56edo

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← 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 (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

Theory

56edo shares its near perfect quality of the classical major third with 28edo, which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of 17edo and 22edo. It has decent approximations to prime harmonics up to 19, but due to the sharpness of its harmonic 3, several intervals of 9 are inconsistent. Therefore, 56edo is not very popular compared to edos like 53edo or 58edo.

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)

As a tuning of other temperaments

In the 5-limit, 56et most notably tempers out the diaschisma, as well as the shibboleth comma. Using the patent val, it tempers out 686/675, 875/864, and 1029/1024 in the 7-limit, 100/99, 245/242, and 385/384 in the 11-limit, and 91/90 and 169/168 in the 13-limit. It supports the diaschismic extension keen in the 7- and 11-limit, and its 13- and 17-limit extension keenic. It also supports hemithirds, superkleismic, and sycamore in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of slendric, mapping 7/6 to an inframinor third of 257.1 ¢, and mapping 9/7 inconsistently to an ultramajor third of 450 ¢.

Another interesting val to consider is 56d (⟨56 89 130 158]), which maps 7/4 sharply to around 986 ¢. This mapping tempers out 50/49 and 64/63 in the 7-limit, providing an alternative to 22edo for pajara. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving 6/5 and 10/9, which are quite out of tune in 22edo. Its approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may want to compress the octave, using tunings such as 145ed6 or 201ed12. It is also an excellent tuning for the 11-limit version of pajara, which additionally tempers out 99/98, 100/99, and 176/175. Finally, it gives an excellent tuning for the no-fives supra temperament tempering out 64/63 and 99/98.

Miscellaneous properties

One step of 56edo is the closest to the syntonic comma, 81/80, of any integer edo's step size by direct approximation, with the number of directly approximated syntonic commas per octave 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. Because it contains 28edo's major third and has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third 5/4 at 18\56, and the Pythagorean major third 81/64 at 19\56. Unfortunately, 56edo does not map the Pythagorean major third 19\56, but instead inconsistently to 20\56, a supermajor third of 428.6 ¢. However, the Pythagorean major third is mapped to 19\56 consistently in 224edo, which is the quadruple of 56edo.

The perfect fifth generates a diatonic scale with a step ratio that is a convergent towards the bronze metallic mean, following 17edo and preceding 185edo.

Subsets and supersets

Since 56 factors into 23 × 7, 56edo has subset edos 2, 4, 7, 8, 14, and 28.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 D
1 21.4 49/48, 55/54, 56/55, 64/63 ^D, vvE♭
2 42.9 28/27, 40/39, 45/44, 50/49, 81/80 ^^D, vE♭
3 64.3 25/24, 36/35, 33/32 ^3D, E♭
4 85.7 19/18, 20/19, 21/20, 22/21 v3D♯, ^E♭
5 107.1 16/15, 17/16, 18/17 vvD♯, ^^E♭
6 128.6 15/14, 13/12, 14/13 vD♯, ^3E♭
7 150.0 12/11 D♯, v3E
8 171.4 10/9, 11/10, 21/19 ^D♯, vvE
9 192.9 19/17, 28/25 ^^D♯, vE
10 214.3 9/8, 17/15 E
11 235.7 8/7 ^E, vvF
12 257.1 7/6 ^^E, vF
13 278.6 13/11, 20/17 F
14 300.0 19/16, 25/21 ^F, vvG♭
15 321.4 6/5 ^^F, vG♭
16 342.9 11/9, 17/14 ^3F, G♭
17 364.3 16/13, 21/17, 26/21 v3F♯, ^G♭
18 385.7 5/4 vvF♯, ^^G♭
19 407.1 14/11, 19/12, 24/19 vF♯, ^3G♭
20 428.6 32/25, 33/26 F♯, v3G
21 450.0 9/7, 13/10 ^F♯, vvG
22 471.4 17/13, 21/16 ^^F♯, vG
23 492.9 4/3 G
24 514.3 35/26 ^G, vvA♭
25 535.7 15/11, 19/14, 26/19, 27/20 ^^G, vA♭
26 557.1 11/8 ^3G, A♭
27 578.6 7/5 v3G♯, ^A♭
28 600.0 17/12, 24/17 vvG♯, ^^A♭
… … … …

* The following table assumes the 19-limit patent val; other approaches are possible. Inconsistent intervals are marked in italics.

Notation

Ups and downs notation

56edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sharp symbol   
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   
Flat symbol
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as 63-EDO.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

Approximation to JI

The following tables show how 15-odd-limit intervals are represented in 56edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 56edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/7, 14/13 0.273 1.3
5/4, 8/5 0.599 2.8
11/6, 12/11 0.637 3.0
15/11, 22/15 1.236 5.8
7/5, 10/7 3.941 18.4
13/10, 20/13 4.214 19.7
7/4, 8/7 4.540 21.2
11/9, 18/11 4.551 21.2
15/8, 16/15 4.588 21.4
13/8, 16/13 4.813 22.5
3/2, 4/3 5.188 24.2
5/3, 6/5 5.787 27.0
11/8, 16/11 5.825 27.2
13/9, 18/13 6.239 29.1
11/10, 20/11 6.424 30.0
9/7, 14/9 6.513 30.4
15/14, 28/15 9.129 42.6
15/13, 26/15 9.402 43.9
7/6, 12/7 9.728 45.4
13/12, 24/13 10.001 46.7
11/7, 14/11 10.365 48.4
9/8, 16/9 10.376 48.4
9/5, 10/9 10.453 48.8
13/11, 22/13 10.638 49.6
15-odd-limit intervals in 56edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/7, 14/13 0.273 1.3
5/4, 8/5 0.599 2.8
11/6, 12/11 0.637 3.0
15/11, 22/15 1.236 5.8
7/5, 10/7 3.941 18.4
13/10, 20/13 4.214 19.7
7/4, 8/7 4.540 21.2
11/9, 18/11 4.551 21.2
15/8, 16/15 4.588 21.4
13/8, 16/13 4.813 22.5
3/2, 4/3 5.188 24.2
5/3, 6/5 5.787 27.0
11/8, 16/11 5.825 27.2
11/10, 20/11 6.424 30.0
15/14, 28/15 9.129 42.6
15/13, 26/15 9.402 43.9
7/6, 12/7 9.728 45.4
13/12, 24/13 10.001 46.7
11/7, 14/11 10.365 48.4
9/8, 16/9 10.376 48.4
13/11, 22/13 10.638 49.6
9/5, 10/9 10.975 51.2
9/7, 14/9 14.916 69.6
13/9, 18/13 15.189 70.9

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* Cents* Associated
ratio* 		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
(250/243) 		Sevond

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

  • Supra7
  • Supra12
  • Subsets of echidnic[16] (6u8d):
    • Frankincense[idiosyncratic term] (this is the original/default tuning): 364.3 - 492.9 - 707.1 - 835.7 - 1200.0
    • Quasi-equipentatonic: 257.1 - 492.9 - 707.1 - 964.3 - 1200.0
    • Sakura-like scale containing phi: 107.1 - 492.9 - 707.1 - 835.7 - 1200.0
  • Subsets of sevond[14]
    • Evened minor pentatonic (approximated from 72edo): 321.4 - 492.9 - 685.7 - 1028.6 - 1200.0

Instruments

Lumatone mappings for 56edo are available.

Music

Bryan Deister
Budjarn Lambeth
Claudi Meneghin
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