59edo: Difference between revisions

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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.

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 ===

Line 16:

== Intervals ==

== Notation ==

Line 92:

[[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 ==

Line 176:

Line 109:

; [[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]]

@@ Line 7: / Line 7: @@
 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 ===
@@ Line 16: / Line 16: @@
 == Intervals ==
-{{Interval table}}
+{{Interval table}}{{Todo|text=ADD 3|inline=1}}
 == Notation ==
@@ Line 92: / Line 92: @@
 [[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 ==
@@ Line 176: / Line 109: @@
 ; [[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]]