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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|62}}
{{ED intro}}


== Theory ==
== Theory ==
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It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[hemimeantone]] temperaments.  
It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[hemimeantone]] temperaments.  


Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].
Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for [[mavling]], a septimal extension of [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 15: Line 15:
Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.  
Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.  


=== Miscellaneous properties ===
=== Miscellany ===
62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.  
62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.  


Line 32: Line 32:
| 0
| 0
| 0.00
| 0.00
| 1/1
| [[1/1]]
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 19.35
| 19.35
| 65/64, 66/65, 78/77, 91/90, 105/104
| [[65/64]], [[66/65]], [[78/77]], [[91/90]], [[105/104]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 38.71
| 38.71
| ''33/32'', 36/35, 45/44, 49/48, 50/49, 55/54, 56/55, ''64/63''
| ''[[33/32]]'', [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[56/55]], ''[[64/63]]''
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 58.06
| 58.06
| ''26/25'', 27/26
| ''[[26/25]]'', [[27/26]]
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 77.42
| 77.42
| 21/20, 22/21, 23/22, 24/23, 25/24, ''28/27''
| [[21/20]], [[22/21]], [[23/22]], [[24/23]], [[25/24]], ''[[28/27]]''
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 96.77
| 96.77
| 17/16, 18/17, 19/18, 20/19
| [[17/16]], [[18/17]], [[19/18]], [[20/19]]
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 116.13
| 116.13
| 15/14, 16/15
| [[15/14]], [[16/15]]
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 135.48
| 135.48
| 13/12, 14/13
| [[13/12]], [[14/13]]
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 154.84
| 154.84
| ''11/10'', 12/11, 23/21
| ''[[11/10]]'', [[12/11]], [[23/21]]
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 174.19
| 174.19
| 21/19
| [[21/19]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 193.55
| 193.55
| ''9/8'', ''10/9'', 19/17, 28/25
| ''[[9/8]]'', ''[[10/9]]'', [[19/17]], [[28/25]]
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 212.90
| 212.90
| 17/15
| [[17/15]]
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 232.26
| 232.26
| 8/7
| [[8/7]]
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
| 13
| 13
| 251.61
| 251.61
| 15/13, 22/19
| [[15/13]], [[22/19]]
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 270.97
| 270.97
| 7/6
| [[7/6]]
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 290.32
| 290.32
| 13/11, 19/16, 20/17
| [[13/11]], [[19/16]], [[20/17]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 309.68
| 309.68
| 6/5
| [[6/5]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 329.03
| 329.03
| 17/14, 23/19
| [[17/14]], [[23/19]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
| 18
| 18
| 348.39
| 348.39
| 11/9, 27/22, 28/23
| [[11/9]], [[27/22]], [[28/23]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 367.74
| 367.74
| 16/13, 21/17, 26/21
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 387.10
| 387.10
| 5/4
| [[5/4]]
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 406.45
| 406.45
| 19/15, 24/19
| [[19/15]], [[24/19]]
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 425.81
| 425.81
| 9/7, 14/11, 23/18, 32/25
| [[9/7]], [[14/11]], [[23/18]], [[32/25]]
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 445.16
| 445.16
| 13/10, 22/17
| [[13/10]], [[22/17]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 464.52
| 464.52
| 17/13, 21/16, 30/23
| [[17/13]], [[21/16]], [[30/23]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
| 25
| 25
| 483.87
| 483.87
| 25/19
| [[25/19]]
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
| 26
| 26
| 503.23
| 503.23
| 4/3
| [[4/3]]
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 522.58
| 522.58
| 19/14, 23/17
| [[19/14]], [[23/17]]
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 541.94
| 541.94
| 11/8, 15/11, 26/19
| [[11/8]], [[15/11]], [[26/19]]
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
| 29
| 29
| 561.29
| 561.29
| 18/13
| [[18/13]]
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|-
|-
| 30
| 30
| 580.65
| 580.65
| 7/5, ''25/18'', 32/23
| [[7/5]], ''[[25/18]]'', [[32/23]]
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|-
|-
| 31
| 31
| 600.00
| 600.00
| 17/12, 24/17
| [[17/12]], [[24/17]]
| {{UDnote|step=10}}
| {{UDnote|step=31}}
|-
|-
| 32
| 32
| 619.35
| 619.35
| 10/7, 23/16, ''36/25''
| [[10/7]], [[23/16]], ''[[36/25]]''
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|-
|-
| 33
| 33
| 638.71
| 638.71
| 13/9
| [[13/9]]
| {{UDnote|step=33}}
| {{UDnote|step=33}}
|-
|-
| 34
| 34
| 658.06
| 658.06
| 16/11, 19/13, 22/15
| [[16/11]], [[19/13]], [[22/15]]
| {{UDnote|step=34}}
| {{UDnote|step=34}}
|-
|-
| 35
| 35
| 677.42
| 677.42
| 28/19, 34/23
| [[28/19]], [[34/23]]
| {{UDnote|step=35}}
| {{UDnote|step=35}}
|-
|-
| 36
| 36
| 696.77
| 696.77
| 3/2
| [[3/2]]
| {{UDnote|step=36}}
| {{UDnote|step=36}}
|-
|-
| 37
| 37
| 716.13
| 716.13
| 38/25
| [[38/25]]
| {{UDnote|step=37}}
| {{UDnote|step=37}}
|-
|-
| 38
| 38
| 735.48
| 735.48
| 23/15, 26/17, 32/21
| [[23/15]], [[26/17]], [[32/21]]
| {{UDnote|step=38}}
| {{UDnote|step=38}}
|-
|-
| 39
| 39
| 754.84
| 754.84
| 17/11, 20/13
| [[17/11]], [[20/13]]
| {{UDnote|step=39}}
| {{UDnote|step=39}}
|-
|-
| 40
| 40
| 774.19
| 774.19
| 11/7, 14/9, 25/16, 36/23
| [[11/7]], [[14/9]], [[25/16]], [[36/23]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
| 41
| 41
| 793.55
| 793.55
| 19/12, 30/19
| [[19/12]], [[30/19]]
| {{UDnote|step=41}}
| {{UDnote|step=41}}
|-
|-
| 42
| 42
| 812.90
| 812.90
| 8/5
| [[8/5]]
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|-
|-
| 43
| 43
| 832.26
| 832.26
| 13/8, 21/13, 34/21
| [[13/8]], [[21/13]], [[34/21]]
| {{UDnote|step=43}}
| {{UDnote|step=43}}
|-
|-
| 44
| 44
| 851.61
| 851.61
| 18/11, 23/14, 44/27
| [[18/11]], [[23/14]], [[44/27]]
| {{UDnote|step=44}}
| {{UDnote|step=44}}
|-
|-
| 45
| 45
| 870.97
| 870.97
| 28/17, 38/23
| [[28/17]], [[38/23]]
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|-
|-
| 46
| 46
| 890.32
| 890.32
| 5/3
| [[5/3]]
| {{UDnote|step=46}}
| {{UDnote|step=46}}
|-
|-
| 47
| 47
| 909.68
| 909.68
| 17/10, 22/13, 32/19
| [[17/10]], [[22/13]], [[32/19]]
| {{UDnote|step=47}}
| {{UDnote|step=47}}
|-
|-
| 48
| 48
| 929.03
| 929.03
| 12/7
| [[12/7]]
| {{UDnote|step=48}}
| {{UDnote|step=48}}
|-
|-
| 49
| 49
| 948.39
| 948.39
| 19/11, 26/15
| [[19/11]], [[26/15]]
| {{UDnote|step=49}}
| {{UDnote|step=49}}
|-
|-
| 50
| 50
| 967.74
| 967.74
| 7/4
| [[7/4]]
| {{UDnote|step=50}}
| {{UDnote|step=50}}
|-
|-
| 51
| 51
| 987.10
| 987.10
| 30/17
| [[30/17]]
| {{UDnote|step=51}}
| {{UDnote|step=51}}
|-
|-
| 52
| 52
| 1006.45
| 1006.45
| ''9/5'', ''16/9'', 25/14, 34/19
| ''[[9/5]]'', ''[[16/9]]'', [[25/14]], [[34/19]]
| {{UDnote|step=52}}
| {{UDnote|step=52}}
|-
|-
| 53
| 53
| 1025.81
| 1025.81
| 38/21
| [[38/21]]
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|-
|-
| 54
| 54
| 1045.16
| 1045.16
| 11/6, ''20/11'', 42/23
| [[11/6]], ''[[20/11]]'', [[42/23]]
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|-
|-
| 55
| 55
| 1064.52
| 1064.52
| 13/7, 24/13
| [[13/7]], [[24/13]]
| {{UDnote|step=55}}
| {{UDnote|step=55}}
|-
|-
| 56
| 56
| 1083.87
| 1083.87
| 15/8, 28/15
| [[15/8]], [[28/15]]
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|-
|-
| 57
| 57
| 1103.23
| 1103.23
| 17/9, 19/10, 32/17, 36/19
| [[17/9]], [[19/10]], [[32/17]], [[36/19]]
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|-
|-
| 58
| 58
| 1122.58
| 1122.58
| 21/11, 23/12, ''27/14'', 40/21, 44/23, 48/25
| [[21/11]], [[23/12]], ''[[27/14]]'', [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|-
|-
| 59
| 59
| 1141.94
| 1141.94
| ''25/13'', 52/27
| ''[[25/13]]'', [[52/27]]
| {{UDnote|step=59}}
| {{UDnote|step=59}}
|-
|-
| 60
| 60
| 1161.29
| 1161.29
| 35/18, 49/25, 55/28, ''63/32'', ''64/33'', 88/45, 96/49, 108/55
| [[35/18]], [[49/25]], [[55/28]], ''[[63/32]]'', ''[[64/33]]'', [[88/45]], [[96/49]], [[108/55]]
| {{UDnote|step=60}}
| {{UDnote|step=60}}
|-
|-
| 61
| 61
| 1180.65
| 1180.65
| 65/33, 77/39, 128/65, 180/91, 208/105
| [[65/33]], [[77/39]], [[128/65]], [[180/91]], [[208/105]]
| {{UDnote|step=61}}
| {{UDnote|step=61}}
|-
|-
| 62
| 62
| 1200.00
| 1200.00
| 2/1
| [[2/1]]
| {{UDnote|step=62}}
| {{UDnote|step=62}}
|}
|}
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
62edo can be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation #Sharp-4|ups and downs]]. This can be done by combining sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp4-szg}}


{{Sharpness-sharp4}}
=== Kite's ups and downs notation ===
62edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[69edo#Sagittal notation|69]] and [[76edo#Sagittal notation|76]], and is a superset of the notation for [[31edo#Sagittal notation|31-EDO]].
This notation uses the same sagittal sequence as edos [[69edo #Sagittal notation|69]] and [[76edo #Sagittal notation|76]], and is a superset of the notation for [[31edo #Sagittal notation|31edo]].


==== Evo flavor ====
==== Evo flavor ====
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</imagemap>
</imagemap>


In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.


=== Armodue notation ===
=== Armodue notation ===
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{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
|-
! colspan="2" | &#35;
! colspan="2" | #
! Cents
! Cents
! Armodue notation
! Armodue notation
Line 785: Line 788:
| 1
| 1
|  
|  
|}
== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[314zpi]]
| 61.9380472360525
| 19.3741981471691
| 6.262952
| 0.952068
| 15.026453
| 62edo
| 1201.20028512448
| 8
| 8
|}
|}


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| [[Gallium]]
| [[Gallium]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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
 
== Octave stretch or compression ==
62edo tunes most simple [[prime]]s flat. [[Octave stretching|Stretching the octave]] of 62edo by the right amount improves its approximations of [[JI]].
 
Some tunings of 62edo which do this include [[ed7|174ed7]] and [[zpi|314zpi]].


== Instruments ==
== Instruments ==
; Fretted instruments
 
* [[Skip fretting system 62 6 11]]
=== Lumatone ===
* [[Lumatone mapping for 62edo]]
 
=== Skip fretting ===
'''[[Skip fretting]] system 62 6 11''' has strings tuned 11\62 apart, while frets are 6\62.
 
On a 4-string bass, here are your open strings:
 
0 11 22 33
 
A good supraminor 3rd is found on the 2nd string, 1st fret. A supermajor third is found on the open 3rd string. The major 6th can be found on the 4th string, 2nd fret.
 
5-string bass
 
51 0 11 22 33
 
This adds an interval of a major 7th (minus an 8ve) at the first string, 1st fret.
 
6-string guitar
 
0 11 22 33 44 55
 
”Major” 020131
 
7-string guitar
 
0 11 22 33 44 55 4
 
 
'''Skip fretting system 62 9 11''' is another 62edo skip fretting system. The 5th is on the 5th string. The major 3rd is on the 2nd string, 1st fret.
{{todo|add illustration|text=Base it off of the diagram from [[User:MisterShafXen/Skip fretting system 62 9 11]]}}
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/UerD0NqBbng ''microtonal improvisation in 62edo''] (2025)
* [https://www.youtube.com/watch?v=ujaUA-uwDvE ''62edo improv''] (2025)
* [https://www.youtube.com/watch?v=3_mwEylLEwU&list=WL&index=338 ''62edo prelude''] (2026)
Retrieved from "https://en.xen.wiki/w/62edo"