-
|-
| 55
| 55
| 1118.644
| 1118.6
| [[40/21]], [[21/11]]
| [[21/11]], [[40/21]]
|
|
|
|-
|-
| 56
| 56
| 1138.983
| 1139.0
| [[625/324]]
| [[27/14]], [[52/27]]
| [[48/25]], [[160/81]], [[35/18]], [[64/33]]
|
|
|-
|-
| 57
| 57
| 1159.322
| 1159.3
| [[125/64]]
| [[39/20]], [[88/45]]  
| [[96/49]]
|
|
|-
|-
| 58
| 58
| 1179.661
| 1179.7
| [[160/81]]
| [[160/81]]
| [[49/25]], [[196/99]]
|
|
|-
|-
| 59
| 59
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| [[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: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"