104edo: Difference between revisions

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''104edo'' divides the octave into 104 parts of size 11.54 ~~cents each. It has two different equally viable 5-limit~~ [[~~val~~|val]]s, and both are useful. The flat major third val, <104 165 241|, tempers out 3125/3072, and supports [[Magic_family|magic temperament]]. The sharp major third val, <104 165 242|, tempers out 2048/2025 and supports [[Diaschismic_family|diaschismic temperament]].

'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each.

104edo ~~with the flat third is especially notable as an excellent tuning for~~ [[~~Magic_family~~|~~magic temperament~~]], ~~providing the~~ [[~~Optimal_patent_val|optimal~~ patent val]] ~~for 11-limit magic~~ and ~~the 13-limit magic extension~~ [[Magic_family~~#Magic-13-limit-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 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[~~Optimal_patent_val~~|~~optimal patent val~~]].

== 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]].

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 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.

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.

~~'''~~17-limit Regular Temperaments'''

== 17-limit Regular Temperaments ==

''todo: clarify this table''

{| class="wikitable"

@@ Line 1: / Line 1: @@
-''104edo'' divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[val|val]]s, and both are useful. The flat major third val, &lt;104 165 241|, tempers out 3125/3072, and supports [[Magic_family|magic temperament]]. The sharp major third val, &lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic_family|diaschismic temperament]].
+'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each.
-edo with the flat third is especially notable as an excellent tuning for [[Magic_family|magic temperament]], providing the [[Optimal_patent_val|optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family#Magic-13-limit-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 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[Optimal_patent_val|optimal patent val]].
+== Theory ==
+edo 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]].
-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.
+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 [[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.
+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.
 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.
-'''17-limit Regular Temperaments'''
+== 17-limit Regular Temperaments ==
+''todo: clarify this table''
 {| class="wikitable"