87edo: Difference between revisions

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Put into table form.
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Added Difference in Cents column.
Line 92: Line 92:
See [[Detempering|detempering]].
See [[Detempering|detempering]].


[91/90, 49/48, 40/39, 28/27, 25/24, 21/20, 35/33, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 28/25, 9/8, 25/22, 8/7, 15/13, 7/6, 75/64, 13/11, 25/21, 6/5, 40/33, 11/9, 16/13, 26/21, 5/4, 44/35, 14/11, 32/25, 9/7, 13/10, 21/16, 33/25, 4/3, 35/26, 27/20, 15/11, 11/8, 18/13, 7/5, 45/32, 64/45, 10/7, 13/9, 16/11, 22/15, 40/27, 52/35, 3/2, 50/33, 32/21, 20/13, 14/9, 25/16, 11/7, 35/22, 8/5, 21/13, 13/8, 18/11, 33/20, 5/3, 42/25, 22/13, 75/44, 12/7, 26/15, 7/4, 44/25, 16/9, 25/14, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 66/35, 21/11, 25/13, 27/14, 39/20, 55/28, 99/50, 2]
In this table, "Difference in Cents" indicates whether 87 is flat (negative) or sharp (positive) of the detempered interval. For example, 15 steps, at 206.89655 cents, corresponds to 9/8 and is 3.0 cents sharp.


{| class="wikitable"
{|class="wikitable"
|-
|-
| | Steps of 87
| | Steps of 87
| | Cents
| | Cents value
| | Detempered Interval
| | Detempered Interval
| | Difference in Cents
|-
|-
| | 1
| | 1
| | 13.79310
| | 13.79310
| | 91/90
| | 91/90
| | -5.3
|-
|-
| | 2
| | 2
| | 27.58621
| | 27.58621
| | 49/48
| | 49/48
| | -8.1
|-
|-
| | 3
| | 3
| | 41.37931
| | 41.37931
| | 40/39
| | 40/39
| | -2.5
|-
|-
| | 4
| | 4
| | 55.17241
| | 55.17241
| | 28/27
| | 28/27
| | -7.8
|-
|-
| | 5
| | 5
| | 68.96552
| | 68.96552
| | 25/24
| | 25/24
| | -1.7
|-
|-
| | 6
| | 6
| | 82.75862
| | 82.75862
| | 21/20
| | 21/20
| | -1.7
|-
|-
| | 7
| | 7
| | 96.55172
| | 96.55172
| | 35/33
| | 35/33
| | -5.3
|-
|-
| | 8
| | 8
| | 110.34483
| | 110.34483
| | 16/15
| | 16/15
| | -1.4
|-
|-
| | 9
| | 9
| | 124.13793
| | 124.13793
| | 14/13
| | 14/13
| | -4.2
|-
|-
| | 10
| | 10
| | 137.93103
| | 137.93103
| | 13/12
| | 13/12
| | -0.6
|-
|-
| | 11
| | 11
| | 151.72414
| | 151.72414
| | 12/11
| | 12/11
| | 1.1
|-
|-
| | 12
| | 12
| | 165.51724
| | 165.51724
| | 11/10
| | 11/10
| | 0.5
|-
|-
| | 13
| | 13
| | 179.31035
| | 179.31035
| | 10/9
| | 10/9
| | -3.1
|-
|-
| | 14
| | 14
| | 193.10345
| | 193.10345
| | 28/25
| | 28/25
| | -3.1
|-
|-
| | 15
| | 15
| | 206.89655
| | 206.89655
| | 9/8
| | 9/8
| | 3.0
|-
|-
| | 16
| | 16
| | 220.68966
| | 220.68966
| | 25/22
| | 25/22
| | -0.6
|-
|-
| | 17
| | 17
| | 234.48276
| | 234.48276
| | 8/7
| | 8/7
| | 3.3
|-
|-
| | 18
| | 18
| | 248.27586
| | 248.27586
| | 15/13
| | 15/13
| | 0.5
|-
|-
| | 19
| | 19
| | 262.06897
| | 262.06897
| | 7/6
| | 7/6
| | -4.8
|-
|-
| | 20
| | 20
| | 275.86207
| | 275.86207
| | 75/64
| | 75/64
| | 1.3
|-
|-
| | 21
| | 21
| | 289.65517
| | 289.65517
| | 13/11
| | 13/11
| | 0.4
|-
|-
| | 22
| | 22
| | 303.44828
| | 303.44828
| | 25/21
| | 25/21
| | 1.6
|-
|-
| | 23
| | 23
| | 317.24138
| | 317.24138
| | 6/5
| | 6/5
| | 1.6
|-
|-
| | 24
| | 24
| | 331.03448
| | 331.03448
| | 40/33
| | 40/33
| | -2.0
|-
|-
| | 25
| | 25
| | 344.82759
| | 344.82759
| | 11/9
| | 11/9
| | -2.6
|-
|-
| | 26
| | 26
| | 358.62069
| | 358.62069
| | 16/13
| | 16/13
| | -0.9
|-
|-
| | 27
| | 27
| | 372.41379
| | 372.41379
| | 26/21
| | 26/21
| | 2.7
|-
|-
| | 28
| | 28
| | 386.20690
| | 386.20690
| | 5/4
| | 5/4
| | -0.1
|-
|-
| | 29
| | 29
| | 400.00000
| | 400.00000
| | 44/35
| | 44/35
| | 3.8
|-
|-
| | 30
| | 30
| | 413.79310
| | 413.79310
| | 14/11
| | 14/11
| | -3.7
|-
|-
| | 31
| | 31
| | 427.58621
| | 427.58621
| | 32/25
| | 32/25
| | 0.2
|-
|-
| | 32
| | 32
| | 441.37931
| | 441.37931
| | 9/7
| | 9/7
| | 6.3
|-
|-
| | 33
| | 33
| | 455.17241
| | 455.17241
| | 13/10
| | 13/10
| | 1.0
|-
|-
| | 34
| | 34
| | 468.96552
| | 468.96552
| | 21/16
| | 21/16
| | -1.8
|-
|-
| | 35
| | 35
| | 482.75862
| | 482.75862
| | 33/25
| | 33/25
| | 2.1
|-
|-
| | 36
| | 36
| | 496.55172
| | 496.55172
| | 4/3
| | 4/3
| | -1.5
|-
|-
| | 37
| | 37
| | 510.34483
| | 510.34483
| | 35/26
| | 35/26
| | -4.3
|-
|-
| | 38
| | 38
| | 524.13793
| | 524.13793
| | 27/20
| | 27/20
| | 4.6
|-
|-
| | 39
| | 39
| | 537.93103
| | 537.93103
| | 15/11
| | 15/11
| | 1.0
|-
|-
| | 40
| | 40
| | 551.72414
| | 551.72414
| | 11/8
| | 11/8
| | 0.4
|-
|-
| | 41
| | 41
| | 565.51724
| | 565.51724
| | 18/13
| | 18/13
| | 2.1
|-
|-
| | 42
| | 42
| | 579.31035
| | 579.31035
| | 7/5
| | 7/5
| | -3.2
|-
|-
| | 43
| | 43
| | 593.10345
| | 593.10345
| | 45/32
| | 45/32
| | 2.9
|-
|-
| | 44
| | 44
| | 606.89655
| | 606.89655
| | 64/45
| | 64/45
| | -2.9
|-
|-
| | 45
| | 45
| | 620.68966
| | 620.68966
| | 10/7
| | 10/7
| | 3.2
|-
|-
| | 46
| | 46
| | 634.48276
| | 634.48276
| | 13/9
| | 13/9
| | -2.1
|-
|-
| | 47
| | 47
| | 648.27586
| | 648.27586
| | 16/11
| | 16/11
| | -0.4
|-
|-
| | 48
| | 48
| | 662.06897
| | 662.06897
| | 22/15
| | 22/15
| | -1.0
|-
|-
| | 49
| | 49
| | 675.86207
| | 675.86207
| | 40/27
| | 40/27
| | -4.6
|-
|-
| | 50
| | 50
| | 689.65517
| | 689.65517
| | 52/35
| | 52/35
| | 4.3
|-
|-
| | 51
| | 51
| | 703.44828
| | 703.44828
| | 3/2
| | 3/2
| | 1.5
|-
|-
| | 52
| | 52
| | 717.24138
| | 717.24138
| | 50/33
| | 50/33
| | -2.1
|-
|-
| | 53
| | 53
| | 731.03448
| | 731.03448
| | 32/21
| | 32/21
| | 1.8
|-
|-
| | 54
| | 54
| | 744.82759
| | 744.82759
| | 20/13
| | 20/13
| | -1.0
|-
|-
| | 55
| | 55
| | 758.62069
| | 758.62069
| | 14/9
| | 14/9
| | -6.3
|-
|-
| | 56
| | 56
| | 772.41379
| | 772.41379
| | 25/16
| | 25/16
| | -0.2
|-
|-
| | 57
| | 57
| | 786.20690
| | 786.20690
| | 11/7
| | 11/7
| | 3.7
|-
|-
| | 58
| | 58
| | 800.00000
| | 800.00000
| | 35/22
| | 35/22
| | -3.8
|-
|-
| | 59
| | 59
| | 813.79310
| | 813.79310
| | 8/5
| | 8/5
| | 0.1
|-
|-
| | 60
| | 60
| | 827.58621
| | 827.58621
| | 21/13
| | 21/13
| | -2.7
|-
|-
| | 61
| | 61
| | 841.37931
| | 841.37931
| | 13/8
| | 13/8
| | 0.9
|-
|-
| | 62
| | 62
| | 855.17241
| | 855.17241
| | 18/11
| | 18/11
| | 2.6
|-
|-
| | 63
| | 63
| | 868.96552
| | 868.96552
| | 33/20
| | 33/20
| | 2.0
|-
|-
| | 64
| | 64
| | 882.75862
| | 882.75862
| | 5/3
| | 5/3
| | -1.6
|-
|-
| | 65
| | 65
| | 896.55172
| | 896.55172
| | 42/25
| | 42/25
| | -1.6
|-
|-
| | 66
| | 66
| | 910.34483
| | 910.34483
| | 22/13
| | 22/13
| | -0.4
|-
|-
| | 67
| | 67
| | 924.13793
| | 924.13793
| | 75/44
| | 75/44
| | 0.9
|-
|-
| | 68
| | 68
| | 937.93103
| | 937.93103
| | 12/7
| | 12/7
| | 4.8
|-
|-
| | 69
| | 69
| | 951.72414
| | 951.72414
| | 26/15
| | 26/15
| | -0.5
|-
|-
| | 70
| | 70
| | 965.51724
| | 965.51724
| | 7/4
| | 7/4
| | -3.3
|-
|-
| | 71
| | 71
| | 979.31035
| | 979.31035
| | 44/25
| | 44/25
| | 0.6
|-
|-
| | 72
| | 72
| | 993.10345
| | 993.10345
| | 16/9
| | 16/9
| | -3.0
|-
|-
| | 73
| | 73
| | 1006.89655
| | 1006.89655
| | 25/14
| | 25/14
| | 3.1
|-
|-
| | 74
| | 74
| | 1020.68966
| | 1020.68966
| | 9/5
| | 9/5
| | 3.1
|-
|-
| | 75
| | 75
| | 1034.48276
| | 1034.48276
| | 20/11
| | 20/11
| | -0.5
|-
|-
| | 76
| | 76
| | 1048.27586
| | 1048.27586
| | 11/6
| | 11/6
| | -1.1
|-
|-
| | 77
| | 77
| | 1062.06897
| | 1062.06897
| | 24/13
| | 24/13
| | 0.6
|-
|-
| | 78
| | 78
| | 1075.86207
| | 1075.86207
| | 13/7
| | 13/7
| | 4.2
|-
|-
| | 79
| | 79
| | 1089.65517
| | 1089.65517
| | 15/8
| | 15/8
| | 1.4
|-
|-
| | 80
| | 80
| | 1103.44828
| | 1103.44828
| | 66/35
| | 66/35
| | 5.3
|-
|-
| | 81
| | 81
| | 1117.24138
| | 1117.24138
| | 21/11
| | 21/11
| | -2.2
|-
|-
| | 82
| | 82
| | 1131.03448
| | 1131.03448
| | 25/13
| | 25/13
| | -1.1
|-
|-
| | 83
| | 83
| | 1144.82759
| | 1144.82759
| | 27/14
| | 27/14
| | 7.8
|-
|-
| | 84
| | 84
| | 1158.62069
| | 1158.62069
| | 39/20
| | 39/20
| | 2.5
|-
|-
| | 85
| | 85
| | 1172.41379
| | 1172.41379
| | 55/28
| | 55/28
| | 3.6
|-
|-
| | 86
| | 86
| | 1186.20690
| | 1186.20690
| | 99/50
| | 99/50
| | 3.6
|-
|-
| | 87
| | 87
| | 1200.00000
| | 1200.00000
| | 2/1
| | 2/1
| | 0.0
|}
|}


Revision as of 13:49, 5 May 2020

The 87 equal temperament, often abbreviated 87-tET, 87-EDO, or 87-ET, is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 cents. It is solid as both a 13-limit (or 15 odd limit) and as a 5-limit system, and of course does well enough in any limit in between. It represents the 13-limit tonality diamond both uniquely and consistently, and is the smallest equal temperament to do so.

87et tempers out 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

87et is a particularly good tuning for rodan temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit POTE generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for clyde temperament is 0.04455 cents sharp of the 7-limit POTE generator.

Rank two temperaments

Periods

per

octave

Generator Cents Associated

ratio

Temperament
1 4\87 55.172 33/32 Sensa
1 10\87 137.931 13/12 Quartemka
1 14\87 193.103 28/25 Luna/hemithirds
1 17\87 234.483 8/7 Rodan
1 23\87 317.241 6/5 Hanson/countercata/metakleismic
1 32\87 441.379 9/7 Clyde
1 38\87 524.138 65/48 Widefourth
1 40\87 551.724 11/8 Emkay
3 23\87 317.241 6/5 Tritikleismic
29 28\87 386.207 5/4 Mystery

87 can serve as a MOS in these:

270&87 <<24 -9 -66 12 27 ... ||

494&87 <<51 -1 -133 11 32 ... ||

13-limit detempering of 87et

See detempering.

In this table, "Difference in Cents" indicates whether 87 is flat (negative) or sharp (positive) of the detempered interval. For example, 15 steps, at 206.89655 cents, corresponds to 9/8 and is 3.0 cents sharp.

Steps of 87 Cents value Detempered Interval Difference in Cents
1 13.79310 91/90 -5.3
2 27.58621 49/48 -8.1
3 41.37931 40/39 -2.5
4 55.17241 28/27 -7.8
5 68.96552 25/24 -1.7
6 82.75862 21/20 -1.7
7 96.55172 35/33 -5.3
8 110.34483 16/15 -1.4
9 124.13793 14/13 -4.2
10 137.93103 13/12 -0.6
11 151.72414 12/11 1.1
12 165.51724 11/10 0.5
13 179.31035 10/9 -3.1
14 193.10345 28/25 -3.1
15 206.89655 9/8 3.0
16 220.68966 25/22 -0.6
17 234.48276 8/7 3.3
18 248.27586 15/13 0.5
19 262.06897 7/6 -4.8
20 275.86207 75/64 1.3
21 289.65517 13/11 0.4
22 303.44828 25/21 1.6
23 317.24138 6/5 1.6
24 331.03448 40/33 -2.0
25 344.82759 11/9 -2.6
26 358.62069 16/13 -0.9
27 372.41379 26/21 2.7
28 386.20690 5/4 -0.1
29 400.00000 44/35 3.8
30 413.79310 14/11 -3.7
31 427.58621 32/25 0.2
32 441.37931 9/7 6.3
33 455.17241 13/10 1.0
34 468.96552 21/16 -1.8
35 482.75862 33/25 2.1
36 496.55172 4/3 -1.5
37 510.34483 35/26 -4.3
38 524.13793 27/20 4.6
39 537.93103 15/11 1.0
40 551.72414 11/8 0.4
41 565.51724 18/13 2.1
42 579.31035 7/5 -3.2
43 593.10345 45/32 2.9
44 606.89655 64/45 -2.9
45 620.68966 10/7 3.2
46 634.48276 13/9 -2.1
47 648.27586 16/11 -0.4
48 662.06897 22/15 -1.0
49 675.86207 40/27 -4.6
50 689.65517 52/35 4.3
51 703.44828 3/2 1.5
52 717.24138 50/33 -2.1
53 731.03448 32/21 1.8
54 744.82759 20/13 -1.0
55 758.62069 14/9 -6.3
56 772.41379 25/16 -0.2
57 786.20690 11/7 3.7
58 800.00000 35/22 -3.8
59 813.79310 8/5 0.1
60 827.58621 21/13 -2.7
61 841.37931 13/8 0.9
62 855.17241 18/11 2.6
63 868.96552 33/20 2.0
64 882.75862 5/3 -1.6
65 896.55172 42/25 -1.6
66 910.34483 22/13 -0.4
67 924.13793 75/44 0.9
68 937.93103 12/7 4.8
69 951.72414 26/15 -0.5
70 965.51724 7/4 -3.3
71 979.31035 44/25 0.6
72 993.10345 16/9 -3.0
73 1006.89655 25/14 3.1
74 1020.68966 9/5 3.1
75 1034.48276 20/11 -0.5
76 1048.27586 11/6 -1.1
77 1062.06897 24/13 0.6
78 1075.86207 13/7 4.2
79 1089.65517 15/8 1.4
80 1103.44828 66/35 5.3
81 1117.24138 21/11 -2.2
82 1131.03448 25/13 -1.1
83 1144.82759 27/14 7.8
84 1158.62069 39/20 2.5
85 1172.41379 55/28 3.6
86 1186.20690 99/50 3.6
87 1200.00000 2/1 0.0

Music

Pianodactyl play by Gene Ward Smith

Retrieved from "https://en.xen.wiki/index.php?title=87edo&oldid=44145"