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Using the flat fifth instead of the sharp one allows for the {{nowrap|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.
Using the flat fifth instead of the sharp one allows for the {{nowrap|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 [[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 & 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]] & 59 temperament with a subminor third generator provides an interesting temperament.


=== Odd harmonics ===
=== Odd harmonics ===
Line 16: Line 16:


== Intervals ==
== Intervals ==
{{Interval table}}
{| 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}}


== Notation ==
== Notation ==
Line 92: Line 459:
[[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.
[[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 [[zpi|296zpi]] 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.
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.


What follows is a comparison of stretched- and compressed-octave 59edo tunings.
== Scales ==
 
; [[Porcupine]] scales
; [[93edt]]
* Porcupine[7]: 8 8 8 11 8 8 8
* Octave size: 1206.62{{c}}
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
Stretching the octave of 59edo by around 6.5{{c}} results in improved primes 3, 7 and 11 but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 8.22{{c}}. The tuning 93edt does this. So does the tuning [[equal tuning|203ed11]] whose octaves are identical within 0.1{{c}}.
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
{{Harmonics in equal|93|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 93edt}}
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')
{{Harmonics in equal|93|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93edt (continued)}}
 
; [[ed6|152ed6]]  
* Octave size: 1204.05{{c}}
Stretching the octave of 59edo by around 4{{c}} results in improved primes 3 and 7, but worse primes 2, 5, 11 and 13. This approximates all harmonics up to 16 within 9.53{{c}}. The tuning 152ed6 does this.
{{Harmonics in equal|152|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed6}}
{{Harmonics in equal|152|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed6 (continued)}}
 
; [[zpi|294zpi]]
* Step size: 20.399{{c}}, octave size: 1203.54{{c}}
Stretching the octave of 59edo by around 3.5{{c}} results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.08{{c}}. The tuning 294zpi does this.
{{Harmonics in cet|20.399|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 294zpi}}
{{Harmonics in cet|20.399|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 294zpi (continued)}}
 
; [[ed12|211ed12]]
* Octave size: 1202.92{{c}}
Stretching the octave of 59edo by around 3{{c}} results in improved primes 3 and 7, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.82{{c}}. The tuning 211ed12 does this.
{{Harmonics in equal|211|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 211ed12}}
{{Harmonics in equal|211|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 211ed12 (continued)}}
 
; [[zpi|295zpi]]
* Step size: 20.342{{c}}, octave size: 1200.18{{c}}
Stretching the octave of 59edo by around a fifth of a cent results in slightly improved primes 11 and 13, but slightly worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 9.97{{c}}. The tuning 294zpi does this. 294zpi shares error equally between the two mappings of harmonic 3, so it is the best [[dual-fifth]] option for 59edo.
{{Harmonics in cet|20.342|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 295zpi}}
{{Harmonics in cet|20.342|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 295zpi (continued)}}
 
; 59edo
* Step size: 20.339{{c}}, octave size: 1200.00{{c}}
Pure-octaves 59edo approximates all harmonics up to 16 within 10.04{{c}}. So does the tuning [[ed5|137ed5]] whose octave is identical within 0.05{{c}}.
{{Harmonics in equal|59|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59edo}}
{{Harmonics in equal|59|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59edo (continued)}}
 
; [[WE|59et, 13-limit WE tuning]]
* Step size: 20.320{{c}}, octave size: 1198.88{{c}}
Compressing the octave of 59edo by around 1{{c}} results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.95{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|20.320|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning}}
{{Harmonics in cet|20.320|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning (continued)}}
 
; [[WE|59et, 7-limit WE tuning]]
* Step size: 20.301{{c}}, octave size: 1197.76{{c}}
Compressing the octave of 59edo by around 2{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.91{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
{{Harmonics in cet|20.301|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, 59et, 7-limit WE tuning}}
{{Harmonics in cet|20.301|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 7-limit WE tuning (continued)}}
 
; [[ed7|166ed7]]
* Octave size: 1197.35{{c}}
Compressing the octave of 59edo by around 2.5{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.71{{c}}. The tuning 166ed7 does this.
{{Harmonics in equal|166|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166ed7}}
{{Harmonics in equal|166|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166ed7 (continued)}}
 
; [[ed12|212ed12]]
* Octave size: 1197.24{{c}}
Compressing the octave of 59edo by around 3{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.26{{c}}. The tuning 212ed12 does this.
{{Harmonics in equal|212|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 212ed12}}
{{Harmonics in equal|212|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 212ed12 (continued)}}
 
; [[zpi|296zpi]]
* Step size: 20.282{{c}}, octave size: 1196.64{{c}}
Compressing the octave of 59edo by around 3.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.09{{c}}. The tuning 296zpi does this.
{{Harmonics in cet|20.282|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 296zpi}}
{{Harmonics in cet|20.282|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 296zpi (continued)}}
 
; [[ed6|153ed6]]  
* Octave size: 1196.18{{c}}
Compressing the octave of 59edo by around 4{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 8.81{{c}}. The tuning 153ed6 does this.
{{Harmonics in equal|153|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 153ed6}}
{{Harmonics in equal|153|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 153ed6 (continued)}}


== Instruments ==
== Instruments ==
Line 176: Line 476:
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [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]
* "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]]
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