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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]]

Latest revision as of 15:00, 16 July 2025

← 71ed5 72ed5 73ed5 →
Prime factorization 23 × 32
Step size 38.6988 ¢ 
Octave 31\72ed5 (1199.66 ¢)
(semiconvergent)
Twelfth 49\72ed5 (1896.24 ¢)
Consistency limit 12
Distinct consistency limit 9

72 equal divisions of the 5th harmonic (abbreviated 72ed5) is a nonoctave tuning system that divides the interval of 5/1 into 72 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 51/72, or the 72nd root of 5.

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 12-integer-limit, but it has a flat tendency, with prime harmonics 2, 3, 7, and 11 all tuned flat. It supports meantone as the number of divisions of the 5th harmonic is multiple of 4.

Harmonics

Approximation of harmonics in 72ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.3 -5.7 -0.7 +0.0 -6.1 -2.0 -1.0 -11.4 -0.3 -10.5 -6.4
Relative (%) -0.9 -14.8 -1.7 +0.0 -15.6 -5.2 -2.6 -29.5 -0.9 -27.3 -16.5
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(0) 		80
(8) 		87
(15) 		93
(21) 		98
(26) 		103
(31) 		107
(35) 		111
(39)
Approximation of harmonics in 72ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.8 -2.4 -5.7 -1.3 +9.8 -11.8 +10.7 -0.7 -7.7 -10.9 -10.4 -6.7
Relative (%) +25.4 -6.1 -14.8 -3.5 +25.3 -30.4 +27.7 -1.7 -20.0 -28.1 -27.0 -17.4
Steps
(reduced) 		115
(43) 		118
(46) 		121
(49) 		124
(52) 		127
(55) 		129
(57) 		132
(60) 		134
(62) 		136
(64) 		138
(66) 		140
(68) 		142
(70)

Subsets and supersets

72 is a largely composite number. Since it factors into primes as 23 × 32, 72ed5 has subset ed5's 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36.

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 38.7 33/32, 36/35, 49/48, 50/49, 64/63
2 77.4 21/20, 22/21, 25/24, 28/27
3 116.1 14/13, 15/14, 16/15
4 154.8 12/11, 13/12
5 193.5 9/8, 10/9
6 232.2 8/7
7 270.9 7/6
8 309.6 6/5
9 348.3 11/9, 16/13
10 387.0 5/4
11 425.7 9/7, 14/11
12 464.4 13/10, 17/13, 21/16
13 503.1 4/3
14 541.8 11/8, 18/13, 26/19
15 580.5 7/5
16 619.2 10/7
17 657.9 16/11, 19/13, 22/15
18 696.6 3/2
19 735.3 20/13, 26/17, 32/21
20 774.0 11/7, 14/9
21 812.7 8/5
22 851.4 13/8, 18/11
23 890.1 5/3
24 928.8 12/7
25 967.5 7/4
26 1006.2 9/5
27 1044.9 11/6
28 1083.6 13/7, 15/8
29 1122.3 17/9, 19/10, 21/11
30 1161.0 35/18, 49/25, 63/32
31 1199.7 2/1
32 1238.4 33/16, 45/22, 49/24, 55/27
33 1277.1 21/10, 25/12
34 1315.8 15/7, 17/8, 19/9
35 1354.5 13/6
36 1393.2 9/4
37 1431.9 16/7
38 1470.6 7/3
39 1509.3 12/5
40 1548.0 22/9
41 1586.7 5/2
42 1625.3 18/7
43 1664.0 21/8
44 1702.7 8/3
45 1741.4 11/4
46 1780.1 14/5
47 1818.8 20/7
48 1857.5 26/9, 38/13
49 1896.2 3/1
50 1934.9 40/13
51 1973.6 25/8
52 2012.3 16/5
53 2051.0 13/4
54 2089.7 10/3
55 2128.4 24/7
56 2167.1 7/2
57 2205.8 18/5, 25/7
58 2244.5 11/3
59 2283.2 15/4
60 2321.9 27/7
61 2360.6 35/9, 63/16
62 2399.3 4/1
63 2438.0 33/8, 45/11, 49/12
64 2476.7 21/5, 25/6
65 2515.4 17/4
66 2554.1 22/5
67 2592.8 9/2
68 2631.5 32/7
69 2670.2 14/3
70 2708.9 19/4, 24/5
71 2747.6 44/9
72 2786.3 5/1

See also

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