56edo

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← 55edo56edo57edo →
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 (abbreviated 56edo or 56ed2), also called 56-tone equal temperament (56tet) or 56 equal temperament (56et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 56 equal parts of about 21.4 ¢ each. Each step represents a frequency ratio of 21/56, or the 56th root of 2.

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 ^4D, v6E
5 107.143 16/15 ^5D, v5E
6 128.571 15/14, 13/12, 14/13 ^6D, v4E
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 ^4F, v6G
18 385.714 5/4 ^5F, v5G
19 407.143 14/11 ^6F, v4G
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 35/26 ^G, vvA♭
25 535.714 27/20, 15/11 ^^G, vA♭
26 557.143 11/8 ^3G, A♭
27 578.571 7/5 ^4G, v6A
28 600.000 45/32, 64/45 ^5G, v5A

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [89 -56 [56 89]] -1.64 1.63 7.64
2.3.5 2048/2025, 1953125/1889568 [56 89 130]] -1.01 1.61 7.50
2.3.5.7 686/675, 875/864, 1029/1024 [56 89 130 157]] -0.352 1.80 8.38
2.3.5.7.11 100/99, 245/242, 385/384, 686/675 [56 89 130 157 194]] -0.618 1.69 7.90
2.3.5.7.11.13 91/90, 100/99, 169/168, 245/242, 385/384 [56 89 130 157 194 207]] -0.299 1.70 7.95

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator
(Reduced)
Cents
(Reduced)
Associated Ratio
(Reduced)
Temperament
1 3\56 64.29 25/24 Sycamore
1 9\56 192.86 28/25 Hemithirds
1 11\56 235.71 8/7 Slendric
1 15\56 321.43 6/5 Superkleismic
1 25\56 535.71 15/11 Maquila (56d) / maquiloid (56)
2 11\56 235.71 8/7 Echidnic
2 23\56
(5\56)
492.86
(107.14)
4/3
(17/16)
Keen / keenic
4 23\56
(5\56)
492.86
(107.14)
4/3
(17/16)
Bidia (7-limit)
7 23\56
(1\56)
492.86
(21.43)
4/3
(?)
Sevond

Scales

Music

Claudi Meneghin

See also