62edo: Difference between revisions

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== Theory ==

~~62edo's~~ [[patent val]] of 62edo 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. 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 ~~relative~~ size differences between 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]].

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=== Subsets and supersets ===

Since 62 factors into ~~{{factorization|62}}~~, 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.

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 ==

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== Notation ==

=== Ups and downs 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]]:

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.

[[Alternative symbols for ups and downs notation]] uses sharps and flats and quarter-tone accidentals combined with arrows, borrowed from extended [[Helmholtz–Ellis 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]].

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== Approximation to JI ==

=== Zeta peak index ===

{~~| class="wikitable center-all"~~

{{ZPI

|-

| zpi = 314

~~! colspan="3"~~ | ~~Tuning~~

| steps = 61.9380472360525

~~! colspan~~=~~"3" | Strength~~

| step size = 19.3741981471691

~~! colspan="2"~~ | ~~Closest edo~~

| tempered height = 6.262952

~~! colspan~~=~~"2" | Integer limit~~

| pure height = 4.11259

|-

| integral = 0.952068

~~! ZPI~~

| gap = 15.026453

~~! Steps per octave~~

| octave = 1201.20028512448

~~! Step size (cents)~~

| consistent = 8

~~! Height~~

| distinct = 8

~~! Integral~~

}}

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[314zpi]]~~

| 61.9380472360525

| 19.3741981471691

| 6.262952

| 0.952068

| 15.026453

| ~~62edo~~

| 1201.20028512448

| 8

|}

== Regular temperament properties ==

Line 873:

Line 856:

| 1

| 3\62

| 58.06

| 58.1

| 27/26

| [[Hemisecordite]]

Line 879:

Line 862:

| 1

| 7\62

| 135.48

| 135.5

| 13/12

| [[Doublethink]]

Line 885:

Line 868:

| 1

| 13\62

| 251.61

| 251.6

| 15/13

| [[Hemimeantone]]

Line 891:

Line 874:

| 1

| 17\62

| 329.03

| 329.0

| 16/11

| [[Mabon]]

|-

| 1

| 29\62

| 561.3

| 18/13

| [[Demivalentine]]

|-

| 2

| 3\62

| 58.06

| 58.1

| 27/26

| [[Semihemisecordite]]

Line 903:

Line 892:

| 2

| 4\62

| 77.42

| 77.4

| 21/20

| [[Semivalentine]]

Line 909:

Line 898:

| 2

| 6\62

| 116.13

| 116.1

| 15/14

| [[Semimiracle]]

Line 915:

Line 904:

| 2

| 26\62

| 503.22

| 503.2

| 4/3

| [[Semimeantone]]

Line 921:

Line 910:

| 31

| 29\62 (1\62)

| 561.29 (19.35)

| 561.3 (19.4)

| 11/8 (196/195)

| [[Kumhar]] (62e)

Line 927:

Line 916:

| 31

| 19\62 (1\62)

| 367.74 (19.35)

| 367.7 (19.4)

| 16/13 (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)

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|62}}
+{{ED intro}}
 == Theory ==
-edo's [[patent val]] of 62edo 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. 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 relative size differences between 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]].
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 62 factors into {{factorization|62}}, 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 ===
 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 ==
@@ Line 349: / Line 349: @@
 == Notation ==
 === Ups and downs notation ===
-edo 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]]:
+edo 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]].
@@ Line 789: / Line 789: @@
 == Approximation to JI ==
 === Zeta peak index ===
-{| class="wikitable center-all"
+{{ZPI
-|-
+| zpi = 314
-! colspan="3" | Tuning
+| steps = 61.9380472360525
-! colspan="3" | Strength
+| step size = 19.3741981471691
-! colspan="2" | Closest edo
+| tempered height = 6.262952
-! colspan="2" | Integer limit
+| pure height = 4.11259
-|-
+| integral = 0.952068
-! ZPI
+| gap = 15.026453
-! Steps per octave
+| octave = 1201.20028512448
-! Step size (cents)
+| consistent = 8
-! Height
+| distinct = 8
-! Integral
+}}
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[314zpi]]
-| 61.9380472360525
-| 19.3741981471691
-| 6.262952
-| 0.952068
-| 15.026453
-| 62edo
-| 1201.20028512448
-| 8
-| 8
-|}
 == Regular temperament properties ==
@@ Line 873: / Line 856: @@
 | 1
 | 3\62
-| 58.06
+| 58.1
 | 27/26
 | [[Hemisecordite]]
@@ Line 879: / Line 862: @@
 | 1
 | 7\62
-| 135.48
+| 135.5
 | 13/12
 | [[Doublethink]]
@@ Line 885: / Line 868: @@
 | 1
 | 13\62
-| 251.61
+| 251.6
 | 15/13
 | [[Hemimeantone]]
@@ Line 891: / Line 874: @@
 | 1
 | 17\62
-| 329.03
+| 329.0
 | 16/11
 | [[Mabon]]
+|-
+| 1
+| 29\62
+| 561.3
+| 18/13
+| [[Demivalentine]]
 |-
 | 2
 | 3\62
-| 58.06
+| 58.1
 | 27/26
 | [[Semihemisecordite]]
@@ Line 903: / Line 892: @@
 | 2
 | 4\62
-| 77.42
+| 77.4
 | 21/20
 | [[Semivalentine]]
@@ Line 909: / Line 898: @@
 | 2
 | 6\62
-| 116.13
+| 116.1
 | 15/14
 | [[Semimiracle]]
@@ Line 915: / Line 904: @@
 | 2
 | 26\62
-| 503.22
+| 503.2
 | 4/3
 | [[Semimeantone]]
@@ Line 921: / Line 910: @@
 | 31
 | 29\62<br>(1\62)
-| 561.29<br>(19.35)
+| 561.3<br>(19.4)
 | 11/8<br>(196/195)
 | [[Kumhar]] (62e)
@@ Line 927: / Line 916: @@
 | 31
 | 19\62<br>(1\62)
-| 367.74<br>(19.35)
+| 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:
+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.
+-string bass
+0 11 22 33
+This adds an interval of a major 7th (minus an 8ve) at the first string, 1st fret.
+-string guitar
+11 22 33 44 55
+”Major” 020131
+-string guitar
+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)