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== 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 ===
=== Odd harmonics ===
Line 23: Line 23:
| 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 64: 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 159: 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 189: 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 208: 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]]
|  
|  
|-
|-
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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 ==
== Notation ==
===Ups and downs notation===
=== Stein–Zimmermann–Gould notation ===
60edo can be notated with [[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).
[[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}}
{{Sharpness-sharp5a}}
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp5|60}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation is a superset of the notations for EDOs [[12edo#Sagittal notation|12]] and [[6edo#Sagittal notation|6]].
This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].


==== Evo flavor ====
==== Evo flavor ====
Line 357: Line 360:
default [[File:60-EDO_Revo_Sagittal.svg]]
default [[File:60-EDO_Revo_Sagittal.svg]]
</imagemap>
</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 ==
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| 9.44
| 9.44
|}
|}
=== Uniform maps ===
{{Uniform map|edo=60}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 448: Line 459:
| 340.0
| 340.0
| 39/32
| 39/32
| [[Houborizic]] (60) / houbor (60e)
| [[Houborizic]] (60) / [[houbor]] (60e)
|-
|-
| 1
| 1
Line 454: Line 465:
| 380.0
| 380.0
| 5/4
| 5/4
| [[Magic]] (60) / witchcraft (60e)
| [[Magic]] (60) / [[witchcraft]] (60e)
|-
|-
| 1
| 1
Line 460: Line 471:
| 580.0
| 580.0
| 7/5
| 7/5
| [[Tritonic]] (60e) / tritoni (60)
| [[Tritonic]] (60e) / [[tritoni]] (60)
|-
|-
| 2
| 2
Line 475: Line 486:
|-
|-
| 2
| 2
| 19\60<br>(11\60)
| 11\60
| 380.0<br>(220.0)
| 220.0
| 5/4<br>(25/22)
| 25/22
| [[Astrology]] (60de) / [[divination]] (60e)
| [[Astrology]] (60de) / [[divination]] (60e)
|-
|-
Line 493: Line 504:
|-
|-
| 5
| 5
| 19\60<br>(5\60)
| 1\60
| 380.0<br>(100.0)
| 20.0
| 5/4<br>(256/245)
| 81/80
| [[Warlock]]
| [[Quintile]] (60)
|-
|-
| 5
| 5
| 25\60<br>(1\60)
| 5\60
| 500.0<br>(20.0)
| 100.0
| 4/3<br>(81/80)
| 256/245
| [[Quintile]] (60)
| [[Warlock]]
|-
|-
| 6
| 6
| 17\60<br>(3\60)
| 3\60
| 340.0<br>(60.0)
| 60.0
| 375/308<br>(1760/1701)
| 1760/1701
| [[Semiseptichrome]] (11-limit, 60e)
| [[Semiseptichrome]] (11-limit, 60e)
|-
|-
| 10
| 10
| 25\60<br>(1\60)
| 1\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(91/90)
| 91/90
| [[Decile]] (60e)<br>[[Decic]] (60) / splendecic (60e) / prodecic (60e)
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
|-
|-
| 12
| 12
| 19\60<br>(1\60)
| 1\60
| 380.0<br>(20.0)
| 20.0
| 5/4<br>(81/80)
| 81/80
| [[Compton]] / comptone (60e)
| [[Compton]] / [[comptone]] (60e)
|-
|-
| 12
| 12
| 12\60<br>(2\60)
| 2\60
| 240.0<br>(40.0)
| 40.0
| 8/7<br>(40/39)
| 40/39
| [[Catnip]] (60cf)
| [[Catnip]] (60cf)
|-
|-
| 15
| 15
| 25\60<br>(3\60)
| 3\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(126/125)
| 91/90
| [[Pentadecal]] (60) / quindecal (60e)
| [[Pentadecal]] / [[quindecal]] (60e)
|-
|-
| 20
| 20
| 25\60<br>(2\60)
| 2\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(99/98)
| 99/98
| [[Degrees]] (60e)
| [[Degrees]] (60e)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Diagrams ==
== Diagrams ==
Line 577: Line 588:


; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/nlKHUDCR3pI ''60edo improv''] (2025)
* [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]]
Retrieved from "https://en.xen.wiki/w/60edo"