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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}}
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