104edo: Difference between revisions

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'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each.  
'''104edo''' divides the [[octave]] into 104 parts of size 11.5385 [[cent|cents]] each.  


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


Line 10: Line 10:
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.
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 ==
== Rank two temperaments==
=== In patent val ===
===In patent val ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all"
! Periods<br>per octave
!Periods<br>per octave
! Generator
!Generator
! Cents
! Cents
! Associated ratio
!Associated ratio
! Temperament
!Temperament
|-
|-
| 1
| rowspan="2" |1
| 33\104
|33\104
| 380.769
|380.769
| 5/4
| 5/4
| [[Magic]] / necromancy / divination
|[[Magic]] / necromancy / divination
|-
|-
| 1
|51\104
| 51\104
|588.462
| 588.462
|7/5
| 7/5
|[[Untriton]]
| [[Untriton]]
|-
|-
| 4
|4
| 9\104
|9\104
| 103.846
|103.846
| 18/17
|18/17
| [[Undim]]
|[[Undim]]
|}
|}


=== In 104c val ===
===In 104c val===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all"
! Periods<br>per octave
!Periods<br>per octave
! Generator<br>(reduced)
!Generator<br>(reduced)
! Cents<br>(reduced)
!Cents<br>(reduced)
! Associated ratio<br>(reduced)
!Associated ratio<br>(reduced)
! Temperament
!Temperament
|-
|-
| 1
| rowspan="3" |1
| 21\104
|21\104
| 242.308
| 242.308
| 147/128
|147/128
| [[Septiquarter]]
|[[Septiquarter]]
|-
|-
| 1
|27\104
| 27\104
|311.538
| 311.538
|6/5
| 6/5
|[[Oolong]]
| [[Oolong]]
|-
|-
| 1
|47\104
| 47\104
| 542.308
| 542.308
| 15/11
| 15/11
| [[Casablanca]] / marrakesh
|[[Casablanca]] / marrakesh
|-
|-
| 2
|2
| 43\104
|43\104
| 496.154
|496.154
| 4/3
|4/3
| [[Diaschismic]]
|[[Diaschismic]]
|-
|-
| 8
|8
| 50\104<br>(2\104)
|50\104<br>(2\104)
| 576.923<br>(23.077)
|576.923<br>(23.077)
| 121/84<br>(78/77)
|121/84<br>(78/77)
| [[Octowerck]] (7- or 11-limit)
|[[Octowerck]] (7- or 11-limit)
|}
|}


== Intervals ==
==Intervals==
{| class="wikitable center-all right-2 left-3 left-4 left-5"
{| class="wikitable center-all"
|-
|-
! rowspan="2"| #
! rowspan="2" |#
! rowspan="2"| Cents
! rowspan="2" |Cents
! colspan="3"| Approximate Ratios
! colspan="3" | Approximate Ratios
|-
|-
! of 2.3.7.11.13.17.19.25<br>Subgroup
!of 2.3.7.11.13.17.19.25<br>Subgroup
! Additional Ratios of 5<br>Tending Sharp (104c Val)
!Additional Ratios of 5<br>Tending Sharp (104c Val)
! Additional Ratios of 5<br>Tending Flat (Patent Val)
!Additional Ratios of 5<br>Tending Flat (Patent Val)
|-
|-
| 0
| 0
| 0.000
|0.000
| [[1/1]]
|[[1/1]]
| [[126/125]]
|[[126/125]]
| [[225/224]], [[100/99]]
|[[225/224]], [[100/99]]
|-
|-
| 1
|1
| 11.538
|11.538
| [[225/224]], [[100/99]]
| [[225/224]], [[100/99]]
|
|
|
|
|-
|-
| 2
|2
| 23.077
|23.077
| [[64/63]]
|[[64/63]]
| [[81/80]], [[225/224]]
|[[81/80]], [[225/224]]
| [[50/49]]
|[[50/49]]
|-
|-
| 3
|3
| 34.615
|34.615
| [[49/48]], [[50/49]]
|[[49/48]], [[50/49]]
|  
|
| [[81/80]], [[126/125]]
|[[81/80]], [[126/125]]
|-
|-
| 4
|4
| 46.154
|46.154
|
|
| [[36/35]], [[50/49]]
|[[36/35]], [[50/49]]
|
|
|-
|-
| 5
|5
| 57.692
|57.692
| [[28/27]], [[33/32]]
|[[28/27]], [[33/32]]
|
|
| [[25/24]], [[36/35]]
|[[25/24]], [[36/35]]
|-
|-
| 6
|6
| 69.231
|69.231
| [[25/24]]
|[[25/24]]
|
|
|
|
|-
|-
| 7
|7
| 80.769
|80.769
| [[22/21]]
|[[22/21]]
| [[25/24]], [[21/20]]
|[[25/24]], [[21/20]]
| [[20/19]]
|[[20/19]]
|-
|-
| 8
|8
| 92.308
|92.308
| [[19/18]]
|[[19/18]]
| [[20/19]]
|
| [[21/20]]
[[20/19]]
|
[[21/20]]
|-
|-
| 9
|9
| 103.846
|103.846
| [[17/16]], [[18/17]]
|[[17/16]], [[18/17]]
| [[16/15]]
|
[[16/15]]
|
|
|-
|-
| 10
|10
| 115.385
|115.385
|
|
|
|
| [[16/15]], [[15/14]]
|[[16/15]], [[15/14]]
|-
|-
| 11
|11
| 126.923
|126.923
| [[14/13]]
|[[14/13]]
| [[15/14]]
|[[15/14]]
|
|
|-
|-
| 12
|12
| 138.462
|138.462
| [[13/12]]
|[[13/12]]
|
|
|
|
|-
|-
| 13
|13
| 150.000
|150.000
| [[12/11]]
|[[12/11]]
|
|
|
|
|-
|-
| 14
|14
| 161.538
|161.538
|
|
| [[11/10]]
|[[11/10]]
|
|
|-
|-
| 15
|15
| 173.077
|173.077
| [[21/19]]
|[[21/19]]
|
|
| [[10/9]], [[11/10]]
|[[10/9]], [[11/10]]
|-
|-
| 16
|16
| 184.615
|184.615
|
|
| [[10/9]]
|[[10/9]]
|
|
|-
|-
| 17
|17
| 196.154
|196.154
| [[28/25]], [[19/17]]
|[[28/25]], [[19/17]]
|
|
|
|
|-
|-
| 18
|18
| 207.692
|207.692
| 9/8
|9/8
| [[17/15]]
|[[17/15]]
|
|
|-
|-
| 19
|19
| 219.231
|219.231
| [[25/22]]
|[[25/22]]
|
|
| [[17/15]]
|[[17/15]]
|-
|-
| 20
|20
| 230.769
|230.769
| [[8/7]]
|[[8/7]]
|
|
|
|
|-
|-
| 21
| 21
| 242.308
|242.308
|
|
|
|
| [[15/13]]
|[[15/13]]
|-
|-
| 22
|22
| 253.846
|253.846
| [[22/19]]
|[[22/19]]
| [[15/13]]
|[[15/13]]
|
|
|-
|-
| 23
|23
| 265.385
|265.385
| [[7/6]]
|[[7/6]]
|
|
|
|
|-
|-
| 24
|24
| 276.923
|276.923
| [[75/64]]
|[[75/64]]
|
|
| [[20/17]]
|[[20/17]]
|-
|-
| 25
| 25
| 288.462
|288.462
| [[32/27]], [[13/11]]
|[[32/27]], [[13/11]]
| [[20/17]]
|[[20/17]]
|
|
|-
|-
| 26
| 26
| 300.000
|300.000
| [[25/21]], [[19/16]]
|[[25/21]], [[19/16]]
|
|
|
|
|-
|-
| 27
|27
| 311.538
|311.538
|
|
| [[6/5]]
|[[6/5]]
|
|
|-
|-
| 28
|28
| 323.077
|323.077
|
|
|
|
| [[6/5]]
|[[6/5]]
|-
|-
| 29
|29
| 334.615
|334.615
| [[17/14]]
|[[17/14]]
|
|
|
|
|-
|-
| 30
|30
| 346.154
|346.154
| [[11/9]], [[39/32]]
|[[11/9]], [[39/32]]
|
|
|
|
|-
|-
| 31
|31
| 357.692
|357.692
| [[27/22]], [[16/13]]
|[[27/22]], [[16/13]]
|
|
|
|
|-
|-
| 32
|32
| 369.231
|369.231
| [[26/21]], [[21/17]]
|[[26/21]], [[21/17]]
|
|
|
|
|-
|-
| 33
| 33
| 380.769
|380.769
|
|
|
|
| [[5/4]]
|[[5/4]]
|-
|-
| 34
|34
| 392.308
|392.308
|
|
| [[5/4]]
|[[5/4]]
|
|
|-
|-
| 35
|35
| 403.846
|403.846
| [[63/50]], [[24/19]]
|[[63/50]], [[24/19]]
| [[19/15]]
|[[19/15]]
|
|
|-
|-
| 36
| 36
| 415.385
|415.385
| [[81/64]], [[14/11]]
|[[81/64]], [[14/11]]
|
|
|
| [[19/15]]
[[19/15]]
|-
|-
| 37
|37
| 426.923
|426.923
| [[32/25]]
|[[32/25]]
|
|
|
|
|-
|-
| 38
|38
| 438.462
|438.462
| [[9/7]]
|[[9/7]]
|
|
|
|
|-
|-
| 39
|39
| 450.000
|450.000
| [[22/17]]
|[[22/17]]
| [[13/10]]
|[[13/10]]
|
|
|-
|-
| 40
|40
| 461.538
|461.538
| [[17/13]]
|[[17/13]]
|
|
| [[13/10]]
|[[13/10]]
|-
|-
| 41
|41
| 473.077
|473.077
| [[21/16]]
|[[21/16]]
|
|
|
|
|-
|-
| 42
|42
| 484.615
|484.615
|
|
|
|
|
|
|-
|-
| 43
|43
| 496.154
|496.154
| [[4/3]]
|[[4/3]]
|
|
|
|
|-
|-
| 44
|44
| 507.692
|507.692
|
|
|
|
|
|
|-
|-
| 45
|45
| 519.231
|519.231
|
|
| [[27/20]]
|[[27/20]]
|
|
|-
|-
| 46
|46
| 530.769
|530.769
| [[19/14]]
|[[19/14]]
|
|
| [[27/20]], [[15/11]]
|[[27/20]], [[15/11]]
|-
|-
| 47
|47
| 542.308
|542.308
| [[26/19]]
|[[26/19]]
| [[15/11]]
|[[15/11]]
|
|
|-
|-
| 48
|48
| 553.846
|553.846
| [[11/8]]
|[[11/8]]
|
|
|
|
|-
|-
| 49
|49
| 565.385
|565.385
| [[18/13]]
|[[18/13]]
|
|
|
|
|-
|-
| 50
|50
| 576.923
|576.923
|
|
| [[7/5]]
|[[7/5]]
|
|
|-
|-
| 51
|51
| 588.462
|588.462
|
|
|
|
| [[45/32]], [[7/5]]
|[[45/32]], [[7/5]]
|-
|-
| 52
|52
| 600.000
|600.000
| [[17/12]], [[24/17]]
|[[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|[[45/32]], [[64/45]]
|
|
|-
|-
Line 412: Line 413:
|…
|…
|}
|}
 
Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13edo]], it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci edos.
{| class="wikitable"
|+Circulating temperaments in 104edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|21:20
|-
|6
|[[2L 6s]]
|18:17
|-
|7
|[[6L 1s]]
|15:14
|-
|8
|[[8edo]]
|equal
|-
|9
|[[5L 4s]]
|12:11
|-
|10
|[[4L 6s]]
|11:10
|-
|11
|[[5L 6s]]
|10:9
|-
|12
|[[8L 4s]]
|9:8
|-
|13
|[[13edo]]
|equal
|-
|14
|[[4L 10s]]
|8:7
|-
|15
|[[14L 1s]]
| rowspan="3" |7:6
|-
|16
|8L 8s
|-
|17
|[[2L 15s]]
|-
|18
|12L 6s
| rowspan="3" |6:5
|-
|19
|[[9L 10s]]
|-
|20
|4L 16s
|-
|21
|20L 1s
| rowspan="5" |5:4
|-
|22
|16L 6s
|-
|23
|[[12L 11s]]
|-
|24
|8L 16s
|-
|25
|4L 21s
|-
|26
|[[26edo]]
|equal
|-
|27
|23L 4s
| rowspan="8" |4:3
|-
|28
|20L 8s
|-
|29
|[[17L 12s]]
|-
|30
|14L 16s
|-
|31
|11L 20s
|-
|32
|8L 24s
|-
|33
|5L 28s
|-
|34
|2L 32s
|-
|35
|34L 1s
| rowspan="17" |3:2
|-
|36
|32L 4s
|-
|37
|30L 7s
|-
|38
|28L 10s
|-
|39
|26L 13s
|-
|40
|24L 16s
|-
|41
|22L 19s
|-
|42
|20L 22s
|-
|43
|18L 25s
|-
|44
|16L 28s
|-
|45
|14L 31s
|-
|46
|12L 34s
|-
|47
|10L 37s
|-
|48
|8L 40s
|-
|49
|6L 43s
|-
|50
|4L 46s
|-
|51
|2L 46s
|-
|52
|[[52edo]]
|equal
|-
|53
|51L 2s
| rowspan="31" |2:1
|-
|54
|50L 4s
|-
|55
|49L 6s
|-
|56
|48L 8s
|-
|57
|47L 10s
|-
|58
|46L 12s
|-
|59
|45L 14s
|-
|60
|44L 16s
|-
|61
|43L 18s
|-
|62
|42L 20s
|-
|63
|41L 22s
|-
|64
|40L 24s
|-
|65
|39L 26s
|-
|66
|38L 28s
|-
|67
|37L 30s
|-
|68
|36L 32s
|-
|69
|35L 34s
|-
|70
|34L 36s
|-
|71
|33L 38s
|-
|72
|32L 40s
|-
|73
|31L 42s
|-
|74
|30L 44s
|-
|75
|29L 46s
|-
|76
|28L 48s
|-
|77
|27L 50s
|-
|78
|26L 52s
|-
|79
|25L 54s
|-
|80
|24L 56s
|-
|81
|23L 58s
|-
|82
|22L 60s
|-
|83
|21L 62s
|}
[[Category:apollo]]
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:diaschismic]]

Revision as of 21:49, 20 April 2021

104edo divides the octave into 104 parts of size 11.5385 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
51\104 588.462 7/5 Untriton
4 9\104 103.846 18/17 Undim

In 104c val

Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated ratio
(reduced) 		Temperament
1 21\104 242.308 147/128 Septiquarter
27\104 311.538 6/5 Oolong
47\104 542.308 15/11 Casablanca / marrakesh
2 43\104 496.154 4/3 Diaschismic
8 50\104
(2\104) 		576.923
(23.077) 		121/84
(78/77) 		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
… … … … …

Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*13edo, it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci edos.

Circulating temperaments in 104edo
Tones Pattern L:s
5 4L 1s 21:20
6 2L 6s 18:17
7 6L 1s 15:14
8 8edo equal
9 5L 4s 12:11
10 4L 6s 11:10
11 5L 6s 10:9
12 8L 4s 9:8
13 13edo equal
14 4L 10s 8:7
15 14L 1s 7:6
16 8L 8s
17 2L 15s
18 12L 6s 6:5
19 9L 10s
20 4L 16s
21 20L 1s 5:4
22 16L 6s
23 12L 11s
24 8L 16s
25 4L 21s
26 26edo equal
27 23L 4s 4:3
28 20L 8s
29 17L 12s
30 14L 16s
31 11L 20s
32 8L 24s
33 5L 28s
34 2L 32s
35 34L 1s 3:2
36 32L 4s
37 30L 7s
38 28L 10s
39 26L 13s
40 24L 16s
41 22L 19s
42 20L 22s
43 18L 25s
44 16L 28s
45 14L 31s
46 12L 34s
47 10L 37s
48 8L 40s
49 6L 43s
50 4L 46s
51 2L 46s
52 52edo equal
53 51L 2s 2:1
54 50L 4s
55 49L 6s
56 48L 8s
57 47L 10s
58 46L 12s
59 45L 14s
60 44L 16s
61 43L 18s
62 42L 20s
63 41L 22s
64 40L 24s
65 39L 26s
66 38L 28s
67 37L 30s
68 36L 32s
69 35L 34s
70 34L 36s
71 33L 38s
72 32L 40s
73 31L 42s
74 30L 44s
75 29L 46s
76 28L 48s
77 27L 50s
78 26L 52s
79 25L 54s
80 24L 56s
81 23L 58s
82 22L 60s
83 21L 62s
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