62edo: Difference between revisions

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== Theory ==

62 = 2 × 31 and the [[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. 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 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.

It provides the [[optimal patent val]] for [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.

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! Steps

! Cents

! Approximate ~~Ratios~~*

! Approximate ratios*

! [[Ups and downs notation~~|Ups and Downs Notation~~]]

! [[Ups and downs notation]]

|-

| 0

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| {{UDnote|step=62}}

|}

<nowiki>*~~</nowiki>~~ 23-limit patent val, inconsistent intervals in ''italic''

<nowiki />* 23-limit patent val, inconsistent intervals in ''italic''

== Notation ==

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{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"

|-

! colspan="2" | #

! Cents

! Armodue ~~Notation~~

! Armodue notation

! Associated ~~Ratio~~

! Associated ratio

|-

| 0

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== 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|Comma List]]~~

~~! rowspan="2" | [[Mapping]]~~

~~! rowspan="2" | Optimal 8ve Stretch (¢)~~

~~! colspan="2" | Tuning Error~~

|-

~~! [[TE error|Absolute]] (¢)~~

~~! [[TE simple badness|Relative]] (%)~~

|-

| 2.3.5.7.11.13

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| 1.18

| 6.09

|}

=== Rank-2 temperaments ===

{~~| class="wikitable center~~-~~all left-5"~~

~~! Periods per 8ve~~

! Generator*

! Cents*

! Associated Ratio*

~~! Temperaments~~

|-

| 1

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|-

| 31

| 29\62 (1\62)

| 29\62 (1\62)

| 561.29 (19.35)

| 561.29 (19.35)

| 11/8 (196/195)

| 11/8 (196/195)

| [[Kumhar]] (62e)

|-

| 31

| 19\62 (1\62)

| 19\62 (1\62)

| 367.74 (19.35)

| 367.74 (19.35)

| 16/13 (77/76)

| 16/13 (77/76)

| [[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~~

@@ Line 3: / Line 3: @@
 == Theory ==
-= 2 × 31 and the [[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. 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 &times; 31}} and the [[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. 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.
 It provides the [[optimal patent val]] for [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.
@@ Line 27: / Line 27: @@
 ! Steps
 ! Cents
-! Approximate Ratios*
+! Approximate ratios*
-! [[Ups and downs notation|Ups and Downs Notation]]
+! [[Ups and downs notation]]
 |-
 | 0
@@ Line 345: / Line 345: @@
 | {{UDnote|step=62}}
 |}
-<nowiki>*</nowiki> 23-limit patent val, inconsistent intervals in ''italic''
+<nowiki />* 23-limit patent val, inconsistent intervals in ''italic''
 == Notation ==
@@ Line 365: / Line 365: @@
 {| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
 |-
-! colspan="2" | #
+! colspan="2" | &#35;
 ! Cents
-! Armodue Notation
+! Armodue notation
-! Associated Ratio
+! Associated ratio
 |-
 | 0
@@ Line 751: / Line 751: @@
 == Regular temperament properties ==
 edo is contorted 31edo through the 11-limit.
-{| class="wikitable center-4 center-5 center-6"
+{{comma basis begin}}
-! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|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
@@ Line 788: / Line 780: @@
 | 1.18
 | 6.09
-|}
+{{comma basis end}}
 === Rank-2 temperaments ===
-{| class="wikitable center-all left-5"
+{{rank-2 begin}}
-! Periods<br>per 8ve
-! Generator*
-! Cents*
-! Associated<br>Ratio*
-! Temperaments
 |-
 | 1
@@ Line 847: / Line 834: @@
 |-
 | 31
-| 29\62<br>(1\62)
+| 29\62<br />(1\62)
-| 561.29<br>(19.35)
+| 561.29<br />(19.35)
-| 11/8<br>(196/195)
+| 11/8<br />(196/195)
 | [[Kumhar]] (62e)
 |-
 | 31
-| 19\62<br>(1\62)
+| 19\62<br />(1\62)
-| 367.74<br>(19.35)
+| 367.74<br />(19.35)
-| 16/13<br>(77/76)
+| 16/13<br />(77/76)
 | [[Gallium]]
-|}
+{{rank-2 end}}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+{{orf}}