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! Cents
! Cents
! Approximate Ratios
! Approximate Ratios
! [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 21.429
| 21.429
| [[49/48]], [[64/63]]
| [[49/48]], [[64/63]]
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 42.857
| 42.857
| [[28/27]], [[50/49]], [[81/80]]
| [[28/27]], [[50/49]], [[81/80]]
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 64.286
| 64.286
| [[25/24]], [[36/35]], [[33/32]]
| [[25/24]], [[36/35]], [[33/32]]
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 85.714
| 85.714
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 107.143
| 107.143
| [[16/15]]
| [[16/15]]
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 128.571
| 128.571
| [[15/14]], [[13/12]], [[14/13]]
| [[15/14]], [[13/12]], [[14/13]]
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 150.000
| 150.000
| [[12/11]]
| [[12/11]]
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 171.429
| 171.429
| [[10/9]], [[11/10]]
| [[10/9]], [[11/10]]
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 192.857
| 192.857
| [[28/25]]
| [[28/25]]
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 214.286
| 214.286
| [[9/8]]
| [[9/8]]
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 235.714
| 235.714
| [[8/7]]
| [[8/7]]
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 257.143
| 257.143
| [[7/6]], [[15/13]]
| [[7/6]], [[15/13]]
| {{UDnote|step=12}]
|-
|-
| 13
| 13
| 278.571
| 278.571
| [[75/64]], [[13/11]]
| [[75/64]], [[13/11]]
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 300.000
| 300.000
| [[25/21]]
| [[25/21]]
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 321.429
| 321.429
| [[6/5]]
| [[6/5]]
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 342.857
| 342.857
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 364.286
| 364.286
| [[27/22]], [[16/13]], [[26/21]]
| [[27/22]], [[16/13]], [[26/21]]
| {{UDnote|step=17}}
|-
|-
| 18
| 18
| 385.714
| 385.714
| [[5/4]]
| [[5/4]]
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 407.143
| 407.143
| [[14/11]]
| [[14/11]]
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 428.571
| 428.571
| [[32/25]], [[33/26]]
| [[32/25]], [[33/26]]
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 450.000
| 450.000
| [[9/7]], [[13/10]]
| [[9/7]], [[13/10]]
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 471.429
| 471.429
| [[21/16]]
| [[21/16]]
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 492.857
| 492.857
| [[4/3]]
| [[4/3]]
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 514.286
| 514.286
|  
|  
| {{UDnote|step=24}}
|-
|-
| 25
| 25
Line 123: Line 149:
| 557.143
| 557.143
| [[11/8]]
| [[11/8]]
| {{UDnote|step=25}}
|-
|-
| 27
| 27
| 578.571
| 578.571
| [[7/5]]
| [[7/5]]
| {{UDnote|step=26}}
|-
|-
| 28
| 28
| 600.000
| 600.000
| [[45/32]], [[64/45]]
| [[45/32]], [[64/45]]
| {{UDnote|step=27}}
|-
|-
| …
| …

Revision as of 03:16, 15 June 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

56 equal divisions of the octave (56edo), or 56-tone equal temperament (56tet), 56 equal temperament (56et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 56 equal parts of about 21.4 ¢ each, a size close to the syntonic comma 81/80.

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 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 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. 56edo can be used to tune Barium temperament which sets 56 syntonci commas to the octave.


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)

Intervals

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

# 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 step=12}]
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
26 557.143 11/8 ^^G, vA♭
27 578.571 7/5 ^3G, A♭
28 600.000 45/32, 64/45 v3G♯, ^A♭
… … …

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

See also

Lumatone mapping for 56edo

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