Xenharmonic Wiki

62edo: Difference between revisions

← Older edit
VisualWikitext
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,637 edits
No edit summary
→Intervals: sed -E "s#([0-9]+)/([0-9]+)#\[\[\1/\2\]\]#g" 62EDOintervals.txt > 62EDOintervals.new.txt
 
(61 intermediate revisions by 18 users not shown)
Line 1: Line 1:
'''62edo''' divides the octave into 62 equal parts of 19.35484 cents each.
{{Infobox ET}}
{{ED intro}}


62 = 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit; in the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]. It provides the [[optimal patent val]] for [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone_family#Hemimeantone|hemimeantone]] temperaments.
== Theory ==
{{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|tempers out]] [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the [[17-limit]] [[221/220]], [[273/272]], and [[289/288]]; in the [[19-limit]] [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the size differences between consecutive harmonics are monotonically decreasing for all first 24 harmonics, and 62edo is one of the few [[meantone]] edos that achieve this, great for those who seek higher-limit meantone harmony.
 
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 [[mavling]], a septimal extension of [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].
 
=== Odd harmonics ===
{{Harmonics in equal|62}}
 
=== Subsets and 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.
 
=== 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.
 
The 11 & 62 temperament is called mabon, named so because its associated year length corresponds to an autumnal equinoctial year. In the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.
 
The 15 & 62 temperament, corresponding to the leap day cycle, is [[demivalentine]] in the 13-limit.


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 }} supports hornbostel.
{{harmonics in equal|62}}
== Intervals ==
== Intervals ==
 
{| class="wikitable center-all right-2 left-3"
{| class="wikitable"
|-
! Steps
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.00
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 19.35
| [[65/64]], [[66/65]], [[78/77]], [[91/90]], [[105/104]]
| {{UDnote|step=1}}
|-
| 2
| 38.71
| ''[[33/32]]'', [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[56/55]], ''[[64/63]]''
| {{UDnote|step=2}}
|-
| 3
| 58.06
| ''[[26/25]]'', [[27/26]]
| {{UDnote|step=3}}
|-
| 4
| 77.42
| [[21/20]], [[22/21]], [[23/22]], [[24/23]], [[25/24]], ''[[28/27]]''
| {{UDnote|step=4}}
|-
| 5
| 96.77
| [[17/16]], [[18/17]], [[19/18]], [[20/19]]
| {{UDnote|step=5}}
|-
| 6
| 116.13
| [[15/14]], [[16/15]]
| {{UDnote|step=6}}
|-
| 7
| 135.48
| [[13/12]], [[14/13]]
| {{UDnote|step=7}}
|-
| 8
| 154.84
| ''[[11/10]]'', [[12/11]], [[23/21]]
| {{UDnote|step=8}}
|-
| 9
| 174.19
| [[21/19]]
| {{UDnote|step=9}}
|-
| 10
| 193.55
| ''[[9/8]]'', ''[[10/9]]'', [[19/17]], [[28/25]]
| {{UDnote|step=10}}
|-
| 11
| 212.90
| [[17/15]]
| {{UDnote|step=11}}
|-
| 12
| 232.26
| [[8/7]]
| {{UDnote|step=12}}
|-
| 13
| 251.61
| [[15/13]], [[22/19]]
| {{UDnote|step=13}}
|-
| 14
| 270.97
| [[7/6]]
| {{UDnote|step=14}}
|-
| 15
| 290.32
| [[13/11]], [[19/16]], [[20/17]]
| {{UDnote|step=15}}
|-
| 16
| 309.68
| [[6/5]]
| {{UDnote|step=16}}
|-
| 17
| 329.03
| [[17/14]], [[23/19]]
| {{UDnote|step=18}}
|-
| 18
| 348.39
| [[11/9]], [[27/22]], [[28/23]]
| {{UDnote|step=18}}
|-
| 19
| 367.74
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=19}}
|-
| 20
| 387.10
| [[5/4]]
| {{UDnote|step=20}}
|-
| 21
| 406.45
| [[19/15]], [[24/19]]
| {{UDnote|step=21}}
|-
| 22
| 425.81
| [[9/7]], [[14/11]], [[23/18]], [[32/25]]
| {{UDnote|step=22}}
|-
| 23
| 445.16
| [[13/10]], [[22/17]]
| {{UDnote|step=23}}
|-
| 24
| 464.52
| [[17/13]], [[21/16]], [[30/23]]
| {{UDnote|step=24}}
|-
| 25
| 483.87
| [[25/19]]
| {{UDnote|step=25}}
|-
| 26
| 503.23
| [[4/3]]
| {{UDnote|step=26}}
|-
| 27
| 522.58
| [[19/14]], [[23/17]]
| {{UDnote|step=27}}
|-
| 28
| 541.94
| [[11/8]], [[15/11]], [[26/19]]
| {{UDnote|step=28}}
|-
| 29
| 561.29
| [[18/13]]
| {{UDnote|step=29}}
|-
| 30
| 580.65
| [[7/5]], ''[[25/18]]'', [[32/23]]
| {{UDnote|step=30}}
|-
| 31
| 600.00
| [[17/12]], [[24/17]]
| {{UDnote|step=31}}
|-
| 32
| 619.35
| [[10/7]], [[23/16]], ''[[36/25]]''
| {{UDnote|step=32}}
|-
| 33
| 638.71
| [[13/9]]
| {{UDnote|step=33}}
|-
| 34
| 658.06
| [[16/11]], [[19/13]], [[22/15]]
| {{UDnote|step=34}}
|-
| 35
| 677.42
| [[28/19]], [[34/23]]
| {{UDnote|step=35}}
|-
| 36
| 696.77
| [[3/2]]
| {{UDnote|step=36}}
|-
| 37
| 716.13
| [[38/25]]
| {{UDnote|step=37}}
|-
| 38
| 735.48
| [[23/15]], [[26/17]], [[32/21]]
| {{UDnote|step=38}}
|-
| 39
| 754.84
| [[17/11]], [[20/13]]
| {{UDnote|step=39}}
|-
| 40
| 774.19
| [[11/7]], [[14/9]], [[25/16]], [[36/23]]
| {{UDnote|step=40}}
|-
| 41
| 793.55
| [[19/12]], [[30/19]]
| {{UDnote|step=41}}
|-
| 42
| 812.90
| [[8/5]]
| {{UDnote|step=42}}
|-
| 43
| 832.26
| [[13/8]], [[21/13]], [[34/21]]
| {{UDnote|step=43}}
|-
| 44
| 851.61
| [[18/11]], [[23/14]], [[44/27]]
| {{UDnote|step=44}}
|-
| 45
| 870.97
| [[28/17]], [[38/23]]
| {{UDnote|step=45}}
|-
| 46
| 890.32
| [[5/3]]
| {{UDnote|step=46}}
|-
| 47
| 909.68
| [[17/10]], [[22/13]], [[32/19]]
| {{UDnote|step=47}}
|-
| 48
| 929.03
| [[12/7]]
| {{UDnote|step=48}}
|-
| 49
| 948.39
| [[19/11]], [[26/15]]
| {{UDnote|step=49}}
|-
| 50
| 967.74
| [[7/4]]
| {{UDnote|step=50}}
|-
| 51
| 987.10
| [[30/17]]
| {{UDnote|step=51}}
|-
| 52
| 1006.45
| ''[[9/5]]'', ''[[16/9]]'', [[25/14]], [[34/19]]
| {{UDnote|step=52}}
|-
| 53
| 1025.81
| [[38/21]]
| {{UDnote|step=53}}
|-
| 54
| 1045.16
| [[11/6]], ''[[20/11]]'', [[42/23]]
| {{UDnote|step=54}}
|-
| 55
| 1064.52
| [[13/7]], [[24/13]]
| {{UDnote|step=55}}
|-
| 56
| 1083.87
| [[15/8]], [[28/15]]
| {{UDnote|step=56}}
|-
| 57
| 1103.23
| [[17/9]], [[19/10]], [[32/17]], [[36/19]]
| {{UDnote|step=57}}
|-
| 58
| 1122.58
| [[21/11]], [[23/12]], ''[[27/14]]'', [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=58}}
|-
| 59
| 1141.94
| ''[[25/13]]'', [[52/27]]
| {{UDnote|step=59}}
|-
| 60
| 1161.29
| [[35/18]], [[49/25]], [[55/28]], ''[[63/32]]'', ''[[64/33]]'', [[88/45]], [[96/49]], [[108/55]]
| {{UDnote|step=60}}
|-
|-
! ARMODUE NOMENCLATURE 8;3 RELATION
| 61
| 1180.65
| [[65/33]], [[77/39]], [[128/65]], [[180/91]], [[208/105]]
| {{UDnote|step=61}}
|-
|-
| <ul><li>'''Ɨ''' = Thick (1/8-tone up)</li><li>'''‡''' = Semisharp (1/4-tone up)</li><li>'''b''' = Flat (5/8-tone down)</li><li>'''◊''' = Node (sharp/flat blindspot 1/2-tone)</li><li>'''#''' = Sharp (5/8-tone up)</li><li>'''v''' = Semiflat (1/4-tone down)</li><li>'''⌐''' = Thin (1/8-tone down)</li></ul>
| 62
| 1200.00
| [[2/1]]
| {{UDnote|step=62}}
|}
|}
<nowiki />* 23-limit patent val, inconsistent intervals in ''italic''
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp4-szg}}
=== 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 ===
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 ====
<imagemap>
File:62-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:62-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:62-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:62-EDO_Revo_Sagittal.svg]]
</imagemap>
==== Evo-SZ flavor ====
<imagemap>
File:62-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:62-EDO_Evo-SZ_Sagittal.svg]]
</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.


{| class="wikitable center-1 right-2"
=== Armodue notation ===
; Armodue nomenclature 8;3 relation
* '''Ɨ''' = Thick (1/8-tone up)
* '''‡''' = Semisharp (1/4-tone up)
* '''b''' = Flat (5/8-tone down)
* '''◊''' = Node (sharp/flat blindspot 1/2-tone)
* '''#''' = Sharp (5/8-tone up)
* '''v''' = Semiflat (1/4-tone down)
* '''⌐''' = Thin (1/8-tone down)
 
{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
|-
! #
! colspan="2" | #
! [[Cent]]s
! Cents
! Armodue notation
! Armodue notation
! Approximate intervals
! Associated ratio
|-
|-
| 0
| 0
| 0.000
|
| 0.0
| 1
| 1
|  
|  
|-
|-
| 1
| 1
| 19.355
|
| 19.4
| 1Ɨ
| 1Ɨ
| 90/89
|  
|-
|-
| 2
| 2
| 38.710
|
| 38.7
| 1‡ (9#)
| 1‡ (9#)
| 45/44
|  
|-
|-
| 3
| 3
| 58.065
|
| 58.1
| 2b
| 2b
| 30/29
|  
|-
|-
| 4
| 4
| 77.419
|
| 77.4
| 1◊2
| 1◊2
| 23/22
|  
|-
|-
| 5
| 5
| 96.774
|
| 96.8
| 1#
| 1#
| 37/35, 18/17, 19/18
|  
|-
|-
| 6
| 6
| 116.129
|
| 116.1
| 2v
| 2v
| 31/29, 15/14, 16/15
|  
|-
|-
| 7
| 7
| 135.484
|
| 135.5
| 2⌐
| 2⌐
| 27/25, 13/12, 14/13
|  
|-
|-
| 8
| 8
| 154.839
|
| 154.8
| 2
| 2
| 12/11
| 11/10~12/11
|-
|-
| 9
| 9
| 174.194
|
| 174.2
| 2Ɨ
| 2Ɨ
| 11/10
|
|-
|-
| 10
| 10
| 193.548
|
| 193.5
| 2‡
| 2‡
| 19/17, 9/8, 10/9
|  
|-
|-
| 11
| 11
| 212.903
|
| 212.9
| 3b
| 3b
| 17/15, 9/8
| 8/7
|-
|-
| 12
| 12
| 232.258
|
| 232.3
| 2◊3
| 2◊3
| 8/7
|  
|-
|-
| 13
| 13
| 251.613
|
| 251.6
| 2#
| 2#
| 15/13
|  
|-
|-
| 14
| 14
| 270.968
|  
| 271.0
| 3v
| 3v
| 7/6
|  
|-
|-
| 15
| 15
| 290.323
|
| 290.3
| 3⌐
| 3⌐
|  
|  
|-
|-
| 16
| 16
| 309.677
|
| 309.7
| 3
| 3
| 6/5
| 6/5~7/6
|-
|-
| 17
| 17
| 329.032
|
| 329.0
| 3Ɨ
| 3Ɨ
|  
|  
|-
|-
| 18
| 18
| 348.387
|
| 348.4
| 3‡
| 3‡
| 11/9
|  
|-
|-
| 19
| 19
| 367.742
| ·
| 367.7
| 4b
| 4b
| ·
| 5/4
|-
|-
| 20
| 20
| 387.097
|
| 387.1
| 3◊4
| 3◊4
| 5/4
|  
|-
|-
| 21
| 21
| 406.452
|
| 406.5
| 3#
| 3#
|  
|  
|-
|-
| 22
| 22
| 425.806
|
| 425.8
| 4v (5b)
| 4v (5b)
|  
|  
|-
|-
| 23
| 23
| 445.161
|
| 445.2
| 4⌐
| 4⌐
|  
|  
|-
|-
| 24
| 24
| 464.516
|
| 464.5
| 4
| 4
|  
|  
|-
|-
| 25
| 25
| 483.871
|
| 483.9
| 4Ɨ (5v)
| 4Ɨ (5v)
|  
|  
|-
|-
| 26
| 26
| 503.226
|
| 503.2
| 5⌐ (4‡)
| 5⌐ (4‡)
| 4/3
|  
|-
|-
| 27
| 27
| 522.581
| ·
| 522.6
| 5
| 5
| ·
| 4/3~11/8
|-
|-
| 28
| 28
| 541.935
|
| 541.9
| 5Ɨ
| 5Ɨ
|  
|  
|-
|-
| 29
| 29
| 561.290
|
| 561.3
| 5‡ (4#)
| 5‡ (4#)
|  
|  
|-
|-
| 30
| 30
| 580.645
|
| 580.6
| 6b
| 6b
| 7/5
| 10/7
|-
|-
| 31
| 31
| 600.000
|
| 600.0
| 5◊6
| 5◊6
|  
|  
|-
|-
| 32
| 32
| 619.355
|
| 619.4
| 5#
| 5#
| 10/7
| 7/5
|-
|-
| 33
| 33
| 638.710
|
| 638.7
| 6v
| 6v
|  
|  
|-
|-
| 34
| 34
| 658.065
|
| 658.1
| 6⌐
| 6⌐
|  
|  
|-
|-
| 35
| 35
| 677.419
| ·
| 677.4
| 6
| 6
| ·
| 3/2~16/11
|-
|-
| 36
| 36
| 696.774
|
| 696.8
| 6Ɨ
| 6Ɨ
| |3/2
|  
|-
|-
| 37
| 37
| 716.129
|
| 716.1
| 6‡
| 6‡
|  
|  
|-
|-
| 38
| 38
| 735.484
|
| 735.5
| 7b
| 7b
|  
|  
|-
|-
| 39
| 39
| 754.839
|
| 754.8
| 6◊7
| 6◊7
|  
|  
|-
|-
| 40
| 40
| 774.194
|
| 774.2
| 6#
| 6#
|  
|  
|-
|-
| 41
| 41
| 793.548
|
| 793.5
| 7v
| 7v
|  
|  
|-
|-
| 42
| 42
| 812.903
|
| 812.9
| 7⌐
| 7⌐
| 8/5
|  
|-
|-
| 43
| 43
| 832.258
| ·
| 832.3
| 7
| 7
| ·
| 8/5
|-
|-
| 44
| 44
| 851.613
|
| 851.6
| 7Ɨ
| 7Ɨ
| 18/11
|  
|-
|-
| 45
| 45
| 870.968
|  
| 871.0
| 7‡
| 7‡
|  
|  
|-
|-
| 46
| 46
| 890.323
|
| 890.3
| 8b
| 8b
| 5/3
| 5/3~12/7
|-
|-
| 47
| 47
| 909.677
|
| 909.7
| 7◊8
| 7◊8
|  
|  
|-
|-
| 48
| 48
| 929.032
|
| 929.0
| 7#
| 7#
| 12/7
|  
|-
|-
| 49
| 49
| 948.387
|
| 948.4
| 8v
| 8v
| 26/15
|  
|-
|-
| 50
| 50
| 967.742
|
| 967.7
| 8⌐
| 8⌐
| 7/4
|  
|-
|-
| 51
| 51
| 987.097
|
| 987.1
| 8
| 8
| 16/9
| 7/4
|-
|-
| 52
| 52
| 1006.452
|
| 1006.5
| 8Ɨ
| 8Ɨ
|  
|  
|-
|-
| 53
| 53
| 1025.806
|
| 1025.8
| 8‡
| 8‡
|  
|  
|-
|-
| 54
| 54
| 1045.161
|
| 1045.2
| 9b
| 9b
|  
| 11/6~20/11
|-
|-
| 55
| 55
| 1064.516
|
| 1064.5
| 8◊9
| 8◊9
|  
|  
|-
|-
| 56
| 56
| 1083.871
|
| 1083.9
| 8#
| 8#
|  
|  
|-
|-
| 57
| 57
| 1103.226
|
| 1103.2
| 9v (1b)
| 9v (1b)
|  
|  
|-
|-
| 58
| 58
| 1122.581
|
| 1122.6
| 9⌐
| 9⌐
|  
|  
|-
|-
| 59
| 59
| 1141.936
|
| 1141.9
| 9
| 9
|  
|  
|-
|-
| 60
| 60
| 1161.290
|
| 1161.3
| 9Ɨ (1v)
| 9Ɨ (1v)
|  
|  
|-
|-
| 61
| 61
| 1180.645
|
| 1180.6
| 1⌐ (9‡)
| 1⌐ (9‡)
|  
|  
|-
|-
| 62
| 62
| 1200.000
|
| 1200.0
| 1
| 1
|  
|  
|}
|}


[[Category:Equal divisions of the octave]]
== Regular temperament properties ==
62edo is contorted 31edo through the 11-limit.
{| 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]] (%)
|-
| 2.3.5.7.11.13
| 81/80, 99/98, 121/120, 126/125, 169/168
| {{mapping| 62 98 144 174 214 229 }}
| +1.38
| 1.41
| 7.28
|-
| 2.3.5.7.11.13.17
| 81/80, 99/98, 121/120, 126/125, 169/168, 221/220
| {{mapping| 62 98 144 174 214 229 253 }}
| +1.47
| 1.32
| 6.83
|-
| 2.3.5.7.11.13.17.19
| 81/80, 99/98, 121/120, 126/125, 153/152, 169/168, 209/208
| {{mapping| 62 98 144 174 214 229 253 263 }}
| +1.50
| 1.24
| 6.40
|-
| 2.3.5.7.11.13.17.19.23
| 81/80, 99/98, 121/120, 126/125, 153/152, 161/160, 169/168, 209/208
| {{mapping| 62 98 144 174 214 229 253 263 280 }}
| +1.55
| 1.18
| 6.09
|}
 
=== 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
| 3\62
| 58.1
| 27/26
| [[Hemisecordite]]
|-
| 1
| 7\62
| 135.5
| 13/12
| [[Doublethink]]
|-
| 1
| 13\62
| 251.6
| 15/13
| [[Hemimeantone]]
|-
| 1
| 17\62
| 329.0
| 16/11
| [[Mabon]]
|-
| 1
| 29\62
| 561.3
| 18/13
| [[Demivalentine]]
|-
| 2
| 3\62
| 58.1
| 27/26
| [[Semihemisecordite]]
|-
| 2
| 4\62
| 77.4
| 21/20
| [[Semivalentine]]
|-
| 2
| 6\62
| 116.1
| 15/14
| [[Semimiracle]]
|-
| 2
| 26\62
| 503.2
| 4/3
| [[Semimeantone]]
|-
| 31
| 29\62<br>(1\62)
| 561.3<br>(19.4)
| 11/8<br>(196/195)
| [[Kumhar]] (62e)
|-
| 31
| 19\62<br>(1\62)
| 367.7<br>(19.4)
| 16/13<br>(77/76)
| [[Gallium]]
|}
<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 ==
 
=== 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"