104edo: Difference between revisions

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+rank-2 temperaments
Nobody understands that table; replace with a normal (and shorter) one
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== 17-limit Regular Temperaments ==
== Intervals ==
''todo: clarify this table''
{| class="wikitable center-all right-2 left-3 left-4 left-5"
 
{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | '''2'''
| | '''23.08'''
|-
| | 3
| | 34.615
|-
| | 4
| | 46.15
|-
| | '''5'''
| | '''57.69'''
|-
| | '''7'''
| | '''80.77'''
|-
| | 8
| | 92.31
|-
| | 9
| | 103.85
|-
| | 10
| | 115.385
|-
| | 11
| | 126.92
|-
| | 12
| | 138.46
|-
| | '''13'''
| | '''150'''
|-
| | 14
| | 161.54
|-
| | 15
| | 173.08
|-
| | 16
| | 184.615
|-
| | 17
| | 196.15
|-
| | '''18'''
| | '''207.69'''
|-
|-
| | '''20'''
! rowspan="2"| #
| | '''230.77'''
! rowspan="2"| Cents
! colspan="3"| Approximate Ratios
|-
|-
| | 21
! of 2.3.7.11.13.17.19.25<br>Subgroup
| | 242.31
! Additional Ratios of 5<br>Tending Sharp (104c Val)
! Additional Ratios of 5<br>Tending Flat (Patent Val)
|-
|-
| | 22
| 0
| | 253.85
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
|-
| | '''23'''
| 1
| | '''265.385'''
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
|-
| | '''25'''
| 2
| | '''288.46'''
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
|-
| | 26
| 3
| | 300
| 34.615
| [[49/48]], [[50/49]]
|  
| [[81/80]], [[126/125]]
|-
|-
| | 27
| 4
| | 311.54
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
|-
| | 28
| 5
| | 323.08
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
|-
| | 29
| 6
| | 334.615
| 69.231
| [[25/24]]
|
|
|-
|-
| | '''30'''
| 7
| | '''346.15'''
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
|-
| | 31
| 8
| | 357.69
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
|-
| | 32
| 9
| | 369.23
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
|-
| | 33
| 10
| | 380.77
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
|-
| | 34
| 11
| | 392.31
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
|-
| | 35
| 12
| | 403.85
| 138.462
| [[13/12]]
|
|
|-
|-
| | 36
| 13
| | 415.385
| 150.000
| [[12/11]]
|
|
|-
|-
| | 38
| 14
| | 438.46
| 161.538
|
| [[11/10]]
|
|-
|-
| | 39
| 15
| | 450
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
|-
| | 40
| 16
| | 461.54
| 184.615
|
| [[10/9]]
|
|-
|-
| | '''41'''
| 17
| | '''473.08'''
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
|-
| | '''43'''
| 18
| | '''496.15'''
| 207.692
| 9/8
| [[17/15]]
|
|-
|-
| | '''45'''
| 19
| | '''519.23'''
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
|-
| | 46
| 20
| | 530.77
| 230.769
| [[8/7]]
|
|
|-
|-
| | 47
| 21
| | 542.31
| 242.308
|
|
| [[15/13]]
|-
|-
| | '''48'''
| 22
| | '''553.85'''
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
|-
| | 50
| 23
| | 576.92
| 265.385
| [[7/6]]
|
|
|-
|-
| | 51
| 24
| | 588.45
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
|-
| | 52
| 25
| | 600
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
|-
| | 53
| 26
| | 611.54
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
|-
| | '''54'''
| 27
| | '''623.08'''
| 311.538
|
| [[6/5]]
|
|-
|-
| | 56
| 28
| | 646.15
| 323.077
|
|
| [[6/5]]
|-
|-
| | 57
| 29
| | 657.69
| 334.615
| [[17/14]]
|
|
|-
|-
| | 58
| 30
| | 669.23
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
|-
| | 59
| 31
| | 680.77
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
|-
| | '''61'''
| 32
| | '''703.85'''
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
|-
| | 63
| 33
| | 726.92
| 380.769
|
|
| [[5/4]]
|-
|-
| | 64
| 34
| | 738.46
| 392.308
|
| [[5/4]]
|
|-
|-
| | 65
| 35
| | 750
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
|-
| | '''66'''
| 36
| | '''761.54'''
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
|-
| | 67
| 37
| | 773.08
| 426.923
| [[32/25]]
|
|
|-
|-
| | '''68'''
| 38
| | '''784.615'''
| 438.462
| [[9/7]]
|
|
|-
|-
| | 69
| 39
| | 796.15
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
|-
| | 70
| 40
| | 807.69
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
|-
| | 71
| 41
| | 819.23
| 473.077
| [[21/16]]
|
|
|-
|-
| | 72
| 42
| | 830.77
| 484.615
|
|
|
|-
|-
| | 73
| 43
| | 842.31
| 496.154
| [[4/3]]
|
|
|-
|-
| | '''74'''
| 44
| | '''853.85'''
| 507.692
|
|
|
|-
|-
| | 75
| 45
| | 865.385
| 519.231
|
| [[27/20]]
|
|-
|-
| | 76
| 46
| | 876.92
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
|-
| | 77
| 47
| | 888.46
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
|-
| | 78
| 48
| | 900
| 553.846
| [[11/8]]
|
|
|-
|-
| | 79
| 49
| | 911.54
| 565.385
| [[18/13]]
|
|
|-
|-
| | '''81'''
| 50
| | '''934.615'''
| 576.923
|
| [[7/5]]
|
|-
|-
| | 82
| 51
| | 946.15
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
|-
| | 83
| 52
| | 957.69
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
|-
| | '''84'''
|…
| | '''969.23'''
|…
|-
|…
| | 86
|…
| | 992.31
|…
|-
| | 87
| | 1003.85
|-
| | 88
| | 1015.385
|-
| | 89
| | 1026.92
|-
| | 90
| | 1038.46
|-
| | '''91'''
| | '''1050'''
|-
| | 92
| | 1061.54
|-
| | 93
| | 1073.08
|-
| | 95
| | 1096.15
|-
| | 96
| | 1107.69
|-
| | '''97'''
| | '''1119.23'''
|-
| | 99
| | 1142.31
|-
| | '''100'''
| | '''1153.85'''
|-
| | 101
| | 1165.385
|-
| | '''102'''
| | '''1176.92'''
|}
|}
[[Category:apollo]]
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:diaschismic]]

Revision as of 06:48, 4 September 2020

104edo divides the octave into 104 parts of size 11.54 cents each.

Theory

104edo has two different equally viable 5-limit vals, and both are useful. The flat major third val, ⟨104 165 241] (patent val), tempers out 3125/3072, and supports magic temperament. The sharp major third val, ⟨104 165 242] (104c val), tempers out 2048/2025 and supports diaschismic temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the optimal patent val for 11-limit magic and the 13-limit magic extension necromancy. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

Rank two temperaments

In patent val

Periods
per octave 		Generator Cents Associated
ratio 		Temperament
1 33\104 380.769 5/4 Magic / necromancy / divination
1 51\104 588.462 7/5 Untriton
4 9\104 103.846 Undim

In 104c val

Periods
per octave 		Generator Cents Associated
ratio 		Temperament
1 21\104 242.308 147/128 Septiquarter
1 27\104 311.538 6/5 Oolong
1 47\104 542.308 15/11 Casablanca / marrakesh
2 43\104 496.154 4/3 Diaschismic
8 50\104 576.923 121/84 Octowerck (7- or 11-limit)

Intervals

# Cents Approximate Ratios
of 2.3.7.11.13.17.19.25
Subgroup 		Additional Ratios of 5
Tending Sharp (104c Val) 		Additional Ratios of 5
Tending Flat (Patent Val)
0 0.000 1/1 126/125 225/224, 100/99
1 11.538 225/224, 100/99
2 23.077 64/63 81/80, 225/224 50/49
3 34.615 49/48, 50/49 81/80, 126/125
4 46.154 36/35, 50/49
5 57.692 28/27, 33/32 25/24, 36/35
6 69.231 25/24
7 80.769 22/21 25/24, 21/20 20/19
8 92.308 19/18 20/19 21/20
9 103.846 17/16, 18/17 16/15
10 115.385 16/15, 15/14
11 126.923 14/13 15/14
12 138.462 13/12
13 150.000 12/11
14 161.538 11/10
15 173.077 21/19 10/9, 11/10
16 184.615 10/9
17 196.154 28/25, 19/17
18 207.692 9/8 17/15
19 219.231 25/22 17/15
20 230.769 8/7
21 242.308 15/13
22 253.846 22/19 15/13
23 265.385 7/6
24 276.923 75/64 20/17
25 288.462 32/27, 13/11 20/17
26 300.000 25/21, 19/16
27 311.538 6/5
28 323.077 6/5
29 334.615 17/14
30 346.154 11/9, 39/32
31 357.692 27/22, 16/13
32 369.231 26/21, 21/17
33 380.769 5/4
34 392.308 5/4
35 403.846 63/50, 24/19 19/15
36 415.385 81/64, 14/11 19/15
37 426.923 32/25
38 438.462 9/7
39 450.000 22/17 13/10
40 461.538 17/13 13/10
41 473.077 21/16
42 484.615
43 496.154 4/3
44 507.692
45 519.231 27/20
46 530.769 19/14 27/20, 15/11
47 542.308 26/19 15/11
48 553.846 11/8
49 565.385 18/13
50 576.923 7/5
51 588.462 45/32, 7/5
52 600.000 17/12, 24/17 45/32, 64/45
… … … … …
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