Xenharmonic Wiki

62edo: Difference between revisions

VisualWikitext
Wikispaces>JosephRuhf
**Imported revision 339785034 - Original comment: **
 
→Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 62edo'' (2025)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2012-05-26 21:33:38 UTC</tt>.<br>
 
: The original revision id was <tt>339785034</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
{{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.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[hemimeantone]] temperaments.  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">62 edo, being two parallel tracks of 31, is a 1/10-tone meantone system like 60, 64, 66b, 68b and 70b and the best of all the six since 31 is the "real" 1/5-tone meantone system (33 is nearly equivalent to fifths of 10/9 and 35 barely qualifies as fifths of a tone because its so-called wholetone is 11¢ flat of 10/9 and 30, 32 and 34 aren't even meantones). It is also strong as an 1/8-tone Armodue-Hornbostel system, with the 6th being 35 steps. However, 31 is a "false" quarter-tone system with respect to the same temperament since the Armodue-Hornbostel whole tone is simplest when achieved by a chain of 5 of the septimal minor third at 7/31, which is a good generator for [[Orwell]]. This makes 8/62 an Orwell whole tone as well, so 62 is twice an 1/8-tone system; but this is no real surprise since 1/8-tone systems will have 40 to 80 divisions in the octave as a matter of course.</pre></div>
 
<h4>Original HTML content:</h4>
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]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;62edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;62 edo, being two parallel tracks of 31, is a 1/10-tone meantone system like 60, 64, 66b, 68b and 70b and the best of all the six since 31 is the &amp;quot;real&amp;quot; 1/5-tone meantone system (33 is nearly equivalent to fifths of 10/9 and 35 barely qualifies as fifths of a tone because its so-called wholetone is 11¢ flat of 10/9 and 30, 32 and 34 aren't even meantones). It is also strong as an 1/8-tone Armodue-Hornbostel system, with the 6th being 35 steps. However, 31 is a &amp;quot;false&amp;quot; quarter-tone system with respect to the same temperament since the Armodue-Hornbostel whole tone is simplest when achieved by a chain of 5 of the septimal minor third at 7/31, which is a good generator for &lt;a class="wiki_link" href="/Orwell"&gt;Orwell&lt;/a&gt;. This makes 8/62 an Orwell whole tone as well, so 62 is twice an 1/8-tone system; but this is no real surprise since 1/8-tone systems will have 40 to 80 divisions in the octave as a matter of course.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== 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.
 
== Intervals ==
{| 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 ==
=== 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 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" | &#35;
! Cents
! Armodue notation
! Associated ratio
|-
| 0
|
| 0.0
| 1
|
|-
| 1
|
| 19.4
| 1Ɨ
|
|-
| 2
|
| 38.7
| 1‡ (9#)
|
|-
| 3
|
| 58.1
| 2b
|
|-
| 4
|
| 77.4
| 1◊2
|
|-
| 5
|
| 96.8
| 1#
|
|-
| 6
|
| 116.1
| 2v
|
|-
| 7
|
| 135.5
| 2⌐
|
|-
| 8
|
| 154.8
| 2
| 11/10~12/11
|-
| 9
|
| 174.2
| 2Ɨ
|
|-
| 10
|
| 193.5
| 2‡
|
|-
| 11
|
| 212.9
| 3b
| 8/7
|-
| 12
|
| 232.3
| 2◊3
|
|-
| 13
|
| 251.6
| 2#
|
|-
| 14
|
| 271.0
| 3v
|
|-
| 15
|
| 290.3
| 3⌐
|
|-
| 16
|
| 309.7
| 3
| 6/5~7/6
|-
| 17
|
| 329.0
| 3Ɨ
|
|-
| 18
|
| 348.4
| 3‡
|
|-
| 19
| ·
| 367.7
| 4b
| 5/4
|-
| 20
|
| 387.1
| 3◊4
|
|-
| 21
|
| 406.5
| 3#
|
|-
| 22
|
| 425.8
| 4v (5b)
|
|-
| 23
|
| 445.2
| 4⌐
|
|-
| 24
|
| 464.5
| 4
|
|-
| 25
|
| 483.9
| 4Ɨ (5v)
|
|-
| 26
|
| 503.2
| 5⌐ (4‡)
|
|-
| 27
| ·
| 522.6
| 5
| 4/3~11/8
|-
| 28
|
| 541.9
| 5Ɨ
|
|-
| 29
|
| 561.3
| 5‡ (4#)
|
|-
| 30
|
| 580.6
| 6b
| 10/7
|-
| 31
|
| 600.0
| 5◊6
|
|-
| 32
|
| 619.4
| 5#
| 7/5
|-
| 33
|
| 638.7
| 6v
|
|-
| 34
|
| 658.1
| 6⌐
|
|-
| 35
| ·
| 677.4
| 6
| 3/2~16/11
|-
| 36
|
| 696.8
| 6Ɨ
|
|-
| 37
|
| 716.1
| 6‡
|
|-
| 38
|
| 735.5
| 7b
|
|-
| 39
|
| 754.8
| 6◊7
|
|-
| 40
|
| 774.2
| 6#
|
|-
| 41
|
| 793.5
| 7v
|
|-
| 42
|
| 812.9
| 7⌐
|
|-
| 43
| ·
| 832.3
| 7
| 8/5
|-
| 44
|
| 851.6
| 7Ɨ
|
|-
| 45
|
| 871.0
| 7‡
|
|-
| 46
|
| 890.3
| 8b
| 5/3~12/7
|-
| 47
|
| 909.7
| 7◊8
|
|-
| 48
|
| 929.0
| 7#
|
|-
| 49
|
| 948.4
| 8v
|
|-
| 50
|
| 967.7
| 8⌐
|
|-
| 51
|
| 987.1
| 8
| 7/4
|-
| 52
|
| 1006.5
| 8Ɨ
|
|-
| 53
|
| 1025.8
| 8‡
|
|-
| 54
|
| 1045.2
| 9b
| 11/6~20/11
|-
| 55
|
| 1064.5
| 8◊9
|
|-
| 56
|
| 1083.9
| 8#
|
|-
| 57
|
| 1103.2
| 9v (1b)
|
|-
| 58
|
| 1122.6
| 9⌐
|
|-
| 59
|
| 1141.9
| 9
|
|-
| 60
|
| 1161.3
| 9Ɨ (1v)
|
|-
| 61
|
| 1180.6
| 1⌐ (9‡)
|
|-
| 62
|
| 1200.0
| 1
|
|}
 
== 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 ==
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 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)
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