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The '''60 equal division''' divides the octave into 60 parts of exactly 20 cents each.
{{Infobox ET}}
{{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 supports [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] scores lower in [[badness]] than the [[patent val]], 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 ===
{{Harmonics in equal|60}}
 
=== 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 }}. 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 9: Line 18:
! Degrees
! Degrees
! Cents
! Cents
! Approximate Ratios<br>in the 2.3.5.13.17 subgroup
! Approximate ratios<br>in the 2.3.5.7.13.17 subgroup
! Additional Ratios<br>of 7 and 11 (tending flat, i.e. 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
| [[81/80]], ''[[49/48]]''
| |49/48
|
|-
|-
| | 2
| 2
| | 40
| 40
| |
| [[50/49]], ''[[64/63]]''
| |50/49, 64/63, 33/32
| ''[[33/32]]''
|-
|-
| | 3
| 3
| | 60
| 60
| |25/24
| ''[[25/24]]'', [[28/27]], ''[[36/35]]''
| |28/27, 36/35
|
|-
|-
| | 4
| 4
| | 80
| 80
| |
| [[21/20]]
| |  
|  
|-
|-
| | 5
| 5
| | 100
| 100
| |17/16, 18/17
| [[17/16]], [[18/17]]
| |  
|  
|-
|-
| | 6
| 6
| | 120
| 120
| | 16/15
| [[16/15]], [[15/14]], [[14/13]]
| |15/14, 14/13
|
|-
|-
| | 7
| 7
| | 140
| 140
| |13/12
| [[13/12]]
| |  
|  
|-
|-
| | 8
| 8
| | 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
| | 15/13
| [[8/7]], [[15/13]]
| | 8/7
|  
|-
|-
| | 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
| [[39/32]], [[17/14]]
| | 11/9, 17/14
| [[11/9]]
|-
|-
| | 18
| 18
| | 360
| 360
| | 16/13
| [[16/13]], [[21/17]]
| |27/22, 21/17
| [[27/22]]
|-
|-
| | 19
| 19
| | 380
| 380
| |5/4
| [[5/4]]
| |  
|  
|-
|-
| | 20
| 20
| | 400
| 400
| |81/64
| [[81/64]]
| |
| ''[[33/26]]''
|-
|-
| | 21
| 21
| | 420
| 420
| |  
|  
| | 33/26, 14/11
| [[14/11]]
|-
|-
| | 22
| 22
| | 440
| 440
| |
| [[9/7]]
| | 9/7, 22/17
| [[22/17]]
|-
|-
| | 23
| 23
| | 460
| 460
| | 13/10, 17/13
| ''[[21/16]]'', [[13/10]], [[17/13]]
| |21/16
|  
|-
|-
| | 24
| 24
| | 480
| 480
| |  
|  
| |  
|  
|-
|-
| | 25
| 25
| | 500
| 500
| | 4/3
| [[4/3]]
| |  
|  
|-
|-
| | 26
| 26
| | 520
| 520
| | 27/20
| [[27/20]]
| |  
|  
|-
|-
| | 27
| 27
| | 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]]
| |  
|  
|-
|-
| | 33
| 33
| | 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]]
| |  
|  
|-
|-
| | 36
| 36
| | 720
| 720
| |  
|  
| |  
|  
|-
|-
| | 37
| 37
| | 740
| 740
| | 20/13, 26/17
| ''[[32/21]]'', [[20/13]], [[26/17]]
| |32/21
|  
|-
|-
| | 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
| |160/81
| [[128/81]]
| |
| ''[[52/33]]''
|-
|-
| | 41
| 41
| | 820
| 820
| |8/5
| [[8/5]]
| |  
|  
|-
|-
| | 42
| 42
| | 840
| 840
| | 13/8
| [[13/8]], [[34/21]]
| |44/27, 34/21
| [[44/27]]
|-
|-
| | 43
| 43
| | 860
| 860
| |64/39
| [[64/39]], [[28/17]]
| | 18/11, 28/17
| [[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
| | 26/15
| [[7/4]], [[26/15]]
| | 7/4
|  
|-
|-
| | 49
| 49
| | 980
| 980
| |30/17
| [[30/17]]
| |  
|  
|-
|-
| | 50
| 50
| | 1000
| 1000
| | 16/9
| [[16/9]]
| |  
|  
|-
|-
| | 51
| 51
| | 1020
| 1020
| | 9/5
| [[9/5]]
| |  
|  
|-
|-
| | 52
| 52
| | 1040
| 1040
| |  
|  
| | 11/6, 20/11
| [[11/6]], [[20/11]]
|-
|-
| | 53
| 53
| | 1060
| 1060
| |
| [[24/13]]
| |  
|  
|-
|-
| | 54
| 54
| | 1080
| 1080
| | 15/8
| [[15/8]], [[28/15]], [[13/7]]
| |28/15, 13/7
|
|-
|-
| | 55
| 55
| | 1100
| 1100
| |17/9, 32/17
| [[17/9]], [[32/17]]
| |  
|  
|-
|-
| | 56
| 56
| | 1120
| 1120
| |
| [[40/21]]
| |  
|  
|-
|-
| | 57
| 57
| | 1140
| 1140
| |48/25
| ''[[48/25]]'', [[27/14]], ''[[35/18]]''
| |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
| [[160/81]], ''[[96/49]]''
| |96/49
|
|-
|-
| | 60
| 60
| | 1200
| 1200
| |2/1
| [[2/1]]
| |  
|  
|}
|}


== Compositions ==
== Notation ==
[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/rojqoq.html Rojqoq (So-Called Peace)] [http://micro.soonlabel.com/gene_ward_smith/Others/Sethares/Rojqoq.mp3 play] by William Sethares
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg|60}}


[http://youtu.be/1rrgmP9VYQU Skating On Thin Ice] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/Skating%20On%20Thin%20Ice.mp3 play]
=== 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}}


[http://youtu.be/5GIZOYMkkJ0 My Mother Said So] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/My%20Mother%20Said%20So.mp3 play]
=== Sagittal notation ===
This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].


[http://www.macjams.com/song/62273 Black Salt - White Pepper] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/Black%20Salt%20-%20White%20Pepper.mp3 play]
==== 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>


[http://www.macjams.com/song/64413 NGC 300][http://micro.soonlabel.com/gene_ward_smith/Others/Perry/NGC%20300.mp3 play]
==== 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>


all by Robin Perry
== 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}}


[http://x31eq.com/music/dingshi.mp3 Dingshi] and
== Regular temperament properties ==
Multiple vals are listed since they all provide good temperaments.


[http://x31eq.com/music/jitter.ogg Gene's Jitterbug (ogg)] [http://micro.soonlabel.com/gene_ward_smith/Others/Breed/jitter.mp3 Gene's Jitterbug (mp3)] [http://x31eq.com/music/jitter60.pdf Score]
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 3125/3072, 531441/524288
| {{mapping| 60 95 139 }}
| +1.32
| 1.11
| 5.56
|-
| 2.3.5.7
| 225/224, 245/243, 64827/64000
| {{mapping| 60 95 139 168 }}
| +1.78
| 1.25
| 6.23
|-
| 2.3.5.7.13
| 105/104, 196/195, 245/243, 8281/8192
| {{mapping| 60 95 139 168 222 }}
| +1.45
| 1.29
| 6.46
|-style="border-top: double;"
| 2.3.5.7.11
| 121/120, 225/224, 245/243, 441/440
| {{mapping| 60 95 139 168 207 }} (60e)
| +2.08
| 1.27
| 6.33
|-
| 2.3.5.7.11.13
| 105/104, 121/120, 196/195, 275/273, 325/324
| {{mapping| 60 95 139 168 207 222 }} (60e)
| +1.75
| 1.36
| 6.80
|-style="border-top: double;"
| 2.3.5.7.11
| 100/99, 225/224, 385/384, 3087/3025
| {{mapping| 60 95 139 168 208 }} (60)
| +0.91
| 2.05
| 10.22
|-
| 2.3.5.7.11.13
| 100/99, 105/104, 144/143, 196/195, 1352/1331
| {{mapping| 60 95 139 168 208 222 }} (60)
| +0.79
| 1.89
| 9.44
|}


by Graham Breed
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! 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
| 17\60
| 340.0
| 39/32
| [[Houborizic]] (60) / [[houbor]] (60e)
|-
| 1
| 19\60
| 380.0
| 5/4
| [[Magic]] (60) / [[Magic_extensions#Witchcraft|witchcraft]] (60e)
|-
| 1
| 29\60
| 580.0
| 7/5
| [[Tritonic]] (60e) / [[tritoni]] (60)
|-
| 2
| 1\60
| 20.0
| 81/80
| [[Bicommatic]] (60e)
|-
| 2
| 7\60
| 140.0
| 13/12
| [[Fifive]] / [[fifives]] (60)
|-
| 2
| 19\60<br>(11\60)
| 380.0<br>(220.0)
| 5/4<br>(25/22)
| [[Astrology]] (60de) / [[divination]] (60e)
|-
| 2
| 13\60
| 260.0
| 7/6
| [[Bamity]] (11-limit, 60e)
|-
| 3
| 7\60
| 140.0
| 243/224
| [[Septichrome]]
|-
| 5
| 19\60<br>(5\60)
| 380.0<br>(100.0)
| 5/4<br>(256/245)
| [[Warlock]]
|-
| 5
| 25\60<br>(1\60)
| 500.0<br>(20.0)
| 4/3<br>(81/80)
| [[Quintile]] (60)
|-
| 6
| 17\60<br>(3\60)
| 340.0<br>(60.0)
| 375/308<br>(1760/1701)
| [[Semiseptichrome]] (11-limit, 60e)
|-
| 10
| 25\60<br>(1\60)
| 500.0<br>(20.0)
| 4/3<br>(91/90)
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
|-
| 12
| 19\60<br>(1\60)
| 380.0<br>(20.0)
| 5/4<br>(81/80)
| [[Compton]] / [[comptone]] (60e)
|-
| 12
| 12\60<br>(2\60)
| 240.0<br>(40.0)
| 8/7<br>(40/39)
| [[Catnip]] (60cf)
|-
| 15
| 25\60<br>(3\60)
| 500.0<br>(20.0)
| 4/3<br>(126/125)
| [[Pentadecal]] (60) / [[Cloudy_clan#Quindeca|quindecal]] (60e)
|-
| 20
| 25\60<br>(2\60)
| 500.0<br>(20.0)
| 4/3<br>(99/98)
| [[Degrees]] (60e)
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Images ==
== 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 ==
Due to its highly composite nature, 60edo has an unusually high number of ways it can be subdivided. This means it has multiple good [[skip-fretting]] systems which can be used to create stringed instruments with playable fret spacings that still span the full gamut. Probably the best of these is tuning a 20edo guitar to major thirds, as demonstrated by Robin Perry in the image below. This is very closely related to the [[Kite Guitar]], with tuning accuracy slightly worse in the 11-limit, but far better when ratios of 13, 17 & 19 are added.


[[File:60edoguitar.jpg|alt=60edoguitar.jpg|60edoguitar.jpg]]
[[File:60edoguitar.jpg|alt=60edoguitar.jpg|60edoguitar.jpg]]


Robin Perry guitar
* [[Skip fretting system 60 2 29]]
* [[Skip fretting system 60 3 19]]
* [[Skip fretting system 60 4 17]]
 
* [[Lumatone mapping for 60edo]]
 
== Music ==
; [[Graham Breed]]
* [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]
 
; [[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]]
* [https://youtu.be/1rrgmP9VYQU ''Skating On Thin Ice''] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/Skating%20On%20Thin%20Ice.mp3 play]
* [https://youtu.be/5GIZOYMkkJ0 ''My Mother Said So''] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/My%20Mother%20Said%20So.mp3 play]
* [http://www.macjams.com/song/62273 ''Black Salt - White Pepper''] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/Black%20Salt%20-%20White%20Pepper.mp3 play]
* [http://www.macjams.com/song/64413 ''NGC 300''] [http://micro.soonlabel.com/gene_ward_smith/Others/Perry/NGC%20300.mp3 play]
 
; [[William Sethares]]
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/rojqoq.html ''Rojqoq (So-Called Peace)''] [http://micro.soonlabel.com/gene_ward_smith/Others/Sethares/Rojqoq.mp3 play]
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=MuLl0UUhUK0 ''Absinthe Green'']
* [https://www.youtube.com/watch?v=CNkg1rQE8Zk ''The Well of Sensitivity'']


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[[Category:Equal divisions of the octave]]
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