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104edo: Difference between revisions

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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.
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 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.


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.
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==Intervals==
== Intervals ==


{| class="wikitable center-all"
{| class="wikitable center-all right-2"
|-
|-
! rowspan="2" | #
! rowspan="2" | #
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| 92.308
| 92.308
| [[19/18]]
| [[19/18]]
|
| [[20/19]]
[[20/19]]
| [[21/20]]
|
[[21/20]]
|-
|-
| 9
| 9
| 103.846
| 103.846
| [[17/16]], [[18/17]]
| [[17/16]], [[18/17]]
|
| [[16/15]]
[[16/15]]
|
|
|-
|-
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| [[81/64]], [[14/11]]
| [[81/64]], [[14/11]]
|
|
|
| [[19/15]]
[[19/15]]
|-
|-
| 37
| 37
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|}
|}


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 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"
{| class="wikitable center-all"
|+Circulating temperaments in 104edo
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Tones
! Pattern
! Pattern
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|-
|-
| 8
| 8
| [[8edo]]
| [[8 EDO]]
| equal
| equal
|-
|-
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|-
|-
| 13
| 13
| [[13edo]]
| [[13 EDO]]
| equal
| equal
|-
|-
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| 15
| 15
| [[14L 1s]]
| [[14L 1s]]
| rowspan="3" |7:6
| rowspan="3" | 7:6
|-
|-
| 16
| 16
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| 18
| 18
| 12L 6s
| 12L 6s
| rowspan="3" |6:5
| rowspan="3" | 6:5
|-
|-
| 19
| 19
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| 21
| 21
| 20L 1s
| 20L 1s
| rowspan="5" |5:4
| rowspan="5" | 5:4
|-
|-
| 22
| 22
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|-
|-
| 26
| 26
| [[26edo]]
| [[26 EDO]]
| equal
| equal
|-
|-
| 27
| 27
| 23L 4s
| 23L 4s
| rowspan="8" |4:3
| rowspan="8" | 4:3
|-
|-
| 28
| 28
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| 35
| 35
| 34L 1s
| 34L 1s
| rowspan="17" |3:2
| rowspan="17" | 3:2
|-
|-
| 36
| 36
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|-
|-
| 52
| 52
| [[52edo]]
| [[52 EDO]]
| equal
| equal
|-
|-
| 53
| 53
| 51L 2s
| 51L 2s
| rowspan="31" |2:1
| rowspan="31" | 2:1
|-
|-
| 54
| 54
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