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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|59}}
{{ED intro}}


== Theory ==
== 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 cents), 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'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 [[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.
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.


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


=== Octave stretch ===
=== Subsets and supersets ===
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. The tunings [[203ed11]] or [[ed257/128#59ed257/128|59ed257/128]] could also be used as they are near identical to 93edt.
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.


If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[166ed7]] 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. It is a more accurate choice than 93edt in the [[7-limit]] or no-11s [[13-limit]] due to its better 2/1, but less accurate than 93edt in the [[11-limit]] due to its heavily damaged 11/1.
== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! 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
| [[1/1]]
|
|
|-
| 1
| 20.3
| [[81/80]]
|
|
|-
| 2
| 40.7
| [[40/39]], [[45/44]]
|
|
|-
| 3
| 61.0
| [[27/26]], [[28/27]]
|
|
|-
| 4
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
| 5
| 101.7
| [[17/16]], [[18/17]], [[35/33]]
|
|
|-
| 6
| 122.0
| [[15/14]], [[14/13]]
|
|
|-
| 7
| 142.4
| [[13/12]]
|
|
|-
| 8
| 162.7
| [[11/10]]
|
|
|-
| 9
| 183.1
| [[10/9]]
|
|
|-
| 10
| 203.4
| [[9/8]], [[44/39]]
|
|
|-
| 11
| 223.7
| [[25/22]]
| [[8/7]]
|
|-
| 12
| 244.1
| [[15/13]], [[39/34]]
|
| [[8/7]]
|-
| 13
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
| 14
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
| 15
| 305.1
| [[25/21]]
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
| [[11/9]], [[39/32]], [[128/105]]
| [[16/13]]
|
|-
| 18
| 366.1
| [[21/17]]
|
| [[16/13]]
|-
| 19
| 386.4
| [[5/4]]
|
|
|-
| 20
| 406.8
| [[81/64]]
|
|
|-
| 21
| 427.1
| [[32/25]], [[50/39]]
|
|
|-
| 22
| 447.5
| [[22/17]], [[35/27]], [[128/99]]
|
|
|-
| 23
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
| 24
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
| 25
| 508.5
| [[35/26]]
|
| [[4/3]]
|-
| 26
| 528.8
| [[34/25]]
|
|
|-
| 27
| 549.2
| [[11/8]], [[48/35]]
|
|
|-
| 28
| 569.5
| [[25/18]]
|
|
|-
| 29
| 589.8
| [[45/32]], [[128/91]]
|
|
|-
| 30
| 610.2
| [[64/45]], [[91/64]]
|
|
|-
| 31
| 630.5
| [[36/25]]
|
|
|-
| 32
| 650.8
| [[16/11]], [[35/24]]
|
|
|-
| 33
| 671.2
| [[25/17]]
|
|
|-
| 34
| 691.5
| [[52/35]]
|
| [[3/2]]
|-
| 35
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
| 36
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
| 37
| 752.5
| [[17/11]], [[54/35]], [[99/64]]
|
|
|-
| 38
| 772.9
| [[25/16]], [[39/25]]
|
|
|-
| 39
| 793.2
| [[128/81]]
|
|
|-
| 40
| 813.6
| [[8/5]]
|
|
|-
| 41
| 833.9
| [[34/21]]
|
| [[13/8]]
|-
| 42
| 854.2
| [[18/11]], [[64/39]], [[105/64]]
| [[13/8]]
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
| [[42/25]]
|
|
|-
| 45
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
| 46
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
| 47
| 955.9
| [[26/15]], [[68/39]]
|
| [[7/4]]
|-
| 48
| 976.3
| [[44/25]]
| [[7/4]]
|
|-
| 49
| 996.6
| [[16/9]], [[39/22]]
|
|
|-
| 50
| 1016.9
| [[9/5]]
|
|
|-
| 51
| 1037.3
| [[20/11]]
|
|
|-
| 52
| 1057.6
| [[24/13]]
|
|
|-
| 53
| 1078.0
| [[13/7]], [[28/15]]
|
|
|-
| 54
| 1098.3
| [[17/9]], [[32/17]], [[66/35]]
|
|
|-
| 55
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
| 56
| 1139.0
| [[27/14]], [[52/27]]
|
|
|-
| 57
| 1159.3
| [[39/20]], [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}


There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 59edo’s JI approximations: 293zpi, 294zpi, 295zpi, 296zpi and 297zpi. The main Zeta peak index page details all five tunings.
== Notation ==


=== Subsets and supersets ===
=== Sagittal notation ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]].
==== 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.


== Intervals ==
== Scales ==
{{Interval table}}
; [[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 ==
== Instruments ==
; Lumatone
; Lumatone


See [[Lumatone mapping for 59edo]].  
See [[Lumatone mapping for 59edo]].


== Music ==
== 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]]
; [[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]
* "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]]
; [[Ray Perlner]]

Latest revision as of 21:29, 12 July 2026

← 58edo 59edo 60edo →
Prime factorization 59 (prime)
Step size 20.339 ¢ 
Fifth 35\59 (711.864 ¢)
Semitones (A1:m2) 9:2 (183.1 ¢ : 40.68 ¢)
Dual sharp fifth 35\59 (711.864 ¢)
Dual flat fifth 34\59 (691.525 ¢)
Dual major 2nd 10\59 (203.39 ¢)
Consistency limit 7
Distinct consistency limit 7

59 equal divisions of the octave (abbreviated 59edo or 59ed2), also called 59-tone equal temperament (59tet) or 59 equal temperament (59et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of 21/59, or the 59th root of 2.

Theory

59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its 5/4 is nearly pure (stretched only 0.127 ¢), as the denominator of a convergent to log25. 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 & 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.

As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 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 & 59 temperament with a subminor third generator provides an interesting temperament.

Odd harmonics

Approximation of odd harmonics in 59edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27
Error Absolute (¢) +9.91 +0.13 +7.45 -0.52 -2.17 -6.63 +10.04 -3.26 +7.57 -2.98 +2.23 +0.25 +9.39
Relative (%) +48.7 +0.6 +36.6 -2.6 -10.6 -32.6 +49.3 -16.0 +37.2 -14.7 +11.0 +1.2 +46.2
Steps
(reduced) 		94
(35) 		137
(19) 		166
(48) 		187
(10) 		204
(27) 		218
(41) 		231
(54) 		241
(5) 		251
(15) 		259
(23) 		267
(31) 		274
(38) 		281
(45)

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

Steps Cents Approximate ratios
(2.9.5.21.11.39.17-subgroup) 		Ratios of 3, 7, 13
(tending sharp) 		Ratios of 3, 7, 13
(tending flat)
0 0.0 1/1
1 20.3 81/80
2 40.7 40/39, 45/44
3 61.0 27/26, 28/27
4 81.4 21/20, 22/21
5 101.7 17/16, 18/17, 35/33
6 122.0 15/14, 14/13
7 142.4 13/12
8 162.7 11/10
9 183.1 10/9
10 203.4 9/8, 44/39
11 223.7 25/22 8/7
12 244.1 15/13, 39/34 8/7
13 264.4 7/6, 64/55
14 284.7 20/17, 33/28
15 305.1 25/21
16 325.4
17 345.8 11/9, 39/32, 128/105 16/13
18 366.1 21/17 16/13
19 386.4 5/4
20 406.8 81/64
21 427.1 32/25, 50/39
22 447.5 22/17, 35/27, 128/99
23 467.8 21/16, 64/49
24 488.1 45/34, 85/64 4/3
25 508.5 35/26 4/3
26 528.8 34/25
27 549.2 11/8, 48/35
28 569.5 25/18
29 589.8 45/32, 128/91
30 610.2 64/45, 91/64
31 630.5 36/25
32 650.8 16/11, 35/24
33 671.2 25/17
34 691.5 52/35 3/2
35 711.9 68/45, 128/85 3/2
36 732.2 32/21, 49/32
37 752.5 17/11, 54/35, 99/64
38 772.9 25/16, 39/25
39 793.2 128/81
40 813.6 8/5
41 833.9 34/21 13/8
42 854.2 18/11, 64/39, 105/64 13/8
43 874.6
44 894.9 42/25
45 915.3 17/10, 56/33
46 935.6 12/7, 55/32
47 955.9 26/15, 68/39 7/4
48 976.3 44/25 7/4
49 996.6 16/9, 39/22
50 1016.9 9/5
51 1037.3 20/11
52 1057.6 24/13
53 1078.0 13/7, 28/15
54 1098.3 17/9, 32/17, 66/35
55 1118.6 21/11, 40/21
56 1139.0 27/14, 52/27
57 1159.3 39/20, 88/45
58 1179.7 160/81
59 1200.0 2/1
Todo: complete table

Notation

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 66-EDO.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation144/14381/801053/1024
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation144/14381/801053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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 45 and 52.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35
Evo-SZ flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35

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 stretched-octave version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

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 compressed octaves, then 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
  • Antechinus (nonoctave period)

Instruments

Lumatone

See Lumatone mapping for 59edo.

Music

Bryan Deister
Francium
Budjarn Lambeth
Ray Perlner
Retrieved from "https://en.xen.wiki/index.php?title=59edo&oldid=233735"