Xenharmonic Wiki

104edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
m →Rank two temperaments: table cleanup
Line 1: Line 1:
'''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]]
Retrieved from "https://en.xen.wiki/w/104edo"