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'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[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 right-3 left-5"
=== Prime harmonics ===
! Periods<br>per octave
{{Harmonics in equal|104}}
! Generator
 
! Cents
=== Octave stretch ===
! Associated<br>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]].
 
=== 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
! Temperament
|-
|-
| 1
| 1
| 33\104
| 33\104
| 380.769
| 380.77
| 5/4
| 5/4
| [[Magic]] / necromancy / divination
| [[Magic]] / necromancy / divination
Line 27: Line 88:
| 1
| 1
| 51\104
| 51\104
| 588.462
| 588.46
| 7/5
| 7/5
| [[Untriton]]
| [[Untriton]]
Line 33: Line 94:
| 4
| 4
| 9\104
| 9\104
| 103.846
| 103.85
|  
| 18/17
| [[Undim]]
| [[Undim]]
|}
|}


=== In 104c val ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | 104c val
! Periods<br>per octave
|-
! Generator
! Periods<br />per 8ve
! Cents
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperament
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
|-
| 1
| 1
| 21\104
| 21\104
| 242.308
| 242.31
| 147/128
| 147/128
| [[Septiquarter]]
| [[Septiquarter]]
Line 54: Line 122:
| 1
| 1
| 27\104
| 27\104
| 311.538
| 311.54
| 6/5
| 6/5
| [[Oolong]]
| [[Oolong]]
Line 60: Line 128:
| 1
| 1
| 47\104
| 47\104
| 542.308
| 542.31
| 15/11
| 15/11
| [[Casablanca]] / marrakesh
| [[Casablanca]] / marrakesh
|-
|-
| 2
| 2
| 43\104
| 21\104
| 496.154
| 242.31
| 4/3
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104<br />(9\104)
| 496.15<br />(103.85)
| 4/3<br />(17/16)
| [[Diaschismic]]
| [[Diaschismic]]
|-
|-
| 8
| 8
| 50\104
| 49\104<br />(2\104)
| 576.923
| 565.38<br />(34.62)
| 121/84
| 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


== 17-limit Regular Temperaments ==
== Intervals ==
''todo: clarify this table''
{| class="wikitable center-1 right-2"
 
{| class="wikitable"
|-
|-
! | Degree
! rowspan="2" | #
! | Cents
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
|-
| | '''2'''
! Of 2.3.25.7.11.13.17.19<br>subgroup
| | '''23.08'''
! Additional ratios of 5<br>tending sharp (104c val)
! Additional ratios of 5<br>tending flat (patent val)
|-
|-
| | 3
| 0
| | 34.615
| 0.0
| [[1/1]]
|  
|  
|-
|-
| | 4
| 1
| | 46.15
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
|-
| | '''5'''
| 2
| | '''57.69'''
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
|-
| | '''7'''
| 3
| | '''80.77'''
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
|-
| | 8
| 4
| | 92.31
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
|-
| | 9
| 5
| | 103.85
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
|-
| | 10
| 6
| | 115.385
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
|-
| | 11
| 7
| | 126.92
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
|-
| | 12
| 8
| | 138.46
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
|-
| | '''13'''
| 9
| | '''150'''
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
|-
| | 14
| 10
| | 161.54
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
|-
| | 15
| 11
| | 173.08
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
|-
| | 16
| 12
| | 184.615
| 138.5
| [[13/12]]
|
|
|-
|-
| | 17
| 13
| | 196.15
| 150.0
| [[12/11]]
|
|
|-
|-
| | '''18'''
| 14
| | '''207.69'''
| 161.5
|
| [[11/10]]
|
|-
|-
| | '''20'''
| 15
| | '''230.77'''
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
|-
| | 21
| 16
| | 242.31
| 184.6
|
| [[10/9]]
|
|-
|-
| | 22
| 17
| | 253.85
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
|-
| | '''23'''
| 18
| | '''265.385'''
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
|-
| | '''25'''
| 19
| | '''288.46'''
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
|-
| | 26
| 20
| | 300
| 230.8
| [[8/7]]
|
|
|-
|-
| | 27
| 21
| | 311.54
| 242.3
|-
| [[38/33]]
| | 28
|  
| | 323.08
| [[15/13]]
|-
|-
| | 29
| 22
| | 334.615
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
|-
| | '''30'''
| 23
| | '''346.15'''
| 265.4
| [[7/6]]
|
|
|-
|-
| | 31
| 24
| | 357.69
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
|-
| | 32
| 25
| | 369.23
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
|-
| | 33
| 26
| | 380.77
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
|-
| | 34
| 27
| | 392.31
| 311.5
|
| [[6/5]]
|
|-
|-
| | 35
| 28
| | 403.85
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
|-
| | 36
| 29
| | 415.385
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
|-
| | 38
| 30
| | 438.46
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
|-
| | 39
| 31
| | 450
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
|-
| | 40
| 32
| | 461.54
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
|-
| | '''41'''
| 33
| | '''473.08'''
| 380.8
|
|
| [[5/4]]
|-
|-
| | '''43'''
| 34
| | '''496.15'''
| 392.3
|
| ''[[5/4]]''
|
|-
|-
| | '''45'''
| 35
| | '''519.23'''
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
|-
| | 46
| 36
| | 530.77
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
|-
| | 47
| 37
| | 542.31
| 426.9
| [[32/25]]
|
|
|-
|-
| | '''48'''
| 38
| | '''553.85'''
| 438.5
| [[9/7]]
|
|
|-
|-
| | 50
| 39
| | 576.92
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
|-
| | 51
| 40
| | 588.45
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
|-
| | 52
| 41
| | 600
| 473.1
| [[21/16]]
|
|
|-
|-
| | 53
| 42
| | 611.54
| 484.6
|
|
|
|-
|-
| | '''54'''
| 43
| | '''623.08'''
| 496.2
| [[4/3]]
|
|
|-
|-
| | 56
| 44
| | 646.15
| 507.7
|
|
|
|-
|-
| | 57
| 45
| | 657.69
| 519.2
|
| [[27/20]]
|
|-
|-
| | 58
| 46
| | 669.23
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
|-
| | 59
| 47
| | 680.77
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
|-
| | '''61'''
| 48
| | '''703.85'''
| 553.8
| [[11/8]]
|
|
|-
|-
| | 63
| 49
| | 726.92
| 565.4
| [[18/13]]
|
|
|-
|-
| | 64
| 50
| | 738.46
| 576.9
|
| [[7/5]]
|
|-
|-
| | 65
| 51
| | 750
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
|-
| | '''66'''
| 52
| | '''761.54'''
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
|-
| | 67
| …
| | 773.08
| …
|-
| …
| | '''68'''
| …
| | '''784.615'''
| …
|-
| | 69
| | 796.15
|-
| | 70
| | 807.69
|-
| | 71
| | 819.23
|-
| | 72
| | 830.77
|-
| | 73
| | 842.31
|-
| | '''74'''
| | '''853.85'''
|-
| | 75
| | 865.385
|-
| | 76
| | 876.92
|-
| | 77
| | 888.46
|-
| | 78
| | 900
|-
| | 79
| | 911.54
|-
| | '''81'''
| | '''934.615'''
|-
| | 82
| | 946.15
|-
| | 83
| | 957.69
|-
| | '''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:diaschismic]]
[[Category:Apollo]]
[[Category:edo]]
[[Category:Diaschismic]]
[[Category:magic]]
[[Category:Magic]]
[[Category:necromancy]]
[[Category:Necromancy]]
Retrieved from "https://en.xen.wiki/w/104edo"