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


{{Infobox ET
== Theory ==
| Prime factorization = 59 (prime)
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]].
| Step size = 20.339¢
 
| Sharp fifth = 35\59 (711.9¢)
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.
| Flat fifth = 34\59 (691.5¢)
| Major 2nd = 4\23 (203.4¢)
| Consistency = 5
}}


== Theory ==
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.
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.


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.
=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}


59edo is the 17th [[prime_numbers|prime]] edo.
=== 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
! Steps
! rowspan="2"| Cents
! Cents
! colspan="2"| Approximate Ratios
! Approximate ratios<br>(2.9.5.21.11.39.17-subgroup)
|-
! Ratios of 3, 7, 13<br>(tending sharp)
! 2.9.5.21.11.17 Subgroup
! Ratios of 3, 7, 13<br>(tending flat)
! Full 11-limit in Patent Val
|-
|-
| 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"