62edo: Difference between revisions

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== Theory ==

~~{{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. 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 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.

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 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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== 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 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]]:

===Sagittal 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]].

~~====Evo flavor====~~

==== Evo flavor ====

File:62-EDO_Evo_Sagittal.svg

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</imagemap>

====Revo flavor====

==== Revo flavor ====

File:62-EDO_Revo_Sagittal.svg

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</imagemap>

====Evo-SZ flavor====

==== Evo-SZ flavor ====

File:62-EDO_Evo-SZ_Sagittal.svg

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| [[Gallium]]

|}

<nowiki>*~~</nowiki>~~ [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

==Zeta properties==

===Zeta peak index===

== Zeta properties ==

=== Zeta peak index ===

{| class="wikitable"

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

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

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

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

|-

!ZPI

! colspan="3" | Tuning

!Steps per octave

! colspan="3" | Strength

!Step size (cents)

! colspan="2" | Closest EDO

!Height

! colspan="2" | Integer limit

!Integral

|-

!Gap

! ZPI

!EDO

! Steps per octave

!Octave (cents)

! Step size (cents)

!Consistent

! Height

!Distinct

! Integral

! Gap

! EDO

! Octave (cents)

! Consistent

! Distinct

|-

|[[314zpi]]

| [[314zpi]]

|61.9380472360525

| 61.9380472360525

|19.3741981471691

| 19.3741981471691

|6.262952

| 6.262952

|0.952068

| 0.952068

|15.026453

| 15.026453

|62edo

| 62edo

|1201.20028512448

| 1201.20028512448

|8

| 8

|8

| 8

|}

@@ Line 3: / Line 3: @@
 == Theory ==
-{{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. 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 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.
+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 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 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 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]]:
 {{Sharpness-sharp4}}
-===Sagittal 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]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:62-EDO_Evo_Sagittal.svg
@@ Line 367: / Line 367: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:62-EDO_Revo_Sagittal.svg
@@ Line 379: / Line 378: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:62-EDO_Evo-SZ_Sagittal.svg
@@ Line 901: / Line 899: @@
 | [[Gallium]]
 |}
-<nowiki>*</nowiki> [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-==Zeta properties==
-===Zeta peak index===
+== Zeta properties ==
+=== Zeta peak index ===
 {| class="wikitable"
-! colspan="3" |Tuning
-! colspan="3" |Strength
-! colspan="2" |Closest EDO
-! colspan="2" |Integer limit
 |-
-!ZPI
+! colspan="3" | Tuning
-!Steps per octave
+! colspan="3" | Strength
-!Step size (cents)
+! colspan="2" | Closest EDO
-!Height
+! colspan="2" | Integer limit
-!Integral
+|-
-!Gap
+! ZPI
-!EDO
+! Steps per octave
-!Octave (cents)
+! Step size (cents)
-!Consistent
+! Height
-!Distinct
+! Integral
+! Gap
+! EDO
+! Octave (cents)
+! Consistent
+! Distinct
 |-
-|[[314zpi]]
+| [[314zpi]]
-|61.9380472360525
+| 61.9380472360525
-|19.3741981471691
+| 19.3741981471691
-|6.262952
+| 6.262952
-|0.952068
+| 0.952068
-|15.026453
+| 15.026453
-|62edo
+| 62edo
-|1201.20028512448
+| 1201.20028512448
-|8
+| 8
-|8
+| 8
 |}