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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|60}}
{{ED intro}}
 
== Theory ==
== Theory ==
Since 60 = 5×12, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo|-19 12}}. In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the optimal patent val for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[875/864]], [[245/243]], [[225/224]] and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].
Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|tempers out]] the [[Pythagorean comma]], 531441/524288 ({{monzo| -19 12 }}). In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the [[optimal patent val]] for 5-limit [[magic]]. In the [[7-limit]] it tempers out [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.


=== Odd harmonics ===
=== Odd harmonics ===
Line 8: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
60edo is the 9th [[highly composite edo]], with subset edos {{EDOs| 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 }}.  
60edo is the 9th [[highly composite edo]], with subset edos {{EDOs| 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 }}. In addition, it is of largest consistency among highly composite edos for its size, being consistent in the 9-odd-limit, and all such edos all the way to [[27720edo]] are consistent in only at most 7-odd-limit.


In addition, it is of largest consistency among highly composite edos for its size, being consistent in the 9-odd-limit, and all such edos all the way to [[27720edo]] are consistent in only at most 7-odd-limit.
A step of 60edo is exactly 9 [[dexl]]s, or exactly 41 [[mina]]s.


== Intervals ==
== Intervals ==
Line 17: Line 18:
! Degrees
! Degrees
! Cents
! Cents
! Approximate Ratios<br>in the 2.3.5.7.13.17 subgroup
! Approximate ratios<br>in the 2.3.5.7.13.17 subgroup
! Additional Ratios<br>of 11 (tending flat, 60e val)
! Additional ratios<br>of 11 (tending flat, 60e val)
|-
|-
| 0
| 0
| 0
| 0
| 1/1
| [[1/1]]
|  
|  
|-
|-
| 1
| 1
| 20
| 20
| 81/80, ''49/48''
| [[81/80]], ''[[49/48]]''
|  
|  
|-
|-
| 2
| 2
| 40
| 40
| 50/49, ''64/63''
| [[50/49]], ''[[64/63]]''
| ''33/32''
| ''[[33/32]]''
|-
|-
| 3
| 3
| 60
| 60
| ''25/24'', 28/27, ''36/35''
| ''[[25/24]]'', [[28/27]], ''[[36/35]]''
|  
|  
|-
|-
| 4
| 4
| 80
| 80
| 21/20
| [[21/20]]
|  
|  
|-
|-
| 5
| 5
| 100
| 100
| 17/16, 18/17
| [[17/16]], [[18/17]]
|  
|  
|-
|-
| 6
| 6
| 120
| 120
| 16/15, 15/14, 14/13
| [[16/15]], [[15/14]], [[14/13]]
|  
|  
|-
|-
| 7
| 7
| 140
| 140
| 13/12
| [[13/12]]
|  
|  
|-
|-
Line 63: Line 64:
| 160
| 160
|  
|  
| 12/11, 11/10
| [[12/11]], [[11/10]]
|-
|-
| 9
| 9
| 180
| 180
| 10/9
| [[10/9]]
|  
|  
|-
|-
| 10
| 10
| 200
| 200
| 9/8
| [[9/8]]
|  
|  
|-
|-
| 11
| 11
| 220
| 220
| 17/15
| [[17/15]]
|  
|  
|-
|-
| 12
| 12
| 240
| 240
| 8/7, 15/13
| [[8/7]], [[15/13]]
|  
|  
|-
|-
| 13
| 13
| 260
| 260
| 7/6
| [[7/6]]
|  
|  
|-
|-
| 14
| 14
| 280
| 280
| 20/17
| [[20/17]]
| ''13/11'', 33/28
| [[33/28]]
|-
|-
| 15
| 15
| 300
| 300
| 32/27
| [[32/27]]
|  
| ''[[13/11]]''
|-
|-
| 16
| 16
| 320
| 320
| 6/5
| [[6/5]]
|  
|  
|-
|-
| 17
| 17
| 340
| 340
| 39/32, 17/14
| [[39/32]], [[17/14]]
| 11/9
| [[11/9]]
|-
|-
| 18
| 18
| 360
| 360
| 16/13, 21/17
| [[16/13]], [[21/17]]
| 27/22
| [[27/22]]
|-
|-
| 19
| 19
| 380
| 380
| 5/4
| [[5/4]]
|  
|  
|-
|-
| 20
| 20
| 400
| 400
| 81/64
| [[81/64]]
|  
| ''[[33/26]]''
|-
|-
| 21
| 21
| 420
| 420
|  
|  
| 14/11, ''33/26''
| [[14/11]]
|-
|-
| 22
| 22
| 440
| 440
| 9/7
| [[9/7]]
| 22/17
| [[22/17]]
|-
|-
| 23
| 23
| 460
| 460
| ''21/16'', 13/10, 17/13
| ''[[21/16]]'', [[13/10]], [[17/13]]
|  
|  
|-
|-
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| 25
| 25
| 500
| 500
| 4/3
| [[4/3]]
|  
|  
|-
|-
| 26
| 26
| 520
| 520
| 27/20
| [[27/20]]
|  
|  
|-
|-
Line 158: Line 159:
| 540
| 540
|  
|  
| ''11/8'', 15/11
| ''[[11/8]]'', [[15/11]]
|-
|-
| 28
| 28
| 560
| 560
| 18/13
| [[18/13]]
|  
|  
|-
|-
| 29
| 29
| 580
| 580
| 7/5
| [[7/5]]
|  
|  
|-
|-
| 30
| 30
| 600
| 600
| 17/12, 24/17
| [[17/12]], [[24/17]]
|  
|  
|-
|-
| 31
| 31
| 620
| 620
| 10/7
| [[10/7]]
|  
|  
|-
|-
| 32
| 32
| 640
| 640
| 13/9
| [[13/9]]
|  
|  
|-
|-
Line 188: Line 189:
| 660
| 660
|  
|  
| ''16/11'', 22/15
| ''[[16/11]]'', [[22/15]]
|-
|-
| 34
| 34
| 680
| 680
| 40/27
| [[40/27]]
|  
|  
|-
|-
| 35
| 35
| 700
| 700
| 3/2
| [[3/2]]
|  
|  
|-
|-
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| 37
| 37
| 740
| 740
| ''32/21'', 20/13, 26/17
| ''[[32/21]]'', [[20/13]], [[26/17]]
|  
|  
|-
|-
| 38
| 38
| 760
| 760
| 14/9
| [[14/9]]
| 17/11
| [[17/11]]
|-
|-
| 39
| 39
| 780
| 780
|  
|  
| ''52/33'', 11/7
| [[11/7]]
|-
|-
| 40
| 40
| 800
| 800
| 128/81
| [[128/81]]
|  
| ''[[52/33]]''
|-
|-
| 41
| 41
| 820
| 820
| 8/5
| [[8/5]]
|  
|  
|-
|-
| 42
| 42
| 840
| 840
| 13/8, 34/21
| [[13/8]], [[34/21]]
| 44/27
| [[44/27]]
|-
|-
| 43
| 43
| 860
| 860
| 64/39, 28/17
| [[64/39]], [[28/17]]
| 18/11
| [[18/11]]
|-
|-
| 44
| 44
| 880
| 880
| 5/3
| [[5/3]]
|  
|  
|-
|-
| 45
| 45
| 900
| 900
| 27/16
| [[27/16]]
|  
| ''[[22/13]]''
|-
|-
| 46
| 46
| 920
| 920
| 17/10
| [[17/10]]
| ''22/13'', 56/33
| [[56/33]]
|-
|-
| 47
| 47
| 940
| 940
| 12/7
| [[12/7]]
|  
|  
|-
|-
| 48
| 48
| 960
| 960
| 7/4, 26/15
| [[7/4]], [[26/15]]
|  
|  
|-
|-
| 49
| 49
| 980
| 980
| 30/17
| [[30/17]]
|  
|  
|-
|-
| 50
| 50
| 1000
| 1000
| 16/9
| [[16/9]]
|  
|  
|-
|-
| 51
| 51
| 1020
| 1020
| 9/5
| [[9/5]]
|  
|  
|-
|-
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| 1040
| 1040
|  
|  
| 11/6, 20/11
| [[11/6]], [[20/11]]
|-
|-
| 53
| 53
| 1060
| 1060
| 24/13
| [[24/13]]
|  
|  
|-
|-
| 54
| 54
| 1080
| 1080
| 15/8, 28/15, 13/7
| [[15/8]], [[28/15]], [[13/7]]
|  
|  
|-
|-
| 55
| 55
| 1100
| 1100
| 17/9, 32/17
| [[17/9]], [[32/17]]
|  
|  
|-
|-
| 56
| 56
| 1120
| 1120
| 40/21
| [[40/21]]
|  
|  
|-
|-
| 57
| 57
| 1140
| 1140
| ''48/25'', 27/14, ''35/18''
| ''[[48/25]]'', [[27/14]], ''[[35/18]]''
|  
|  
|-
|-
| 58
| 58
| 1160
| 1160
| 49/25, ''63/32''
| [[49/25]], ''[[63/32]]''
| ''64/33''
| ''[[64/33]]''
|-
|-
| 59
| 59
| 1180
| 1180
| 160/81, ''96/49''
| [[160/81]], ''[[96/49]]''
|  
|  
|-
|-
| 60
| 60
| 1200
| 1200
| 2/1
| [[2/1]]
|  
|  
|}
|}
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg|60}}
=== Kite's ups and downs notation ===
60edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Sharpness-sharp5a}}
=== Sagittal notation ===
This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].
==== Evo flavor ====
<imagemap>
File:60-EDO_Evo_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 80 190 106 [[45927/45056]]
rect 190 80 310 106 [[46/45]]
default [[File:60-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:60-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 628 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[45927/45056]]
rect 190 80 310 106 [[46/45]]
default [[File:60-EDO_Revo_Sagittal.svg]]
</imagemap>
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|60}}
{{Q-odd-limit intervals|59.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 60e val mapping}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 332: Line 372:
|-
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 342: Line 382:
| 2.3.5
| 2.3.5
| 3125/3072, 531441/524288
| 3125/3072, 531441/524288
| [{{val| 60 95 139 }}]
| {{mapping| 60 95 139 }}
| +1.32
| +1.32
| 1.11
| 1.11
Line 349: Line 389:
| 2.3.5.7
| 2.3.5.7
| 225/224, 245/243, 64827/64000
| 225/224, 245/243, 64827/64000
| [{{val| 60 95 139 168 }}]
| {{mapping| 60 95 139 168 }}
| +1.78
| +1.78
| 1.25
| 1.25
Line 356: Line 396:
| 2.3.5.7.13
| 2.3.5.7.13
| 105/104, 196/195, 245/243, 8281/8192
| 105/104, 196/195, 245/243, 8281/8192
| [{{val| 60 95 139 168 222 }}]
| {{mapping| 60 95 139 168 222 }}
| +1.45
| +1.45
| 1.29
| 1.29
| 6.46
| 6.46
|-
|-style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11
| style="border-top: double;" | 121/120, 225/224, 245/243, 441/440
| 121/120, 225/224, 245/243, 441/440
| style="border-top: double;" | [{{val| 60 95 139 168 207 }}] (60e)
| {{mapping| 60 95 139 168 207 }} (60e)
| style="border-top: double;" | +2.08
| +2.08
| style="border-top: double;" | 1.27
| 1.27
| style="border-top: double;" | 6.33
| 6.33
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 105/104, 121/120, 196/195, 275/273, 325/324
| 105/104, 121/120, 196/195, 275/273, 325/324
| [{{val| 60 95 139 168 207 222 }}] (60e)
| {{mapping| 60 95 139 168 207 222 }} (60e)
| +1.75
| +1.75
| 1.36
| 1.36
| 6.80
| 6.80
|-
|-style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11
| style="border-top: double;" | 100/99, 225/224, 385/384, 3087/3025
| 100/99, 225/224, 385/384, 3087/3025
| style="border-top: double;" | [{{val| 60 95 139 168 208 }}] (60)
| {{mapping| 60 95 139 168 208 }} (60)
| style="border-top: double;" | +0.91
| +0.91
| style="border-top: double;" | 2.05
| 2.05
| style="border-top: double;" | 10.22
| 10.22
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 100/99, 105/104, 144/143, 196/195, 1352/1331
| 100/99, 105/104, 144/143, 196/195, 1352/1331
| [{{val| 60 95 139 168 208 222 }}] (60)
| {{mapping| 60 95 139 168 208 222 }} (60)
| +0.79
| +0.79
| 1.89
| 1.89
| 9.44
| 9.44
|}
|}
=== Uniform maps ===
{{Uniform map|edo=60}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
 
{| class="wikitable center-all left-5"
{| class="wikitable center-1 center-2"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! Names
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 7\60
| 140.0
| 13/12
| [[Quintannic]] (60e)
|-
| 1
| 13\60
| 260.0
| 7/6
| [[Superpelog]] (7-limit, 60bbccdd)
|-
|-
| 1
| 1
| 17\60
| 17\60
| [[Amity]]
| 340.0
| 39/32
| [[Houborizic]] (60) / [[houbor]] (60e)
|-
|-
| 1
| 1
| 19\60
| 19\60
| [[Magic]]
| 380.0
| 5/4
| [[Magic]] (60) / [[witchcraft]] (60e)
|-
|-
| 1
| 1
| 29\60
| 29\60
| [[Tritonic]]
| 580.0
| 7/5
| [[Tritonic]] (60e) / [[tritoni]] (60)
|-
| 2
| 1\60
| 20.0
| 81/80
| [[Bicommatic]] (60e)
|-
|-
| 2
| 2
| 7\60
| 7\60
| [[Crepuscular]]/[[Fifive]]
| 140.0
| 13/12
| [[Fifive]] / [[fifives]] (60)
|-
|-
| 2
| 2
| 11\60
| 11\60
| [[Astrology]]/[[Divination]]
| 220.0
| 25/22
| [[Astrology]] (60de) / [[divination]] (60e)
|-
|-
| 2
| 2
| 13\60
| 13\60
| [[Bamity]]
| 260.0
| 7/6
| [[Bamity]] (11-limit, 60e)
|-
| 3
| 7\60
| 140.0
| 243/224
| [[Septichrome]]
|-
|-
| 5
| 5
| 1\60
| 1\60
| [[Pental]]
| 20.0
| 81/80
| [[Quintile]] (60)
|-
| 5
| 5\60
| 100.0
| 256/245
| [[Warlock]]
|-
| 6
| 3\60
| 60.0
| 1760/1701
| [[Semiseptichrome]] (11-limit, 60e)
|-
| 10
| 1\60
| 20.0
| 91/90
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
|-
|-
| 12
| 12
| 1\60
| 1\60
| [[Compton]]
| 20.0
| 81/80
| [[Compton]] / [[comptone]] (60e)
|-
|-
| 12
| 12
| 2\60
| 2\60
| [[Catnip]]
| 40.0
| 40/39
| [[Catnip]] (60cf)
|-
| 15
| 3\60
| 20.0
| 91/90
| [[Pentadecal]] / [[quindecal]] (60e)
|-
| 20
| 2\60
| 20.0
| 99/98
| [[Degrees]] (60e)
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Diagrams ==
== Diagrams ==
[[File:60edo_wheel_with_cents_values.png|alt=60edo wheel with cents values.png|560x560px|60edo wheel with cents values.png]]
[[File:60edo_wheel_with_cents_values.png|alt=60edo wheel with cents values.png|560x560px|60edo wheel with cents values.png]] {{todo|annotate}}


[[File:blue_60edo.png|alt=blue_60edo.png|blue_60edo.png]]
[[File:blue_60edo.png|alt=blue_60edo.png|blue_60edo.png]]
== Octave stretch or compression ==
What follows is a comparison of compressed- and stretched-octave 60edo tunings.
60edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as a no-11 17-limit equal temperament. With the right amount of stretch we can find better harmonics 3, 5, and 7 at the expense of somewhat less accurate approximations of 2 and 13. Tunings such as [[155ed6]], [[95edt]] or [[zpi|301zpi]] make good options for this.
== Scales ==
* [[5- to 10-tone scales in 60edo]]
* Amulet{{idiosyncratic}} (approximated from [[25edo]], subset of [[magic]]): 5 2 5 5 2 5 7 5 5 2 5 7 5
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 6 8 20 5 21
** 7-tone pelog: 6 8 12 8 5 14 7
** 5-tone slendro: 12 12 12 12 12


== Instruments ==
== Instruments ==
Line 445: Line 576:
[[File:60edoguitar.jpg|alt=60edoguitar.jpg|60edoguitar.jpg]]
[[File:60edoguitar.jpg|alt=60edoguitar.jpg|60edoguitar.jpg]]


*[[Lumatone mapping for 60edo]]
* [[Skip fretting system 60 2 29]]
*[[Skip fretting system 60 2 29]]
* [[Skip fretting system 60 3 19]]
*[[Skip fretting system 60 3 19]]
* [[Skip fretting system 60 4 17]]
*[[Skip fretting system 60 4 17]]
 
* [[Lumatone mapping for 60edo]]


== Music ==
== Music ==
Line 454: Line 586:
* [http://x31eq.com/music/dingshi.mp3 ''Dingshi'']
* [http://x31eq.com/music/dingshi.mp3 ''Dingshi'']
* ''Gene's Jitterbug'' [http://x31eq.com/music/jitter.ogg (ogg)] [http://micro.soonlabel.com/gene_ward_smith/Others/Breed/jitter.mp3 (mp3)] [http://x31eq.com/music/jitter60.pdf Score]
* ''Gene's Jitterbug'' [http://x31eq.com/music/jitter.ogg (ogg)] [http://micro.soonlabel.com/gene_ward_smith/Others/Breed/jitter.mp3 (mp3)] [http://x31eq.com/music/jitter60.pdf Score]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/nlKHUDCR3pI ''60edo improv''] (2025-05-16)
* [https://www.youtube.com/shorts/VA_P26_3dTk ''60edo improv''] (2025-11-22)


; [[Robin Perry]]
; [[Robin Perry]]
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* [https://www.youtube.com/watch?v=CNkg1rQE8Zk ''The Well of Sensitivity'']
* [https://www.youtube.com/watch?v=CNkg1rQE8Zk ''The Well of Sensitivity'']


[[Category:60edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Highly composite]]
[[Category:Catnip]]
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