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62edo: Difference between revisions

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


== Theory ==
== Theory ==
62 = 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it 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. Interestingly, the relative size differences of consecutive harmonics are well preserved 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.  
{{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 [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|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 septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].
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{{Harmonics in equal|62}}
{{Harmonics in equal|62}}


=== Miscellaneous properties ===
=== 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.  
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 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 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 an unnamed extension to [[valentine]] in the 13-limit.
The 15 & 62 temperament, corresponding to the leap day cycle, is [[demivalentine]] in the 13-limit.


== Intervals ==
== Intervals ==
{{Interval table}}
{| class="wikitable center-all right-2 left-3"
|-
! 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=10}}
|-
| 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}}
|-
| 61
| 1180.65
| 65/33, 77/39, 128/65, 180/91, 208/105
| {{UDnote|step=61}}
|-
| 62
| 1200.00
| 2/1
| {{UDnote|step=62}}
|}
<nowiki />* 23-limit patent val, inconsistent intervals in ''italic''


== Notation ==
== Notation ==
=== Ups and downs notation ===
62edo can be notated with [[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}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats and quarter-tone accidentals combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
=== 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]].
==== 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.
=== Armodue notation ===
=== Armodue notation ===
Armodue nomenclature 8;3 relation
; Armodue nomenclature 8;3 relation
* '''Ɨ''' = Thick (1/8-tone up)
* '''Ɨ''' = Thick (1/8-tone up)
* '''‡''' = Semisharp (1/4-tone up)
* '''‡''' = Semisharp (1/4-tone up)
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* '''⌐''' = Thin (1/8-tone down)
* '''⌐''' = Thin (1/8-tone down)


{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
|-
! colspan="2" | #
! colspan="2" | &#35;
! [[Cent]]s
! Cents
! Armodue Notation
! Armodue notation
! Associated Ratio
! Associated ratio
|-
|-
| 0
| 0
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|  
|  
|}
|}
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 314
| steps = 61.9380472360525
| step size = 19.3741981471691
| tempered height = 6.262952
| pure height = 4.11259
| integral = 0.952068
| gap = 15.026453
| octave = 1201.20028512448
| consistent = 8
| distinct = 8
}}


== Regular temperament properties ==
== Regular temperament properties ==
62edo is contorted 31edo through the 11-limit.
62edo is contorted 31edo through the 11-limit.
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal <br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
| 3\62
| 3\62
| 58.06
| 58.1
| 27/26
| 27/26
| [[Hemisecordite]]
| [[Hemisecordite]]
Line 476: Line 862:
| 1
| 1
| 7\62
| 7\62
| 135.48
| 135.5
| 13/12
| 13/12
| [[Doublethink]]
| [[Doublethink]]
Line 482: Line 868:
| 1
| 1
| 13\62
| 13\62
| 251.61
| 251.6
| 15/13
| 15/13
| [[Hemimeantone]]
| [[Hemimeantone]]
Line 488: Line 874:
| 1
| 1
| 17\62
| 17\62
| 329.03
| 329.0
| 16/11
| 16/11
| [[Mabon]]
| [[Mabon]]
|-
| 1
| 29\62
| 561.3
| 18/13
| [[Demivalentine]]
|-
|-
| 2
| 2
| 3\62
| 3\62
| 58.06
| 58.1
| 27/26
| 27/26
| [[Semihemisecordite]]
| [[Semihemisecordite]]
Line 500: Line 892:
| 2
| 2
| 4\62
| 4\62
| 77.42
| 77.4
| 21/20
| 21/20
| [[Semivalentine]]
| [[Semivalentine]]
Line 506: Line 898:
| 2
| 2
| 6\62
| 6\62
| 116.13
| 116.1
| 15/14
| 15/14
| [[Semimiracle]]
| [[Semimiracle]]
Line 512: Line 904:
| 2
| 2
| 26\62
| 26\62
| 503.22
| 503.2
| 4/3
| 4/3
| [[Semimeantone]]
| [[Semimeantone]]
Line 518: Line 910:
| 31
| 31
| 29\62<br>(1\62)
| 29\62<br>(1\62)
| 561.29<br>(19.35)
| 561.3<br>(19.4)
| 11/8<br>(196/195)
| 11/8<br>(196/195)
| [[Kumhar]] (62e)
| [[Kumhar]] (62e)
Line 524: Line 916:
| 31
| 31
| 19\62<br>(1\62)
| 19\62<br>(1\62)
| 367.74<br>(19.35)
| 367.7<br>(19.4)
| 16/13<br>(77/76)
| 16/13<br>(77/76)
| [[Gallium]]
| [[Gallium]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== 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)
Retrieved from "https://en.xen.wiki/w/62edo"