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'''104edo''' divides the [[octave]] into 104 parts of size 11.5385 [[cent|cents]] each.
{{Infobox ET}}
{{ED intro}}


==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 is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.<!-- Add commas in 2.3.7.11.13.17.19 as well --> It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 & 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth.


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.
Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.


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.
If prime 5 is desired, 104edo 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 [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament.


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&amp;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.
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 also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.


== Rank two temperaments==
104edo 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]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit.
===In patent val ===
 
{| class="wikitable center-all"
=== Prime harmonics ===
!Periods<br>per octave
{{Harmonics in equal|104}}
!Generator
 
! Cents
=== Octave stretch ===
!Associated ratio
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]].
!Temperament
 
=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2<sup>3</sup> × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperament
|-
|-
| rowspan="2" |1
| 1
|33\104
| 33\104
|380.769
| 380.77
| 5/4
| 5/4
|[[Magic]] / necromancy / divination
| [[Magic]] / necromancy / divination
|-
|-
|51\104
| 1
|588.462
| 51\104
|7/5
| 588.46
|[[Untriton]]
| 7/5
| [[Untriton]]
|-
|-
|4
| 4
|9\104
| 9\104
|103.846
| 103.85
|18/17
| 18/17
|[[Undim]]
| [[Undim]]
|}
|}


===In 104c val===
{| class="wikitable center-all left-5"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | 104c val
!Periods<br>per octave
|-
!Generator<br>(reduced)
! Periods<br />per 8ve
!Cents<br>(reduced)
! Generator*
!Associated ratio<br>(reduced)
! Cents*
!Temperament
! Associated<br />ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
|-
| rowspan="3" |1
| 1
|21\104
| 21\104
| 242.308
| 242.31
|147/128
| 147/128
|[[Septiquarter]]
| [[Septiquarter]]
|-
|-
|27\104
| 1
|311.538
| 27\104
|6/5
| 311.54
|[[Oolong]]
| 6/5
| [[Oolong]]
|-
|-
|47\104
| 1
| 542.308
| 47\104
| 542.31
| 15/11
| 15/11
|[[Casablanca]] / marrakesh
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
|-
|2
| 2
|43\104
| 43\104<br />(9\104)
|496.154
| 496.15<br />(103.85)
|4/3
| 4/3<br />(17/16)
|[[Diaschismic]]
| [[Diaschismic]]
|-
|-
|8
| 8
|50\104<br>(2\104)
| 49\104<br />(2\104)
|576.923<br>(23.077)
| 565.38<br />(34.62)
|121/84<br>(78/77)
| 168/121<br />(55/54)
|[[Octowerck]] (7- or 11-limit)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104<br />(1\104)
| 496.15<br />(11.54)
| 4/3<br />(225/224)
| [[Bosonic]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


==Intervals==
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-1 right-2"
|-
|-
! 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 2.3.25.7.11.13.17.19<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
| 0.0
|[[1/1]]
| [[1/1]]
|[[126/125]]
|  
|[[225/224]], [[100/99]]
|  
|-
|-
|1
| 1
|11.538
| 11.5
| [[225/224]], [[100/99]]
| [[144/143]], [[169/168]]
|
| ''[[91/90]]'', [[121/120]]
|
| [[105/104]], [[196/195]]
|-
|-
|2
| 2
|23.077
| 23.1
|[[64/63]]
| [[64/63]], [[99/98]]
|[[81/80]], [[225/224]]
| [[81/80]], [[100/99]], ''[[105/104]]''
|[[50/49]]
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
|-
|3
| 3
|34.615
| 34.6
|[[49/48]], [[50/49]]
| [[49/48]], [[50/49]]
|
| [[55/54]]
|[[81/80]], [[126/125]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
|-
|4
| 4
|46.154
| 46.2
|
|
|[[36/35]], [[50/49]]
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|
|-
|-
|5
| 5
|57.692
| 57.7
|[[28/27]], [[33/32]]
| [[28/27]], [[33/32]]
|
| ''[[26/25]]''
|[[25/24]], [[36/35]]
| ''[[25/24]]'', ''[[36/35]]''
|-
|-
|6
| 6
|69.231
| 69.2
|[[25/24]]
| [[25/24]], [[26/25]], [[27/26]]
|
|
|
|
|-
|-
|7
| 7
|80.769
| 80.8
|[[22/21]]
| [[22/21]]
|[[25/24]], [[21/20]]
| [[21/20]], ''[[25/24]]''
|[[20/19]]
| ''[[20/19]]'', ''[[26/25]]''
|-
|-
|8
| 8
|92.308
| 92.3
|[[19/18]]
| [[19/18]]
|
| [[20/19]]
[[20/19]]
| ''[[21/20]]''
|
[[21/20]]
|-
|-
|9
| 9
|103.846
| 103.8
|[[17/16]], [[18/17]]
| [[17/16]], [[18/17]]
|
| ''[[16/15]]''
[[16/15]]
|
|
|-
|-
|10
| 10
|115.385
| 115.4
|
|
|
|
|[[16/15]], [[15/14]]
| [[16/15]], [[15/14]]
|-
|-
|11
| 11
|126.923
| 126.9
|[[14/13]]
| [[14/13]]
|[[15/14]]
| ''[[15/14]]''
|
|
|-
|-
|12
| 12
|138.462
| 138.5
|[[13/12]]
| [[13/12]]
|
|
|
|
|-
|-
|13
| 13
|150.000
| 150.0
|[[12/11]]
| [[12/11]]
|
|
|
|
|-
|-
|14
| 14
|161.538
| 161.5
|
|
|[[11/10]]
| [[11/10]]
|
|
|-
|-
|15
| 15
|173.077
| 173.1
|[[21/19]]
| [[21/19]]
|
|
|[[10/9]], [[11/10]]
| ''[[10/9]]'', ''[[11/10]]''
|-
|-
|16
| 16
|184.615
| 184.6
|
|
|[[10/9]]
| [[10/9]]
|
|
|-
|-
|17
| 17
|196.154
| 196.2
|[[28/25]], [[19/17]]
| [[19/17]], [[28/25]]
|
|
|
|
|-
|-
|18
| 18
|207.692
| 207.7
|9/8
| [[9/8]]
|[[17/15]]
| ''[[17/15]]''
|
|
|-
|-
|19
| 19
|219.231
| 219.2
|[[25/22]]
| [[25/22]]
|
|
|[[17/15]]
| [[17/15]]
|-
|-
|20
| 20
|230.769
| 230.8
|[[8/7]]
| [[8/7]]
|
|
|
|
|-
|-
| 21
| 21
|242.308
| 242.3
|
| [[38/33]]
|
|  
|[[15/13]]
| [[15/13]]
|-
|-
|22
| 22
|253.846
| 253.8
|[[22/19]]
| [[22/19]]
|[[15/13]]
| ''[[15/13]]''
|
|
|-
|-
|23
| 23
|265.385
| 265.4
|[[7/6]]
| [[7/6]]
|
|
|
|
|-
|-
|24
| 24
|276.923
| 276.9
|[[75/64]]
| [[75/64]]
|
|
|[[20/17]]
| [[20/17]]
|-
|-
| 25
| 25
|288.462
| 288.5
|[[32/27]], [[13/11]]
| [[13/11]], [[32/27]]
|[[20/17]]
| ''[[20/17]]''
|
|
|-
|-
| 26
| 26
|300.000
| 300.0
|[[25/21]], [[19/16]]
| [[19/16]], [[25/21]]
|
|
|
|
|-
|-
|27
| 27
|311.538
| 311.5
|
|
|[[6/5]]
| [[6/5]]
|
|
|-
|-
|28
| 28
|323.077
| 323.1
|
|
|
|
|[[6/5]]
| ''[[6/5]]'', ''[[40/33]]''
|-
|-
|29
| 29
|334.615
| 334.6
|[[17/14]]
| [[17/14]]
|
| [[40/33]]
|
|
|-
|-
|30
| 30
|346.154
| 346.2
|[[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
|
|
|-
|-
|31
| 31
|357.692
| 357.7
|[[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
|
|
|
|
|-
|-
|32
| 32
|369.231
| 369.2
|[[26/21]], [[21/17]]
| [[21/17]], [[26/21]]
|
|
|
|
|-
|-
| 33
| 33
|380.769
| 380.8
|
|
|
|
|[[5/4]]
| [[5/4]]
|-
|-
|34
| 34
|392.308
| 392.3
|
|
|[[5/4]]
| ''[[5/4]]''
|
|
|-
|-
|35
| 35
|403.846
| 403.8
|[[63/50]], [[24/19]]
| [[24/19]], [[63/50]]
|[[19/15]]
| [[19/15]]
|
|
|-
|-
| 36
| 36
|415.385
| 415.4
|[[81/64]], [[14/11]]
| [[14/11]]
|
|
|
[[19/15]]
| ''[[19/15]]''
|-
|-
|37
| 37
|426.923
| 426.9
|[[32/25]]
| [[32/25]]
|
|
|
|
|-
|-
|38
| 38
|438.462
| 438.5
|[[9/7]]
| [[9/7]]
|
|
|
|
|-
|-
|39
| 39
|450.000
| 450.0
|[[22/17]]
| [[22/17]]
|[[13/10]]
| [[13/10]]
|
|
|-
|-
|40
| 40
|461.538
| 461.5
|[[17/13]]
| [[17/13]]
|
|
|[[13/10]]
| ''[[13/10]]''
|-
|-
|41
| 41
|473.077
| 473.1
|[[21/16]]
| [[21/16]]
|
|
|
|
|-
|-
|42
| 42
|484.615
| 484.6
|
|
|
|
|
|
|-
|-
|43
| 43
|496.154
| 496.2
|[[4/3]]
| [[4/3]]
|
|
|
|
|-
|-
|44
| 44
|507.692
| 507.7
|
|
|
|
|
|
|-
|-
|45
| 45
|519.231
| 519.2
|
|
|[[27/20]]
| [[27/20]]
|
|
|-
|-
|46
| 46
|530.769
| 530.8
|[[19/14]]
| [[19/14]]
|
|
|[[27/20]], [[15/11]]
| ''[[27/20]]'', ''[[15/11]]''
|-
|-
|47
| 47
|542.308
| 542.3
|[[26/19]]
| [[26/19]]
|[[15/11]]
| [[15/11]]
|
|
|-
|-
|48
| 48
|553.846
| 553.8
|[[11/8]]
| [[11/8]]
|
|
|
|
|-
|-
|49
| 49
|565.385
| 565.4
|[[18/13]]
| [[18/13]]
|
|
|
|
|-
|-
|50
| 50
|576.923
| 576.9
|
|
|[[7/5]]
| [[7/5]]
|
|
|-
|-
|51
| 51
|588.462
| 588.5
|
|
|
|
|[[45/32]], [[7/5]]
| ''[[7/5]]'', [[45/32]]
|-
|-
|52
| 52
|600.000
| 600.0
|[[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.
 
{| class="wikitable"
[[Category:Apollo]]
|+Circulating temperaments in 104edo
[[Category:Diaschismic]]
!Tones
[[Category:Magic]]
!Pattern
[[Category:Necromancy]]
!L:s
|-
|5
|[[4L 1s]]
|21:20
|-
|6
|[[2L 4s]]
|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:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]
Retrieved from "https://en.xen.wiki/w/104edo"