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{{Infobox ET}}
{{Infobox ET}}
'''[[Ed5|Division of the 5th harmonic]] into 56 equal parts''' (56ed5) is related to [[24edo|24 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 49.7556 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning is also a [[hyperpyth]], tempering out 135/133, 171/169, 225/221, and 1521/1445 in the patent val.
{{ED intro}}


==Intervals==
== Theory ==
56ed5 is related to 24edo, but with the 5th harmonic rather than the [[2/1|octave]] being just. The octave is about 5.87 cents compressed. This tuning has a [[meantone]] fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning is also a [[hyperpyth]], tempering out 135/133, 171/169, 225/221, and 1521/1445 in the patent val.
 
=== Harmonics ===
{{Harmonics in equal|56|5|1|intervals=integer|columns=11}}
{{Harmonics in equal|56|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 56ed5 (continued)}}
 
=== Subsets and supersets ===
Since 56 factors into primes as {{nowrap| 2<sup>3</sup> × 7 }}, 56ed5 contains subset ed5's {{EDs|equave=5| 2, 4, 7, 8, 14, and 28 }}.
 
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+ Intervals of 56ed5
|+ Intervals of 56ed5
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximated ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| 1/1
| |
|-
|-
| | 1
| 1
| | 49.7556
| 49.8
| | 36/35, 35/34
| 35/34, 36/35
| |
|-
|-
| | 2
| 2
| | 99.5112
| 99.5
| | [[18/17]]
| 18/17
| |
|-
|-
| | 3
| 3
| | 149.2668
| 149.3
| | [[12/11]]
| 12/11
| |
|-
|-
| | 4
| 4
| | 199.0224
| 199.0
| | [[55/49]]
| 55/49
| |
|-
|-
| | 5
| 5
| | 248.7780
| 248.8
| | [[15/13]]
| 15/13
| |
|-
|-
| | 6
| 6
| | 298.5336
| 298.5
| | [[19/16]]
| 19/16
| |
|-
|-
| | 7
| 7
| | 348.2892
| 348.3
| | [[11/9]]
| 11/9
| |
|-
|-
| | 8
| 8
| | 398.0448
| 398.0
| | 34/27
| 5/4
| | pseudo-[[5/4]]
|-
|-
| | 9
| 9
| | 447.8004
| 447.8
| | 35/27
| 35/27
| |
|-
|-
| | 10
| 10
| | 497.5560
| 497.6
| | [[4/3]]
| 4/3
| |
|-
|-
| | 11
| 11
| | 547.3116
| 547.3
| | 70/51
| 70/51
| |
|-
|-
| | 12
| 12
| | 597.0672
| 597.1
| | [[24/17]]
| 24/17
| |
|-
|-
| | 13
| 13
| | 646.8228
| 646.8
| |
|  
| |  
|-
|-
| | 14
| 14
| | 696.5784
| 696.6
| |
| 3/2
| | meantone fifth <br>(pseudo-[[3/2]])
|-
|-
| | 15
| 15
| | 746.3340
| 746.3
| | [[20/13]]
| 20/13
| |
|-
|-
| | 16
| 16
| | 796.0896
| 796.1
| | [[19/12]]
| 19/12
| |
|-
|-
| | 17
| 17
| | 845.8452
| 845.8
| | 44/27, 75/46
| 44/27, 75/46
| |
|-
|-
| | 18
| 18
| | 895.6008
| 895.6
| | 57/34
| 5/3
| | pseudo-[[5/3]]
|-
|-
| | 19
| 19
| | 945.3564
| 945.4
| | [[19/11]]
| 19/11
| |
|-
|-
| | 20
| 20
| | 995.1120
| 995.1
| | [[16/9]]
| 9/5, 16/9
| | pseudo-[[9/5]]
|-
|-
| | 21
| 21
| | 1044.8676
| 1044.9
| | 64/35
| 64/35
| |
|-
|-
| | 22
| 22
| | 1094.6232
| 1094.6
| | [[32/17]]
| 32/17
| |
|-
|-
| | 23
| 23
| | 1144.3788
| 1144.4
| |
|  
| |  
|-
|-
| | 24
| 24
| | 1194.1344
| 1194.1
| | 255/128
| 2/1
| | pseudo-[[octave]]
|-
|-
| | 25
| 25
| | 1243.8901
| 1243.9
| | 80/39, 39/19
| 39/19, 80/39
| |
|-
|-
| | 26
| 26
| | 1293.6457
| 1293.6
| | [[19/18|19/9]]
| 19/9
| |
|-
|-
| | 27
| 27
| | 1343.4013
| 1343.4
| | 50/23
| 50/23
| |
|-
|-
| | 28
| 28
| | 1393.1569
| 1393.2
| | [[19/17|38/17]], 85/38
| 38/17, 85/38
| | meantone major second plus an octave
|-
|-
| | 29
| 29
| | 1442.9125
| 1442.9
| | 23/10
| 23/10
| |
|-
|-
| | 30
| 30
| | 1492.6681
| 1492.7
| | 45/19
| 45/19
| |
|-
|-
| | 31
| 31
| | 1542.4237
| 1542.4
| | 39/16
| 39/16
| |
|-
|-
| | 32
| 32
| | 1592.1793
| 1592.2
| | 128/51
| 5/2
| | pseudo-[[5/2]]
|-
|-
| | 33
| 33
| | 1641.9349
| 1641.9
| |
| 13/5
| | pseudo-[[13/5]]
|-
|-
| | 34
| 34
| | 1691.6905
| 1691.7
| | 85/32
| 85/32
| |
|-
|-
| | 35
| 35
| | 1741.4461
| 1741.4
| | 175/64
| 175/64
| |
|-
|-
| | 36
| 36
| | 1791.2017
| 1791.2
| | [[45/32|45/16]]
| 45/16
| |
|-
|-
| | 37
| 37
| | 1840.9573
| 1841.0
| | 55/19
| 55/19
| |
|-
|-
| | 38
| 38
| | 1890.7129
| 1890.7
| | 170/57
| 3/1
| | pseudo-[[3/1]]
|-
|-
| | 39
| 39
| | 1940.4685
| 1940.5
| | 46/15, 135/44
| 46/15, 135/44
| |
|-
|-
| | 40
| 40
| | 1990.2241
| 1990.2
| | [[30/19|60/19]]
| 60/19
| |
|-
|-
| | 41
| 41
| | 2039.9797
| 2040.0
| | [[13/4]]
| 13/4
| |
|-
|-
| | 42
| 42
| | 2089.7353
| 2089.7
| |
| 10/3
| | meantone major sixth plus an octave <br>(pseudo-[[10/3]])
|-
|-
| | 43
| 43
| | 2139.4909
| 2139.5
| |
| 17/5
| | pseudo-[[17/10|17/5]]
|-
|-
| | 44
| 44
| | 2189.2465
| 2189.2
| | 85/24
| 85/24
| |
|-
|-
| | 45
| 45
| | 2239.0021
| 2239.0
| | 51/14
| 51/14
| |
|-
|-
| | 46
| 46
| | 2288.7577
| 2288.8
| | [[15/4]]
| 15/4, 19/5
| | pseudo-[[19/10|19/5]]
|-
|-
| | 47
| 47
| | 2338.5133
| 2338.5
| | [[27/14|27/7]]
| 27/7
| |
|-
|-
| | 48
| 48
| | 2388.2689
| 2388.3
| | 135/34
| 4/1
| | pseudo-[[4/1]]
|-
|-
| | 49
| 49
| | 2438.0245
| 2438.0
| | [[45/44|45/11]]
| 45/11
| |
|-
|-
| | 50
| 50
| | 2487.7801
| 2487.8
| | [[20/19|80/19]]
| 21/5
| | pseudo-[[21/20|21/5]]
|-
|-
| | 51
| 51
| | 2537.5357
| 2537.5
| | [[13/3]]
| 13/3
| |
|-
|-
| | 52
| 52
| | 2587.2913
| 2587.3
| | [[49/44|49/11]]
| 49/11
| |
|-
|-
| | 53
| 53
| | 2637.0469
| 2637.0
| | [[55/48|55/12]]
| 55/12
| |
|-
|-
| | 54
| 54
| | 2686.8025
| 2686.8
| | 85/18
| 85/18
| |
|-
|-
| | 55
| 55
| | 2736.5581
| 2736.6
| | [[17/14|34/7]]
| 34/7
| |
|-
|-
| | 56
| 56
| | 2786.3137
| 2786.3
| | '''exact [[5/1]]'''
| 5/1
| | just major third plus two octaves
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 56
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 56
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}
{{todo|expand}}

Revision as of 10:19, 27 May 2025

← 55ed5 56ed5 57ed5 →
Prime factorization 23 × 7
Step size 49.7556 ¢ 
Octave 24\56ed5 (1194.13 ¢) (→ 3\7ed5)
Twelfth 38\56ed5 (1890.71 ¢) (→ 19\28ed5)
Consistency limit 6
Distinct consistency limit 6

56 equal divisions of the 5th harmonic (abbreviated 56ed5) is a nonoctave tuning system that divides the interval of 5/1 into 56 equal parts of about 49.8 ¢ each. Each step represents a frequency ratio of 51/56, or the 56th root of 5.

Theory

56ed5 is related to 24edo, but with the 5th harmonic rather than the octave being just. The octave is about 5.87 cents compressed. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning is also a hyperpyth, tempering out 135/133, 171/169, 225/221, and 1521/1445 in the patent val.

Harmonics

Approximation of harmonics in 56ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.9 -11.2 -11.7 +0.0 -17.1 +14.6 -17.6 -22.5 -5.9 -21.6 -23.0
Relative (%) -11.8 -22.6 -23.6 +0.0 -34.4 +29.3 -35.4 -45.2 -11.8 -43.4 -46.2
Steps
(reduced) 		24
(24) 		38
(38) 		48
(48) 		56
(0) 		62
(6) 		68
(12) 		72
(16) 		76
(20) 		80
(24) 		83
(27) 		86
(30)
Approximation of harmonics in 56ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -12.3 +8.7 -11.2 -23.5 +20.8 +21.4 -22.4 -11.7 +3.3 +22.3 -4.9 +20.9
Relative (%) -24.7 +17.5 -22.6 -47.2 +41.9 +43.0 -45.1 -23.6 +6.7 +44.8 -9.9 +42.0
Steps
(reduced) 		89
(33) 		92
(36) 		94
(38) 		96
(40) 		99
(43) 		101
(45) 		102
(46) 		104
(48) 		106
(50) 		108
(52) 		109
(53) 		111
(55)

Subsets and supersets

Since 56 factors into primes as 23 × 7, 56ed5 contains subset ed5's 2, 4, 7, 8, 14, and 28.

Intervals

Intervals of 56ed5
# Cents Approximated ratios
0 0.0 1/1
1 49.8 35/34, 36/35
2 99.5 18/17
3 149.3 12/11
4 199.0 55/49
5 248.8 15/13
6 298.5 19/16
7 348.3 11/9
8 398.0 5/4
9 447.8 35/27
10 497.6 4/3
11 547.3 70/51
12 597.1 24/17
13 646.8
14 696.6 3/2
15 746.3 20/13
16 796.1 19/12
17 845.8 44/27, 75/46
18 895.6 5/3
19 945.4 19/11
20 995.1 9/5, 16/9
21 1044.9 64/35
22 1094.6 32/17
23 1144.4
24 1194.1 2/1
25 1243.9 39/19, 80/39
26 1293.6 19/9
27 1343.4 50/23
28 1393.2 38/17, 85/38
29 1442.9 23/10
30 1492.7 45/19
31 1542.4 39/16
32 1592.2 5/2
33 1641.9 13/5
34 1691.7 85/32
35 1741.4 175/64
36 1791.2 45/16
37 1841.0 55/19
38 1890.7 3/1
39 1940.5 46/15, 135/44
40 1990.2 60/19
41 2040.0 13/4
42 2089.7 10/3
43 2139.5 17/5
44 2189.2 85/24
45 2239.0 51/14
46 2288.8 15/4, 19/5
47 2338.5 27/7
48 2388.3 4/1
49 2438.0 45/11
50 2487.8 21/5
51 2537.5 13/3
52 2587.3 49/11
53 2637.0 55/12
54 2686.8 85/18
55 2736.6 34/7
56 2786.3 5/1
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