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'''Division of the 5th harmonic into 72 equal parts''' (72ed5) is related to [[31edo]], but with the 5/1 rather than the 2/1 being just. The octave is slightly compressed (about 0.3372 cents) and the step size is about 38.6988 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.
{{Infobox ET}}
{{ED intro}}


{| class="wikitable"
== Theory ==
72ed5 is related to [[31edo]], but with the 5/1 rather than the [[2/1]] being just. The octave is slightly compressed (about 0.3372 cents). Like 31edo, 72ed5 is [[consistent]] through the [[integer limit|12-integer-limit]], but it has a flat tendency, with [[prime harmonic]]s 2, [[3/1|3]], [[7/1|7]], and [[11/1|11]] all tuned flat. It [[support]]s [[meantone]] as the number of divisions of the 5th harmonic is multiple of 4.
 
=== Harmonics ===
{{Harmonics in equal|72|5|1|intervals=integer|columns=11}}
{{Harmonics in equal|72|5|1|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 72ed5 (continued)}}
 
=== Subsets and supersets ===
72 is a [[largely composite]] number. Since it factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72ed5 has subset ed5's {{EDs|equave=5| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}.
 
== Intervals ==
{| class="wikitable center-1 right-2 mw-collapsible"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 38.6988
| 38.7
| |
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
| |
|-
|-
| | 2
| 2
| | 77.3976
| 77.4
| |
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
| |
|-
|-
| | 3
| 3
| | 116.0964
| 116.1
| | [[16/15]], [[15/14]]
| [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 4
| 4
| | 154.7952
| 154.8
| |
| [[12/11]], [[13/12]]
| |
|-
|-
| | 5
| 5
| | 193.4940
| 193.5
| |
| [[9/8]], [[10/9]]
| |
|-
|-
| | 6
| 6
| | 232.1928
| 232.2
| | [[8/7]]
| [[8/7]]
| |
|-
|-
| | 7
| 7
| | 270.8916
| 270.9
| |
| [[7/6]]
| |
|-
|-
| | 8
| 8
| | 309.5904
| 309.6
| |
| [[6/5]]
| |
|-
|-
| | 9
| 9
| | 348.2892
| 348.3
| | [[11/9]]
| [[11/9]], [[16/13]]
| |
|-
|-
| | 10
| 10
| | 386.9880
| 387.0
| | [[5/4]]
| [[5/4]]
| |
|-
|-
| | 11
| 11
| | 425.6868
| 425.7
| |
| [[9/7]], [[14/11]]
| |
|-
|-
| | 12
| 12
| | 464.3856
| 464.4
| | 17/13
| [[13/10]], [[17/13]], [[21/16]]
| |
|-
|-
| | 13
| 13
| | 503.0844
| 503.1
| |
| [[4/3]]
| | pseudo-[[4/3]]
|-
|-
| | 14
| 14
| | 541.7832
| 541.8
| |
| [[11/8]], [[18/13]], [[26/19]]
| |
|-
|-
| | 15
| 15
| | 580.4820
| 580.5
| | [[7/5]]
| [[7/5]]
| |
|-
|-
| | 16
| 16
| | 619.1808
| 619.2
| |
| [[10/7]]
| |
|-
|-
| | 17
| 17
| | 657.8796
| 657.9
| |
| [[16/11]], [[19/13]], [[22/15]]
| |
|-
|-
| | 18
| 18
| | 696.5784
| 696.6
| |
| [[3/2]]
| | meantone fifth <br>(pseudo-[[3/2]])
|-
|-
| | 19
| 19
| | 735.2772
| 735.3
| |
| [[20/13]], [[26/17]], [[32/21]]
| |
|-
|-
| | 20
| 20
| | 773.9760
| 774.0
| | [[25/16]]
| [[11/7]], [[14/9]]
| |
|-
|-
| | 21
| 21
| | 812.6748
| 812.7
| | [[8/5]]
| [[8/5]]
| |
|-
|-
| | 22
| 22
| | 851.3736
| 851.4
| |
| [[13/8]], [[18/11]]
| |
|-
|-
| | 23
| 23
| | 890.0724
| 890.1
| |
| [[5/3]]
| | pseudo-[[5/3]]
|-
|-
| | 24
| 24
| | 928.7712
| 928.8
| | 65/38
| [[12/7]]
| |
|-
|-
| | 25
| 25
| | 967.4700
| 967.5
| |
| [[7/4]]
| |
|-
|-
| | 26
| 26
| | 1006.1688
| 1006.2
| |
| [[9/5]]
| |
|-
|-
| | 27
| 27
| | 1044.8676
| 1044.9
| | [[11/6]]
| [[11/6]]
| |
|-
|-
| | 28
| 28
| | 1083.5664
| 1083.6
| |
| [[13/7]], [[15/8]]
| | pseudo-[[15/8]]
|-
|-
| | 29
| 29
| | 1122.2652
| 1122.3
| |
| [[17/9]], [[19/10]], [[21/11]]
| |
|-
|-
| | 30
| 30
| | 1160.9640
| 1161.0
| | 45/23
| [[35/18]], [[49/25]], [[63/32]]
| |
|-
|-
| | 31
| 31
| | 1199.6628
| 1199.7
| | [[octave|2/1]]
| [[2/1]]
| |
|-
|-
| | 32
| 32
| | 1238.3617
| 1238.4
| |
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
| |
|-
|-
| | 33
| 33
| | 1277.0605
| 1277.1
| | 23/11
| [[21/10]], [[25/12]]
| |
|-
|-
| | 34
| 34
| | 1315.7593
| 1315.8
| |
| [[15/7]], [[17/8]], [[19/9]]
| |
|-
|-
| | 35
| 35
| | 1354.4581
| 1354.5
| |
| [[13/6]]
| |
|-
|-
| | 36
| 36
| | 1393.1569
| 1393.2
| | [[19/17|38/17]], 85/38
| [[9/4]]
| | meantone major second plus an octave
|-
|-
| | 37
| 37
| | 1431.8557
| 1431.9
| |
| [[16/7]]
| |
|-
|-
| | 38
| 38
| | 1470.5545
| 1470.6
| |
| [[7/3]]
| |
|-
|-
| | 39
| 39
| | 1509.2533
| 1509.3
| | 55/23
| [[12/5]]
| |
|-
|-
| | 40
| 40
| | 1547.9521
| 1548.0
| |
| [[22/9]]
| |
|-
|-
| | 41
| 41
| | 1586.6509
| 1586.7
| | [[5/2]]
| [[5/2]]
| |
|-
|-
| | 42
| 42
| | 1625.3497
| 1625.3
| | 23/9
| [[18/7]]
| |
|-
|-
| | 43
| 43
| | 1664.0485
| 1664.0
| |
| [[21/8]]
| |
|-
|-
| | 44
| 44
| | 1702.7473
| 1702.7
| |
| [[8/3]]
| | pseudo-[[8/3]]
|-
|-
| | 45
| 45
| | 1741.4461
| 1741.4
| | [[15/11|30/11]]
| [[11/4]]
| |
|-
|-
| | 46
| 46
| | 1780.1449
| 1780.1
| |
| [[14/5]]
| |
|-
|-
| | 47
| 47
| | 1818.8437
| 1818.8
| |
| [[20/7]]
| |
|-
|-
| | 48
| 48
| | 1857.5425
| 1857.5
| | [[19/13|38/13]]
| [[26/9]], [[38/13]]
| |
|-
|-
| | 49
| 49
| | 1896.2413
| 1896.2
| |
| [[3/1]]
| | pseudo-[[3/1]]
|-
|-
| | 50
| 50
| | 1934.9401
| 1934.9
| |
| [[40/13]]
| |
|-
|-
| | 51
| 51
| | 1973.6389
| 1973.6
| | [[25/16|25/8]]
| [[25/8]]
| |
|-
|-
| | 52
| 52
| | 2012.3377
| 2012.3
| | [[16/5]]
| [[16/5]]
| |
|-
|-
| | 53
| 53
| | 2051.0365
| 2051.0
| |
| [[13/4]]
| |
|-
|-
| | 54
| 54
| | 2089.7353
| 2089.7
| |
| [[10/3]]
| | meantone major sixth plus an octave <br>(pseudo-[[10/3]])
|-
|-
| | 55
| 55
| | 2128.4341
| 2128.4
| |
| [[24/7]]
| |
|-
|-
| | 56
| 56
| | 2167.1329
| 2167.1
| |
| [[7/2]]
| |
|-
|-
| | 57
| 57
| | 2205.8317
| 2205.8
| | 25/7
| [[18/5]], [[25/7]]
| |
|-
|-
| | 58
| 58
| | 2244.5305
| 2244.5
| |
| [[11/3]]
| |
|-
|-
| | 59
| 59
| | 2283.2293
| 2283.2
| |
| [[15/4]]
| | pseudo-[[15/4]]
|-
|-
| | 60
| 60
| | 2321.9281
| 2321.9
| | 65/17
| [[27/7]]
| |
|-
|-
| | 61
| 61
| | 2360.6269
| 2360.6
| |
| [[35/9]], [[63/16]]
| |
|-
|-
| | 62
| 62
| | 2399.3257
| 2399.3
| | [[4/1]]
| [[4/1]]
| |
|-
|-
| | 63
| 63
| | 2438.0245
| 2438.0
| | [[45/44|45/11]]
| [[33/8]], [[45/11]], [[49/12]]
| |
|-
|-
| | 64
| 64
| | 2476.7233
| 2476.7
| |
| [[21/5]], [[25/6]]
| |
|-
|-
| | 65
| 65
| | 2515.4221
| 2515.4
| |
| [[17/4]]
| |
|-
|-
| | 66
| 66
| | 2554.1209
| 2554.1
| | 35/8
| [[22/5]]
| |
|-
|-
| | 67
| 67
| | 2592.8197
| 2592.8
| |
| [[9/2]]
| |
|-
|-
| | 68
| 68
| | 2631.5185
| 2631.5
| |
| [[32/7]]
| |
|-
|-
| | 69
| 69
| | 2670.2173
| 2670.2
| | 14/3
| [[14/3]]
| |
|-
|-
| | 70
| 70
| | 2708.9161
| 2708.9
| |
| [[19/4]], [[24/5]]
| |
|-
|-
| | 71
| 71
| | 2747.6149
| 2747.6
| |
| [[44/9]]
| |
|-
|-
| | 72
| 72
| | 2786.3137
| 2786.3
| | '''exact [[5/1]]'''
| [[5/1]]
| | just major third plus two octaves
|}
|}


[[Category:Ed5]]
== See also ==
[[Category:Edonoi]]
* [[18edf]] – relative edf
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[80ed6]] – relative ed6
* [[87ed7]] – relative ed7
* [[107ed11]] – relative ed11
* [[111ed12]] – relative ed12
* [[138ed22]] – relative ed22
* [[204ed96]] – close to the zeta-optimized tuning for 31edo
* [[39cET]]
 
[[Category:31edo]]
Retrieved from "https://en.xen.wiki/w/72ed5"