56edo: Difference between revisions

Line 3:

== Theory ==

It shares ~~it's~~ near perfect major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]].

56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo.

56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[~~Sycamore family #Septimal sycamore|~~sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.

56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.

=== Prime harmonics ===

=== Subsets and supersets ===

56edo has subset edos {{EDOs|1, 2, 4, 7, 8, 14, 28}}.

Since 56 factors into 2<sup>3</sup> × 7, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}.

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the unrounded value being 55.7976. Barium temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.

== Intervals ==

~~The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible.~~

{| class="wikitable center-all right-2 left-3"

! #

! Cents

! Approximate Ratios

! Approximate Ratios<nowiki>*</nowiki>

! [[Ups and downs notation]]

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

|-

| 0

Line 31:

Line 29:

| 1

| 21.429

| [[49/48]], [[64/63]]

| ''[[49/48]]'', [[64/63]]

| {{UDnote|step=1}}

|-

| 2

| 42.857

| [[28/27]], [[50/49]], [[81/80]]

| ''[[28/27]]'', [[50/49]], ''[[81/80]]''

| {{UDnote|step=2}}

|-

| 3

| 64.286

| [[25/24]], [[36/35]], [[33/32]]

| [[25/24]], ''[[36/35]]'', ''[[33/32]]''

| {{UDnote|step=3}}

|-

Line 66:

Line 64:

| 8

| 171.429

| [[10/9]], [[11/10]]

| ''[[10/9]]'', [[11/10]]

| {{UDnote|step=8}}

|-

Line 131:

Line 129:

| 21

| 450.000

| [[9/7]], [[13/10]]

| ''[[9/7]]'', [[13/10]]

| {{UDnote|step=21}}

|-

Line 151:

Line 149:

| 25

| 535.714

| [[27/20]], [[15/11]]

| ''[[27/20]]'', [[15/11]]

| {{UDnote|step=25}}

|-

Line 174:

Line 172:

| …

|}

<nowiki>*</nowiki> The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible. Inconsistent intervals are marked ''italic''.

== Commas ==

Line 186:

Line 185:

== Music ==

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=xWKa59qDkXQ Prelude & Fugue in Pajara]

* [https://www.youtube.com/watch?v=xWKa59qDkXQ ''Prelude & Fugue in Pajara''] (2020) – in pajara, 56edo tuning

* [https://www.youtube.com/watch?v=3oO1SIVWBgI Mirror Canon in F]

* [https://www.youtube.com/watch?v=3oO1SIVWBgI ''Mirror Canon in F''] (2020)

* [https://www.youtube.com/watch?v=s1h083BRWXU Canon 3 in 1 on a Ground]

* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)

== See also ==

* [[Lumatone mapping for 56edo]]

[[Lumatone mapping for 56edo]]

~~[[Category:56edo| ]] ~~

~~[[Category:Equal divisions of the octave|##]] ~~

[[Category:Hemithirds]]

[[Category:Keen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-It shares it's near perfect major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]].
+edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo.
-edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[Sycamore family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.
+edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.
 === Prime harmonics ===
-{{harmonics in equal|56}}
+{{Harmonics in equal|56}}
 === Subsets and supersets ===
-edo has subset edos {{EDOs|1, 2, 4, 7, 8, 14, 28}}.
+Since 56 factors into 2<sup>3</sup> × 7, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}.
 One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the unrounded value being 55.7976. Barium temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.
 == Intervals ==
-The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible.
 {| class="wikitable center-all right-2 left-3"
 ! #
 ! Cents
-! Approximate Ratios
+! Approximate Ratios<nowiki>*</nowiki>
-! [[Ups and downs notation]]
+! [[Ups and downs notation|Ups and Downs Notation]]
 |-
 | 0
@@ Line 31: / Line 29: @@
 | 1
 | 21.429
-| [[49/48]], [[64/63]]
+| ''[[49/48]]'', [[64/63]]
 | {{UDnote|step=1}}
 |-
 | 2
 | 42.857
-| [[28/27]], [[50/49]], [[81/80]]
+| ''[[28/27]]'', [[50/49]], ''[[81/80]]''
 | {{UDnote|step=2}}
 |-
 | 3
 | 64.286
-| [[25/24]], [[36/35]], [[33/32]]
+| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
 | {{UDnote|step=3}}
 |-
@@ Line 66: / Line 64: @@
 | 8
 | 171.429
-| [[10/9]], [[11/10]]
+| ''[[10/9]]'', [[11/10]]
 | {{UDnote|step=8}}
 |-
@@ Line 131: / Line 129: @@
 | 21
 | 450.000
-| [[9/7]], [[13/10]]
+| ''[[9/7]]'', [[13/10]]
 | {{UDnote|step=21}}
 |-
@@ Line 151: / Line 149: @@
 | 25
 | 535.714
-| [[27/20]], [[15/11]]
+| ''[[27/20]]'', [[15/11]]
 | {{UDnote|step=25}}
 |-
@@ Line 174: / Line 172: @@
 | …
 |}
+<nowiki>*</nowiki> The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible. Inconsistent intervals are marked ''italic''.
 == Commas ==
@@ Line 186: / Line 185: @@
 == Music ==
 ; [[Claudi Meneghin]]
-* [https://www.youtube.com/watch?v=xWKa59qDkXQ Prelude & Fugue in Pajara]
+* [https://www.youtube.com/watch?v=xWKa59qDkXQ ''Prelude & Fugue in Pajara''] (2020) – in pajara, 56edo tuning
-* [https://www.youtube.com/watch?v=3oO1SIVWBgI Mirror Canon in F]
+* [https://www.youtube.com/watch?v=3oO1SIVWBgI ''Mirror Canon in F''] (2020)
-* [https://www.youtube.com/watch?v=s1h083BRWXU Canon 3 in 1 on a Ground]
+* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)
 == See also ==
+* [[Lumatone mapping for 56edo]]
-[[Lumatone mapping for 56edo]]
-[[Category:56edo| ]] <!-- main article -->
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
 [[Category:Hemithirds]]
 [[Category:Keen]]