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'''104edo''' divides the [[octave]] into 104 parts of size 11.5385 [[cent|cents]] each.  
'''104 EDO''' divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s 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 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 [[Magic_family #Necromancy|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 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.
{{Primes in edo|104|prec=2}}
 
104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{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 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.
Line 10: Line 13:
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"
{| class="wikitable center-all"
!Periods<br>per octave
! Periods <br> per octave
!Generator
! Generator
! Cents
! Cents
!Associated ratio
! Associated ratio
!Temperament
! Temperament
|-
|-
| rowspan="2" |1
| rowspan="2" | 1
|33\104
| 33\104
|380.769
| 380.769
| 5/4
| 5/4
|[[Magic]] / necromancy / divination
| [[Magic]] / necromancy / divination
|-
|-
|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"
{| 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
|-
|-
| rowspan="3" |1
| rowspan="3" | 1
|21\104
| 21\104
| 242.308
| 242.308
|147/128
| 147/128
|[[Septiquarter]]
| [[Septiquarter]]
|-
|-
|27\104
| 27\104
|311.538
| 311.538
|6/5
| 6/5
|[[Oolong]]
| [[Oolong]]
|-
|-
|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"
{| 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]]
[[20/19]]
Line 141: Line 148:
[[21/20]]
[[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]]
|
|
|-
|-
|…
| …
|…
| …
|…
| …
|…
| …
|…
| …
|}
|}
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.
 
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"
{| class="wikitable"
|+Circulating temperaments in 104edo
|+Circulating temperaments in 104edo
!Tones
! Tones
!Pattern
! Pattern
!L:s
! L:s
|-
|-
|5
| 5
|[[4L 1s]]
| [[4L 1s]]
|21:20
| 21:20
|-
|-
|6
| 6
|[[2L 4s]]
| [[2L 4s]]
|18:17
| 18:17
|-
|-
|7
| 7
|[[6L 1s]]
| [[6L 1s]]
|15:14
| 15:14
|-
|-
|8
| 8
|[[8edo]]
| [[8edo]]
|equal
| equal
|-
|-
|9
| 9
|[[5L 4s]]
| [[5L 4s]]
|12:11
| 12:11
|-
|-
|10
| 10
|[[4L 6s]]
| [[4L 6s]]
|11:10
| 11:10
|-
|-
|11
| 11
|[[5L 6s]]
| [[5L 6s]]
|10:9
| 10:9
|-
|-
|12
| 12
|[[8L 4s]]
| [[8L 4s]]
|9:8
| 9:8
|-
|-
|13
| 13
|[[13edo]]
| [[13edo]]
|equal
| equal
|-
|-
|14
| 14
|[[4L 10s]]
| [[4L 10s]]
|8:7
| 8:7
|-
|-
|15
| 15
|[[14L 1s]]
| [[14L 1s]]
| rowspan="3" |7:6
| rowspan="3" |7:6
|-
|-
|16
| 16
|8L 8s
| 8L 8s
|-
|-
|17
| 17
|[[2L 15s]]
| [[2L 15s]]
|-
|-
|18
| 18
|12L 6s
| 12L 6s
| rowspan="3" |6:5
| rowspan="3" |6:5
|-
|-
|19
| 19
|[[9L 10s]]
| [[9L 10s]]
|-
|-
|20
| 20
|4L 16s
| 4L 16s
|-
|-
|21
| 21
|20L 1s
| 20L 1s
| rowspan="5" |5:4
| rowspan="5" |5:4
|-
|-
|22
| 22
|16L 6s
| 16L 6s
|-
|-
|23
| 23
|[[12L 11s]]
| [[12L 11s]]
|-
|-
|24
| 24
|8L 16s
| 8L 16s
|-
|-
|25
| 25
|4L 21s
| 4L 21s
|-
|-
|26
| 26
|[[26edo]]
| [[26edo]]
|equal
| equal
|-
|-
|27
| 27
|23L 4s
| 23L 4s
| rowspan="8" |4:3
| rowspan="8" |4:3
|-
|-
|28
| 28
|20L 8s
| 20L 8s
|-
|-
|29
| 29
|[[17L 12s]]
| [[17L 12s]]
|-
|-
|30
| 30
|14L 16s
| 14L 16s
|-
|-
|31
| 31
|11L 20s
| 11L 20s
|-
|-
|32
| 32
|8L 24s
| 8L 24s
|-
|-
|33
| 33
|5L 28s
| 5L 28s
|-
|-
|34
| 34
|2L 32s
| 2L 32s
|-
|-
|35
| 35
|34L 1s
| 34L 1s
| rowspan="17" |3:2
| rowspan="17" |3:2
|-
|-
|36
| 36
|32L 4s
| 32L 4s
|-
|-
|37
| 37
|30L 7s
| 30L 7s
|-
|-
|38
| 38
|28L 10s
| 28L 10s
|-
|-
|39
| 39
|26L 13s
| 26L 13s
|-
|-
|40
| 40
|24L 16s
| 24L 16s
|-
|-
|41
| 41
|22L 19s
| 22L 19s
|-
|-
|42
| 42
|20L 22s
| 20L 22s
|-
|-
|43
| 43
|18L 25s
| 18L 25s
|-
|-
|44
| 44
|16L 28s
| 16L 28s
|-
|-
|45
| 45
|14L 31s
| 14L 31s
|-
|-
|46
| 46
|12L 34s
| 12L 34s
|-
|-
|47
| 47
|10L 37s
| 10L 37s
|-
|-
|48
| 48
|8L 40s
| 8L 40s
|-
|-
|49
| 49
|6L 43s
| 6L 43s
|-
|-
|50
| 50
|4L 46s
| 4L 46s
|-
|-
|51
| 51
|2L 46s
| 2L 46s
|-
|-
|52
| 52
|[[52edo]]
| [[52edo]]
|equal
| equal
|-
|-
|53
| 53
|51L 2s
| 51L 2s
| rowspan="31" |2:1
| rowspan="31" |2:1
|-
|-
|54
| 54
|50L 4s
| 50L 4s
|-
|-
|55
| 55
|49L 6s
| 49L 6s
|-
|-
|56
| 56
|48L 8s
| 48L 8s
|-
|-
|57
| 57
|47L 10s
| 47L 10s
|-
|-
|58
| 58
|46L 12s
| 46L 12s
|-
|-
|59
| 59
|45L 14s
| 45L 14s
|-
|-
|60
| 60
|44L 16s
| 44L 16s
|-
|-
|61
| 61
|43L 18s
| 43L 18s
|-
|-
|62
| 62
|42L 20s
| 42L 20s
|-
|-
|63
| 63
|41L 22s
| 41L 22s
|-
|-
|64
| 64
|40L 24s
| 40L 24s
|-
|-
|65
| 65
|39L 26s
| 39L 26s
|-
|-
|66
| 66
|38L 28s
| 38L 28s
|-
|-
|67
| 67
|37L 30s
| 37L 30s
|-
|-
|68
| 68
|36L 32s
| 36L 32s
|-
|-
|69
| 69
|35L 34s
| 35L 34s
|-
|-
|70
| 70
|34L 36s
| 34L 36s
|-
|-
|71
| 71
|33L 38s
| 33L 38s
|-
|-
|72
| 72
|32L 40s
| 32L 40s
|-
|-
|73
| 73
|31L 42s
| 31L 42s
|-
|-
|74
| 74
|30L 44s
| 30L 44s
|-
|-
|75
| 75
|29L 46s
| 29L 46s
|-
|-
|76
| 76
|28L 48s
| 28L 48s
|-
|-
|77
| 77
|27L 50s
| 27L 50s
|-
|-
|78
| 78
|26L 52s
| 26L 52s
|-
|-
|79
| 79
|25L 54s
| 25L 54s
|-
|-
|80
| 80
|24L 56s
| 24L 56s
|-
|-
|81
| 81
|23L 58s
| 23L 58s
|-
|-
|82
| 82
|22L 60s
| 22L 60s
|-
|-
|83
| 83
|21L 62s
| 21L 62s
|}
|}
[[Category:apollo]]
 
[[Category:diaschismic]]
[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:Magic]]
[[Category:necromancy]]
[[Category:Necromancy]]
Retrieved from "https://en.xen.wiki/w/104edo"