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== Theory ==
== 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 ===
=== Prime harmonics ===
{{harmonics in equal|56}}
{{Harmonics in equal|56}}


=== Subsets and supersets ===
=== 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.  
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 ==
== 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"
{| class="wikitable center-all right-2 left-3"
! #
! #
! Cents
! Cents
! Approximate Ratios
! Approximate Ratios<nowiki>*</nowiki>
! [[Ups and downs notation]]
! [[Ups and downs notation|Ups and Downs Notation]]
|-
|-
| 0
| 0
Line 31: Line 29:
| 1
| 1
| 21.429
| 21.429
| [[49/48]], [[64/63]]
| ''[[49/48]]'', [[64/63]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 42.857
| 42.857
| [[28/27]], [[50/49]], [[81/80]]
| ''[[28/27]]'', [[50/49]], ''[[81/80]]''
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 64.286
| 64.286
| [[25/24]], [[36/35]], [[33/32]]
| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
Line 66: Line 64:
| 8
| 8
| 171.429
| 171.429
| [[10/9]], [[11/10]]
| ''[[10/9]]'', [[11/10]]
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
Line 131: Line 129:
| 21
| 21
| 450.000
| 450.000
| [[9/7]], [[13/10]]
| ''[[9/7]]'', [[13/10]]
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
Line 151: Line 149:
| 25
| 25
| 535.714
| 535.714
| [[27/20]], [[15/11]]
| ''[[27/20]]'', [[15/11]]
| {{UDnote|step=25}}
| {{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 ==
== Commas ==
Line 186: Line 185:
== Music ==
== Music ==
; [[Claudi Meneghin]]
; [[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 ==
== 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:Hemithirds]]
[[Category:Keen]]
[[Category:Keen]]

Revision as of 05:33, 17 July 2023

← 55edo 56edo 57edo →
Prime factorization 23 × 7
Step size 21.4286 ¢ 
Fifth 33\56 (707.143 ¢)
Semitones (A1:m2) 7:3 (150 ¢ : 64.29 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

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 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 ⟨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

Approximation of prime harmonics in 56edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +5.19 -0.60 -4.54 +5.82 -4.81 +2.19 +2.49 -6.85 -1.01 -9.32
Relative (%) +0.0 +24.2 -2.8 -21.2 +27.2 -22.5 +10.2 +11.6 -31.9 -4.7 -43.5
Steps
(reduced) 		56
(0) 		89
(33) 		130
(18) 		157
(45) 		194
(26) 		207
(39) 		229
(5) 		238
(14) 		253
(29) 		272
(48) 		277
(53)

Subsets and supersets

Since 56 factors into 23 × 7, 56edo has subset 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

# Cents Approximate Ratios* Ups and Downs Notation
0 0.000 1/1 D
1 21.429 49/48, 64/63 ^D, vvE♭
2 42.857 28/27, 50/49, 81/80 ^^D, vE♭
3 64.286 25/24, 36/35, 33/32 ^3D, E♭
4 85.714 21/20, 22/21 v3D♯, ^E♭
5 107.143 16/15 vvD♯, ^^E♭
6 128.571 15/14, 13/12, 14/13 vD♯, ^3E♭
7 150.000 12/11 D♯, v3E
8 171.429 10/9, 11/10 ^D♯, vvE
9 192.857 28/25 ^^D♯, vE
10 214.286 9/8 E
11 235.714 8/7 ^E, vvF
12 257.143 7/6, 15/13 ^^E, vF
13 278.571 75/64, 13/11 F
14 300.000 25/21 ^F, vvG♭
15 321.429 6/5 ^^F, vG♭
16 342.857 11/9, 39/32 ^3F, G♭
17 364.286 27/22, 16/13, 26/21 v3F♯, ^G♭
18 385.714 5/4 vvF♯, ^^G♭
19 407.143 14/11 vF♯, ^3G♭
20 428.571 32/25, 33/26 F♯, v3G
21 450.000 9/7, 13/10 ^F♯, vvG
22 471.429 21/16 ^^F♯, vG
23 492.857 4/3 G
24 514.286 ^G, vvA♭
25 535.714 27/20, 15/11 ^^G, vA♭
26 557.143 11/8 ^3G, A♭
27 578.571 7/5 v3G♯, ^A♭
28 600.000 45/32, 64/45 vvG♯, ^^A♭
… … … …

* The following table assumes the patent val ⟨56 89 130 157 194 207]. Other approaches are possible. Inconsistent intervals are marked italic.

Commas

  • 5-limit commas: 2048/2025, [-5 -10 9⟩;
  • 7-limit commas: 686/675, 875/864, 1029/1024
  • 11-limit commas: 100/99, 245/242, 385/384, 686/675

Scales

Music

Claudi Meneghin

See also

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