104edo: Difference between revisions
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'''104 | The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. | ||
== Theory == | == Theory == | ||
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. | ||
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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 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. | ||
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&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. | 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&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 === | ||
{{Primes in edo|104}} | |||
=== | == Regular temperament properties == | ||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all" | ==== In patent val ==== | ||
{| class="wikitable center-all left-5" | |||
! Periods <br> per octave | ! Periods <br> per octave | ||
! Generator | ! Generator | ||
| Line 24: | Line 26: | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| 1 | |||
| 33\104 | | 33\104 | ||
| 380.769 | | 380.769 | ||
| Line 30: | Line 32: | ||
| [[Magic]] / necromancy / divination | | [[Magic]] / necromancy / divination | ||
|- | |- | ||
| 1 | |||
| 51\104 | | 51\104 | ||
| 588.462 | | 588.462 | ||
| Line 42: | Line 45: | ||
|} | |} | ||
=== In 104c val === | ==== In 104c val ==== | ||
{| class="wikitable center-all" | {| class="wikitable center-all left-5" | ||
! Periods <br> per octave | ! Periods <br> per octave | ||
! Generator <br> (reduced) | ! Generator <br> (reduced) | ||
| Line 51: | Line 54: | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| 1 | |||
| 21\104 | | 21\104 | ||
| 242.308 | | 242.308 | ||
| Line 57: | Line 60: | ||
| [[Septiquarter]] | | [[Septiquarter]] | ||
|- | |- | ||
| 1 | |||
| 27\104 | | 27\104 | ||
| 311.538 | | 311.538 | ||
| Line 62: | Line 66: | ||
| [[Oolong]] | | [[Oolong]] | ||
|- | |- | ||
| 1 | |||
| 47\104 | | 47\104 | ||
| 542.308 | | 542.308 | ||
| Line 417: | Line 422: | ||
|} | |} | ||
== Scales == | |||
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. | 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. | ||