104edo: Difference between revisions

Line 4:

| Fifth = 61\104 (703.85¢)

| Major 2nd = 18\104 (208¢)

| ~~Minor 2nd~~ = 7~~\104~~ (81¢)

| Semitones = 11:7 (127¢ : 81¢)

~~| Augmented 1sn = 11\104 (127¢~~)

}}

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 about 11.5 [[cent]]s each.

== 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.

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 supports [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] 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 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.

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]], [[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&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]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. 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 ===

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Line 129:

|-

| 2

| 43\104

| 21\104

| 496.15

| 242.31

| 4/3

| 121/105

| [[Semiseptiquarter]]

|-

| 2

| 43\104 (9\104)

| 496.15 (103.85)

| 4/3 (17/16)

| [[Diaschismic]]

|-

Line 140:

Line 144:

| 565.38 (34.62)

| 168/121 (55/54)

| [[Octowerck]]

| [[Octowerck]] / octowerckis

|}

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Line 485:

== 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{{clarify}}. 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"

@@ Line 4: / Line 4: @@
 | Fifth = 61\104 (703.85¢)
 | Major 2nd = 18\104 (208¢)
-| Minor 2nd = 7\104 (81¢)
+| Semitones = 11:7 (127¢ : 81¢)
-| Augmented 1sn = 11\104  (127¢)
 }}
 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 about 11.5 [[cent]]s each.
 == Theory ==
-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.
+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.
-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.
+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-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.
-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.
+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.
-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.
+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.
 === Prime harmonics ===
@@ Line 131: / Line 129: @@
 |-
 | 2
-| 43\104
+| 21\104
-| 496.15
+| 242.31
-| 4/3
+| 121/105
+| [[Semiseptiquarter]]
+|-
+| 2
+| 43\104<br>(9\104)
+| 496.15<br>(103.85)
+| 4/3<br>(17/16)
 | [[Diaschismic]]
 |-
@@ Line 140: / Line 144: @@
 | 565.38 <br> (34.62)
 | 168/121 <br> (55/54)
-| [[Octowerck]]
+| [[Octowerck]] / octowerckis
 |}
@@ Line 481: / Line 485: @@
 == 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{{clarify}}. 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"