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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|104}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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 [[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-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 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]], [[352/351]] 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]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. 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.
 
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.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|104}}
{{Harmonics in equal|104}}
=== Octave stretch ===
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 ===
=== Subsets and supersets ===
Since 104 factors into 2<sup>3</sup> × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.  
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 ==
== Regular temperament properties ==
Line 32: Line 37:
| {{monzo| 165 -104 }}
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| {{mapping| 104 165 }}
| &minus;0.597
| −0.597
| 0.596
| 0.596
| 5.17
| 5.17
Line 39: Line 44:
| 2048/2025, {{monzo| 0 22 -15 }}
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| {{mapping| 104 165 242 }} (104c)
| &minus;1.258
| −1.258
| 1.054
| 1.054
| 9.14
| 9.14
Line 46: Line 51:
| 126/125, 2048/2025, 117649/116640
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| {{mapping| 104 165 242 292 }} (104c)
| &minus;0.980
| −0.980
| 1.032
| 1.032
| 8.95
| 8.95
Line 53: Line 58:
| 126/125, 176/175, 896/891, 14641/14580
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| {{mapping| 104 165 242 292 360 }} (104c)
| &minus;0.930
| −0.930
| 0.929
| 0.929
| 8.05
| 8.05
Line 60: Line 65:
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &minus;0.855
| −0.855
| 0.864
| 0.864
| 7.49
| 7.49
Line 151: Line 156:
| [[Bosonic]]
| [[Bosonic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | &#35;
! 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]]
Line 220: Line 224:
|-
|-
| 9
| 9
| 103.846
| 103.8
| [[17/16]], [[18/17]]
| [[17/16]], [[18/17]]
| ''[[16/15]]''
| ''[[16/15]]''
Line 226: Line 230:
|-
|-
| 10
| 10
| 115.385
| 115.4
|
|
|
|
Line 232: Line 236:
|-
|-
| 11
| 11
| 126.923
| 126.9
| [[14/13]]
| [[14/13]]
| ''[[15/14]]''
| ''[[15/14]]''
Line 238: Line 242:
|-
|-
| 12
| 12
| 138.462
| 138.5
| [[13/12]]
| [[13/12]]
|
|
Line 244: Line 248:
|-
|-
| 13
| 13
| 150.000
| 150.0
| [[12/11]]
| [[12/11]]
|
|
Line 250: Line 254:
|-
|-
| 14
| 14
| 161.538
| 161.5
|
|
| [[11/10]]
| [[11/10]]
Line 256: Line 260:
|-
|-
| 15
| 15
| 173.077
| 173.1
| [[21/19]]
| [[21/19]]
|
|
Line 262: Line 266:
|-
|-
| 16
| 16
| 184.615
| 184.6
|
|
| [[10/9]]
| [[10/9]]
Line 268: Line 272:
|-
|-
| 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]]''
Line 280: Line 284:
|-
|-
| 19
| 19
| 219.231
| 219.2
| [[25/22]]
| [[25/22]]
|
|
Line 286: Line 290:
|-
|-
| 20
| 20
| 230.769
| 230.8
| [[8/7]]
| [[8/7]]
|
|
Line 292: Line 296:
|-
|-
| 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]]''
Line 304: Line 308:
|-
|-
| 23
| 23
| 265.385
| 265.4
| [[7/6]]
| [[7/6]]
|
|
Line 310: Line 314:
|-
|-
| 24
| 24
| 276.923
| 276.9
| [[75/64]]
| [[75/64]]
|
|
Line 316: Line 320:
|-
|-
| 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]]
Line 334: Line 338:
|-
|-
| 28
| 28
| 323.077
| 323.1
|
|
|
|
| [[6/5]], ''[[40/33]]''
| ''[[6/5]]'', ''[[40/33]]''
|-
|-
| 29
| 29
| 334.615
| 334.6
| [[17/14]]
| [[17/14]]
| [[40/33]]
| [[40/33]]
Line 346: Line 350:
|-
|-
| 30
| 30
| 346.154
| 346.2
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 352: Line 356:
|-
|-
| 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
|
|
|
|
Line 370: Line 374:
|-
|-
| 34
| 34
| 392.308
| 392.3
|
|
| ''[[5/4]]''
| ''[[5/4]]''
Line 376: Line 380:
|-
|-
| 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]]
|
|
Line 394: Line 398:
|-
|-
| 38
| 38
| 438.462
| 438.5
| [[9/7]]
| [[9/7]]
|
|
Line 400: Line 404:
|-
|-
| 39
| 39
| 450.000
| 450.0
| [[22/17]]
| [[22/17]]
| [[13/10]]
| [[13/10]]
Line 406: Line 410:
|-
|-
| 40
| 40
| 461.538
| 461.5
| [[17/13]]
| [[17/13]]
|
|
Line 412: Line 416:
|-
|-
| 41
| 41
| 473.077
| 473.1
| [[21/16]]
| [[21/16]]
|
|
Line 418: Line 422:
|-
|-
| 42
| 42
| 484.615
| 484.6
|
|
|
|
Line 424: Line 428:
|-
|-
| 43
| 43
| 496.154
| 496.2
| [[4/3]]
| [[4/3]]
|
|
Line 430: Line 434:
|-
|-
| 44
| 44
| 507.692
| 507.7
|
|
|
|
Line 436: Line 440:
|-
|-
| 45
| 45
| 519.231
| 519.2
|
|
| [[27/20]]
| [[27/20]]
Line 442: Line 446:
|-
|-
| 46
| 46
| 530.769
| 530.8
| [[19/14]]
| [[19/14]]
|
|
Line 448: Line 452:
|-
|-
| 47
| 47
| 542.308
| 542.3
| [[26/19]]
| [[26/19]]
| [[15/11]]
| [[15/11]]
Line 454: Line 458:
|-
|-
| 48
| 48
| 553.846
| 553.8
| [[11/8]]
| [[11/8]]
|
|
Line 460: Line 464:
|-
|-
| 49
| 49
| 565.385
| 565.4
| [[18/13]]
| [[18/13]]
|
|
Line 466: Line 470:
|-
|-
| 50
| 50
| 576.923
| 576.9
|
|
| [[7/5]]
| [[7/5]]
Line 472: Line 476:
|-
|-
| 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]]''
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