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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|60}}
{{ED intro}}


== Theory ==
== Theory ==
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]], tempering out 3125/3072. 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.
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}}
 
== Notation ==
60edo can be notated using [[ups and downs notation]] using [[Helmholtz-Ellis]] accidentals:
{{Sharpness-sharp5|60}}
 
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with three arrows.


=== Odd harmonics ===
=== Odd harmonics ===
Line 27: 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]]
|  
|  
|-
|-
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| 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 168: 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 198: 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]]
|  
|  
|-
|-
Line 217: Line 208:
| 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]]
|  
|  
|-
|-
Line 293: Line 284:
| 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 ==
Multiple vals are listed since they all provide good temperaments.
Multiple vals are listed since they all provide good temperaments.


{{comma basis begin}}
{| 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
| 2.3.5
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| 1.89
| 1.89
| 9.44
| 9.44
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| 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
| 1
Line 410: Line 456:
| 340.0
| 340.0
| 39/32
| 39/32
| [[Houborizic]] (60) / houbor (60e)
| [[Houborizic]] (60) / [[houbor]] (60e)
|-
|-
| 1
| 1
Line 416: Line 462:
| 380.0
| 380.0
| 5/4
| 5/4
| [[Magic]] (60) / witchcraft (60e)
| [[Magic]] (60) / [[Magic_extensions#Witchcraft|witchcraft]] (60e)
|-
|-
| 1
| 1
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| 580.0
| 580.0
| 7/5
| 7/5
| [[Tritonic]] (60e) / tritoni (60)
| [[Tritonic]] (60e) / [[tritoni]] (60)
|-
| 2
| 1\60
| 20.0
| 81/80
| [[Bicommatic]] (60e)
|-
|-
| 2
| 2
Line 431: Line 483:
|-
|-
| 2
| 2
| 19\60<br />(11\60)
| 19\60<br>(11\60)
| 380.0<br />(220.0)
| 380.0<br>(220.0)
| 5/4<br />(25/22)
| 5/4<br>(25/22)
| [[Astrology]] (60de) / [[divination]] (60e)
| [[Astrology]] (60de) / [[divination]] (60e)
|-
|-
Line 449: Line 501:
|-
|-
| 5
| 5
| 19\60<br />(5\60)
| 19\60<br>(5\60)
| 380.0<br />(100.0)
| 380.0<br>(100.0)
| 5/4<br />(256/245)
| 5/4<br>(256/245)
| [[Warlock]]
| [[Warlock]]
|-
|-
| 5
| 5
| 25\60<br />(1\60)
| 25\60<br>(1\60)
| 500.0<br />(20.0)
| 500.0<br>(20.0)
| 4/3<br />(81/80)
| 4/3<br>(81/80)
| [[Pental (temperament)|Pental]] (60)
| [[Quintile]] (60)
|-
|-
| 6
| 6
| 17\60<br />(3\60)
| 17\60<br>(3\60)
| 340.0<br />(60.0)
| 340.0<br>(60.0)
| 375/308<br />(1760/1701)
| 375/308<br>(1760/1701)
| [[Semiseptichrome]] (11-limit, 60e)
| [[Semiseptichrome]] (11-limit, 60e)
|-
|-
| 10
| 10
| 25\60<br />(1\60)
| 25\60<br>(1\60)
| 500.0<br />(20.0)
| 500.0<br>(20.0)
| 4/3<br />(91/90)
| 4/3<br>(91/90)
| [[Decal]] (60e)<br />[[Decic]] (60) / splendecic (60e) / prodecic (60e)
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
|-
|-
| 12
| 12
| 19\60<br />(1\60)
| 19\60<br>(1\60)
| 380.0<br />(20.0)
| 380.0<br>(20.0)
| 5/4<br />(81/80)
| 5/4<br>(81/80)
| [[Compton]] / comptone (60e)
| [[Compton]] / [[comptone]] (60e)
|-
|-
| 12
| 12
| 12\60<br />(2\60)
| 12\60<br>(2\60)
| 240.0<br />(40.0)
| 240.0<br>(40.0)
| 8/7<br />(40/39)
| 8/7<br>(40/39)
| [[Catnip]] (60cf)
| [[Catnip]] (60cf)
|-
|-
| 15
| 15
| 25\60<br />(3\60)
| 25\60<br>(3\60)
| 500.0<br />(20.0)
| 500.0<br>(20.0)
| 4/3<br />(126/125)
| 4/3<br>(126/125)
| [[Pentadecal]] (60) / quindecal (60e)
| [[Pentadecal]] (60) / [[Cloudy_clan#Quindeca|quindecal]] (60e)
|-
|-
| 20
| 20
| 25\60<br />(2\60)
| 25\60<br>(2\60)
| 500.0<br />(20.0)
| 500.0<br>(20.0)
| 4/3<br />(99/98)
| 4/3<br>(99/98)
| [[Degrees]] (60e)
| [[Degrees]] (60e)
{{rank-2 end}}
|}
{{orf}}
<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


== 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 ==
== Scales ==
* [[5- to 10-tone scales in 60edo]]
* [[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 511: Line 573:
[[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 520: Line 583:
* [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]]
Line 535: Line 602:


[[Category:Listen]]
[[Category:Listen]]
[[Category:Catnip]]
Retrieved from "https://en.xen.wiki/w/60edo"