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{{Infobox ET}}
{{Infobox ET}}
'''[[Ed7|Division of the 7th harmonic]] into 87 equal parts''' (87ed7) is related to [[31edo|31 edo]], but with the 7/1 rather than the 2/1 being just. The octave is slightly stretched (about 0.3862 cents) and the step size is about 38.7221 cents.
{{ED intro}}
 
== Theory ==
87ed7 is related to [[31edo]], but with the 7/1 rather than the [[2/1]] being just. The octave is slightly stretched (about 0.3862{{c}}). Like 31edo, 87ed7 is [[consistent]] through the [[integer limit|12-integer-limit]].  
 
=== Harmonics ===
{{Harmonics in equal|87|7|1|intervals=integer|columns=11}}
{{Harmonics in equal|87|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 87ed7 (continued)}}
 
=== Subsets and supersets ===
Since 87 factors into primes as {{nowrap| 3 × 29 }}, 87ed7 contains [[3ed7]] and [[29ed7]] as subset ed7's.


== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{{Interval table}}
|+ Intervals of 87ed7
|-
! | degree
! | cents value
! | corresponding <br>JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| |
|-
| | 1
| | 38.7221
| |
| |
|-
| | 2
| | 77.4443
| |
| |
|-
| | 3
| | 116.1664
| | 77/72, [[15/14]]
| |
|-
| | 4
| | 154.8885
| |
| |
|-
| | 5
| | 193.6107
| | [[19/17]], 85/76
| |
|-
| | 6
| | 232.3328
| | [[8/7]]
| |
|-
| | 7
| | 271.0550
| |
| |
|-
| | 8
| | 309.7771
| |
| |
|-
| | 9
| | 348.4992
| | [[11/9]], [[49/40]]
| |
|-
| | 10
| | 387.2214
| | [[5/4]]
| |
|-
| | 11
| | 425.9435
| |
| |
|-
| | 12
| | 464.6656
| | 98/75
| |
|-
| | 13
| | 503.3878
| | 91/68, 75/56
| |
|-
| | 14
| | 542.1099
| |
| |
|-
| | 15
| | 580.8321
| | [[7/5]]
| |
|-
| | 16
| | 619.5542
| | [[10/7]]
| |
|-
| | 17
| | 658.2763
| |
| |
|-
| | 18
| | 696.9985
| | 112/75, 121/81, 136/91, 187/125
| |
|-
| | 19
| | 735.7206
| |
| |
|-
| | 20
| | 774.4427
| |
| |
|-
| | 21
| | 813.1649
| | [[8/5]]
| |
|-
| | 22
| | 851.8870
| |
| |
|-
| | 23
| | 890.6091
| |
| |
|-
| | 24
| | 929.3313
| | 65/38
| |
|-
| | 25
| | 968.0534
| |
| |
|-
| | 26
| | 1006.7756
| |
| |
|-
| | 27
| | 1045.4977
| | 64/35
| |
|-
| | 28
| | 1084.2198
| |
| |
|-
| | 29
| | 1122.9420
| |
| |
|-
| | 30
| | 1161.6641
| | 88/45, 96/49, 49/25
| |
|-
| | 31
| | 1200.3862
| | [[Octave|2/1]]
| |
|-
| | 32
| | 1239.1084
| |
| |
|-
| | 33
| | 1277.8305
| | [[22/21|44/21]]
| |
|-
| | 34
| | 1316.5527
| |
| |
|-
| | 35
| | 1355.2748
| |
| |
|-
| | 36
| | 1393.9969
| | [[19/17|38/17]], 85/38
| |
|-
| | 37
| | 1432.7191
| |
| |
|-
| | 38
| | 1471.4412
| |
| |
|-
| | 39
| | 1510.1633
| |
| |
|-
| | 40
| | 1548.8855
| |
| |
|-
| | 41
| | 1587.6076
| |
| |
|-
| | 42
| | 1626.3297
| | [[32/25|64/25]]
| |
|-
| | 43
| | 1665.0519
| |
| |
|-
| | 44
| | 1703.7740
| |
| |
|-
| | 45
| | 1742.4962
| | [[26/19|52/19]]
| |
|-
| | 46
| | 1781.2183
| |
| |
|-
| | 47
| | 1819.9404
| |
| |
|-
| | 48
| | 1858.6626
| | [[19/13|38/13]]
| |
|-
| | 49
| | 1897.3847
| |
| |
|-
| | 50
| | 1936.1068
| |
| |
|-
| | 51
| | 1974.8290
| | [[25/16|25/8]]
| |
|-
| | 52
| | 2013.5511
| |
| |
|-
| | 53
| | 2052.2733
| |
| |
|-
| | 54
| | 2090.9954
| |
| |
|-
| | 55
| | 2129.7175
| |
| |
|-
| | 56
| | 2168.4397
| |
| |
|-
| | 57
| | 2207.1618
| | [[34/19|68/19]]
| |
|-
| | 58
| | 2245.8839
| |
| |
|-
| | 59
| | 2284.6061
| |
| |
|-
| | 60
| | 2323.3282
| | 65/17
| |
|-
| | 61
| | 2362.0503
| |
| |
|-
| | 62
| | 2400.7725
| |
| |
|-
| | 63
| | 2439.4946
| | [[45/44|45/11]]
| |
|-
| | 64
| | 2478.2168
| |
| |
|-
| | 65
| | 2516.9389
| |
| |
|-
| | 66
| | 2555.6610
| | 35/8
| |
|-
| | 67
| | 2594.3832
| |
| |
|-
| | 68
| | 2633.1053
| |
| |
|-
| | 69
| | 2671.8274
| |
| |
|-
| | 70
| | 2710.5496
| |
| |
|-
| | 71
| | 2749.2717
| |
| |
|-
| | 72
| | 2787.9939
| | [[5/1]]
| |
|-
| | 73
| | 2826.7160
| |
| |
|-
| | 74
| | 2865.4381
| |
| |
|-
| | 75
| | 2904.1603
| | 75/14
| |
|-
| | 76
| | 2942.8824
| |
| |
|-
| | 77
| | 2981.6045
| |
| |
|-
| | 78
| | 3020.3267
| | [[10/7|40/7]], 63/11
| |
|-
| | 79
| | 3059.0488
| |
| |
|-
| | 80
| | 3097.7709
| |
| |
|-
| | 81
| | 3136.4931
| | [[49/32|49/8]]
| |
|-
| | 82
| | 3175.2152
| |
| |
|-
| | 83
| | 3213.9374
| |
| |
|-
| | 84
| | 3252.6595
| | 98/15, [[18/11|72/11]]
| |
|-
| | 85
| | 3291.3816
| |
| |
|-
| | 86
| | 3330.1038
| |
| |
|-
| | 87
| | 3368.8259
| | '''exact [[7/1]]'''
| | [[7/4|harmonic seventh]] plus two octaves
|}
 
== Harmonics ==
{{Harmonics in equal|87|7|1|intervals=prime}}
{{Harmonics in equal|87|7|1|intervals=prime|collapsed=1|start=12}}


== See also ==
* [[18edf]] – relative edf
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6
* [[107ed11]] – relative ed11
* [[111ed12]] – relative ed12
* [[138ed22]] – relative ed22
* [[204ed96]] – close to the zeta-optimized tuning for 31edo
* [[39cET]]


{{stub}}
[[Category:31edo]]
[[Category:Ed7]]
[[Category:Edonoi]]

Latest revision as of 15:01, 16 July 2025

← 86ed7 87ed7 88ed7 →
Prime factorization 3 × 29
Step size 38.7221 ¢ 
Octave 31\87ed7 (1200.39 ¢)
(semiconvergent)
Twelfth 49\87ed7 (1897.38 ¢)
Consistency limit 12
Distinct consistency limit 9

87 equal divisions of the 7th harmonic (abbreviated 87ed7) is a nonoctave tuning system that divides the interval of 7/1 into 87 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 71/87, or the 87th root of 7.

Theory

87ed7 is related to 31edo, but with the 7/1 rather than the 2/1 being just. The octave is slightly stretched (about 0.3862 ¢). Like 31edo, 87ed7 is consistent through the 12-integer-limit.

Harmonics

Approximation of harmonics in 87ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.4 -4.6 +0.8 +1.7 -4.2 +0.0 +1.2 -9.1 +2.1 -8.0 -3.8
Relative (%) +1.0 -11.8 +2.0 +4.3 -10.8 +0.0 +3.0 -23.6 +5.3 -20.8 -9.8
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(80) 		87
(0) 		93
(6) 		98
(11) 		103
(16) 		107
(20) 		111
(24)
Approximation of harmonics in 87ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.5 +0.4 -2.9 +1.5 +12.8 -8.8 +13.8 +2.5 -4.6 -7.7 -7.2 -3.4
Relative (%) +32.3 +1.0 -7.5 +4.0 +32.9 -22.6 +35.7 +6.3 -11.8 -19.8 -18.5 -8.8
Steps
(reduced) 		115
(28) 		118
(31) 		121
(34) 		124
(37) 		127
(40) 		129
(42) 		132
(45) 		134
(47) 		136
(49) 		138
(51) 		140
(53) 		142
(55)

Subsets and supersets

Since 87 factors into primes as 3 × 29, 87ed7 contains 3ed7 and 29ed7 as subset ed7's.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 38.7
2 77.4 23/22
3 116.2 31/29
4 154.9 23/21, 35/32
5 193.6 19/17, 28/25
6 232.3 8/7
7 271.1 41/35
8 309.8
9 348.5 11/9
10 387.2 5/4
11 425.9 23/18, 32/25
12 464.7 17/13
13 503.4
14 542.1 26/19, 41/30
15 580.8 7/5
16 619.6 10/7
17 658.3 19/13, 41/28
18 697
19 735.7 26/17
20 774.4 25/16, 36/23
21 813.2 8/5
22 851.9 18/11
23 890.6
24 929.3 41/24
25 968.1 7/4
26 1006.8 34/19
27 1045.5
28 1084.2
29 1122.9
30 1161.7
31 1200.4 2/1
32 1239.1
33 1277.8 23/11
34 1316.6 15/7
35 1355.3 35/16
36 1394 38/17
37 1432.7 16/7
38 1471.4
39 1510.2
40 1548.9 22/9
41 1587.6 5/2
42 1626.3 23/9, 41/16
43 1665.1 34/13
44 1703.8
45 1742.5 41/15
46 1781.2 14/5
47 1819.9 20/7
48 1858.7 38/13, 41/14
49 1897.4
50 1936.1
51 1974.8 25/8
52 2013.6 16/5
53 2052.3 36/11
54 2091
55 2129.7 41/12
56 2168.4 7/2
57 2207.2
58 2245.9
59 2284.6
60 2323.3
61 2362.1
62 2400.8 4/1
63 2439.5
64 2478.2
65 2516.9 30/7
66 2555.7 35/8
67 2594.4
68 2633.1 32/7
69 2671.8
70 2710.5
71 2749.3
72 2788 5/1
73 2826.7 41/8
74 2865.4
75 2904.2
76 2942.9
77 2981.6 28/5
78 3020.3 40/7
79 3059 41/7
80 3097.8
81 3136.5
82 3175.2 25/4
83 3213.9 32/5
84 3252.7
85 3291.4
86 3330.1
87 3368.8 7/1

See also

Retrieved from "https://en.xen.wiki/index.php?title=87ed7&oldid=204629"