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


== Theory ==
== Theory ==
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.


{{Primes in edo|104|prec=2}}
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 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.
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 EDO 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.
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.


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.
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.


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.
=== Prime harmonics ===
{{Harmonics in equal|104}}


== Rank two temperaments ==
=== 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]].


=== In patent val ===
=== 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 }}.


{| class="wikitable center-all"
== Regular temperament properties ==
! Periods <br> per octave
{| class="wikitable center-4 center-5 center-6"
! Generator
|-
! Cents
! rowspan="2" | [[Subgroup]]
! Associated ratio
! 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
|-
|-
| rowspan="2" | 1
| 1
| 33\104
| 33\104
| 380.769
| 380.77
| 5/4
| 5/4
| [[Magic]] / necromancy / divination
| [[Magic]] / necromancy / divination
|-
|-
| 1
| 51\104
| 51\104
| 588.462
| 588.46
| 7/5
| 7/5
| [[Untriton]]
| [[Untriton]]
Line 37: Line 94:
| 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"
 
|+ style="font-size: 105%;" | 104c val
{| class="wikitable center-all"
|-
! Periods <br> per octave
! Periods<br />per 8ve
! Generator <br> (reduced)
! Generator*
! Cents <br> (reduced)
! Cents*
! Associated ratio <br> (reduced)
! Associated<br />ratio*
! Temperament
! Temperament
|-
|-
| rowspan="3" | 1
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 21\104
| 242.308
| 242.31
| 147/128
| 147/128
| [[Septiquarter]]
| [[Septiquarter]]
|-
|-
| 1
| 27\104
| 27\104
| 311.538
| 311.54
| 6/5
| 6/5
| [[Oolong]]
| [[Oolong]]
|-
|-
| 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 <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-1 right-2"
{| class="wikitable center-all 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
|
|
|
|
Line 159: Line 236:
|-
|-
| 11
| 11
| 126.923
| 126.9
| [[14/13]]
| [[14/13]]
| [[15/14]]
| ''[[15/14]]''
|
|
|-
|-
| 12
| 12
| 138.462
| 138.5
| [[13/12]]
| [[13/12]]
|
|
Line 171: Line 248:
|-
|-
| 13
| 13
| 150.000
| 150.0
| [[12/11]]
| [[12/11]]
|
|
Line 177: Line 254:
|-
|-
| 14
| 14
| 161.538
| 161.5
|
|
| [[11/10]]
| [[11/10]]
Line 183: Line 260:
|-
|-
| 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]]
Line 195: 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]]''
|
|
|-
|-
| 19
| 19
| 219.231
| 219.2
| [[25/22]]
| [[25/22]]
|
|
Line 213: Line 290:
|-
|-
| 20
| 20
| 230.769
| 230.8
| [[8/7]]
| [[8/7]]
|
|
Line 219: 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]]''
|
|
|-
|-
| 23
| 23
| 265.385
| 265.4
| [[7/6]]
| [[7/6]]
|
|
Line 237: Line 314:
|-
|-
| 24
| 24
| 276.923
| 276.9
| [[75/64]]
| [[75/64]]
|
|
Line 243: 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 261: Line 338:
|-
|-
| 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]]
|
|
Line 279: 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 297: Line 374:
|-
|-
| 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]]
|
|
Line 321: Line 398:
|-
|-
| 38
| 38
| 438.462
| 438.5
| [[9/7]]
| [[9/7]]
|
|
Line 327: Line 404:
|-
|-
| 39
| 39
| 450.000
| 450.0
| [[22/17]]
| [[22/17]]
| [[13/10]]
| [[13/10]]
Line 333: Line 410:
|-
|-
| 40
| 40
| 461.538
| 461.5
| [[17/13]]
| [[17/13]]
|
|
| [[13/10]]
| ''[[13/10]]''
|-
|-
| 41
| 41
| 473.077
| 473.1
| [[21/16]]
| [[21/16]]
|
|
Line 345: Line 422:
|-
|-
| 42
| 42
| 484.615
| 484.6
|
|
|
|
Line 351: Line 428:
|-
|-
| 43
| 43
| 496.154
| 496.2
| [[4/3]]
| [[4/3]]
|
|
Line 357: Line 434:
|-
|-
| 44
| 44
| 507.692
| 507.7
|
|
|
|
Line 363: Line 440:
|-
|-
| 45
| 45
| 519.231
| 519.2
|
|
| [[27/20]]
| [[27/20]]
Line 369: Line 446:
|-
|-
| 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]]
Line 381: Line 458:
|-
|-
| 48
| 48
| 553.846
| 553.8
| [[11/8]]
| [[11/8]]
|
|
Line 387: Line 464:
|-
|-
| 49
| 49
| 565.385
| 565.4
| [[18/13]]
| [[18/13]]
|
|
Line 393: Line 470:
|-
|-
| 50
| 50
| 576.923
| 576.9
|
|
| [[7/5]]
| [[7/5]]
Line 399: 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]]''
|
|
|-
|-
Line 415: Line 492:
| …
| …
| …
| …
|}
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.
{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| 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
| [[26 EDO]]
| 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
| [[52 EDO]]
| 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]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Necromancy]]
[[Category:Necromancy]]

Latest revision as of 23:01, 11 May 2026

← 103edo 104edo 105edo →
Prime factorization 23 × 13
Step size 11.5385 ¢ 
Fifth 61\104 (703.846 ¢)
Semitones (A1:m2) 11:7 (126.9 ¢ : 80.77 ¢)
Consistency limit 3
Distinct consistency limit 3

104 equal divisions of the octave (abbreviated 104edo or 104ed2), also called 104-tone equal temperament (104tet) or 104 equal temperament (104et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 104 equal parts of about 11.5 ¢ each. Each step represents a frequency ratio of 21/104, or the 104th root of 2.

Theory

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, it tempers out 352/351, 364/363, 896/891, 2197/2187, 10648/10647, 16807/16731, 20449/20412, 21632/21609, and 26411/26364. 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 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.

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

If prime 5 is desired, 104edo has two different equally viable 5-limit vals, and both are useful. The flat major third val, ⟨104 165 241] (patent val), tempers out 3125/3072, and supports magic temperament. The sharp major third 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.

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

Approximation of prime harmonics in 104edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.89 -5.54 +0.40 +2.53 +1.78 -1.11 +2.49 -5.20 -2.65 -2.73
Relative (%) +0.0 +16.4 -48.1 +3.5 +21.9 +15.4 -9.6 +21.6 -45.0 -23.0 -23.6
Steps
(reduced) 		104
(0) 		165
(61) 		241
(33) 		292
(84) 		360
(48) 		385
(73) 		425
(9) 		442
(26) 		470
(54) 		505
(89) 		515
(99)

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

Since 104 factors into primes as 23 × 13, 104edo has subset edos 2, 4, 8, 13, 26, and 52.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [165 -104⟩ [⟨104 165]] −0.597 0.596 5.17
2.3.5 2048/2025, [0 22 -15⟩ [⟨104 165 242]] (104c) −1.258 1.054 9.14
2.3.5.7 126/125, 2048/2025, 117649/116640 [⟨104 165 242 292]] (104c) −0.980 1.032 8.95
2.3.5.7.11 126/125, 176/175, 896/891, 14641/14580 [⟨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 [⟨104 165 242 292 360 385]] (104c) −0.855 0.864 7.49

Rank-2 temperaments

Patent val
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 33\104 380.77 5/4 Magic / necromancy / divination
1 51\104 588.46 7/5 Untriton
4 9\104 103.85 18/17 Undim
104c val
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 11\104 126.92 27/25 Mowgli
1 21\104 242.31 147/128 Septiquarter
1 27\104 311.54 6/5 Oolong
1 47\104 542.31 15/11 Casablanca / marrakesh
2 21\104 242.31 121/105 Semiseptiquarter
2 43\104
(9\104) 		496.15
(103.85) 		4/3
(17/16) 		Diaschismic
8 49\104
(2\104) 		565.38
(34.62) 		168/121
(55/54) 		Octowerck / octowerckis
26 43\104
(1\104) 		496.15
(11.54) 		4/3
(225/224) 		Bosonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Intervals

# Cents Approximate ratios
Of 2.3.25.7.11.13.17.19
subgroup 		Additional ratios of 5
tending sharp (104c val) 		Additional ratios of 5
tending flat (patent val)
0 0.0 1/1
1 11.5 144/143, 169/168 91/90, 121/120 105/104, 196/195
2 23.1 64/63, 99/98 81/80, 100/99, 105/104 50/49, 55/54, 91/90, 121/120
3 34.6 49/48, 50/49 55/54 40/39, 45/44, 81/80, 126/125
4 46.2 36/35, 40/39, 45/44, 50/49
5 57.7 28/27, 33/32 26/25 25/24, 36/35
6 69.2 25/24, 26/25, 27/26
7 80.8 22/21 21/20, 25/24 20/19, 26/25
8 92.3 19/18 20/19 21/20
9 103.8 17/16, 18/17 16/15
10 115.4 16/15, 15/14
11 126.9 14/13 15/14
12 138.5 13/12
13 150.0 12/11
14 161.5 11/10
15 173.1 21/19 10/9, 11/10
16 184.6 10/9
17 196.2 19/17, 28/25
18 207.7 9/8 17/15
19 219.2 25/22 17/15
20 230.8 8/7
21 242.3 38/33 15/13
22 253.8 22/19 15/13
23 265.4 7/6
24 276.9 75/64 20/17
25 288.5 13/11, 32/27 20/17
26 300.0 19/16, 25/21
27 311.5 6/5
28 323.1 6/5, 40/33
29 334.6 17/14 40/33
30 346.2 11/9, 39/32
31 357.7 16/13, 27/22
32 369.2 21/17, 26/21
33 380.8 5/4
34 392.3 5/4
35 403.8 24/19, 63/50 19/15
36 415.4 14/11 19/15
37 426.9 32/25
38 438.5 9/7
39 450.0 22/17 13/10
40 461.5 17/13 13/10
41 473.1 21/16
42 484.6
43 496.2 4/3
44 507.7
45 519.2 27/20
46 530.8 19/14 27/20, 15/11
47 542.3 26/19 15/11
48 553.8 11/8
49 565.4 18/13
50 576.9 7/5
51 588.5 7/5, 45/32
52 600.0 17/12, 24/17 45/32, 64/45
… … … … …
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