104edo: Difference between revisions

Line 9:

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]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|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.

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 {{nowrap| 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 ===

=== 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 2<sup>3</sup> × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

Since 104 factors into primes as {{nowrap| 2<sup>3</sup> × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==

@@ Line 9: / Line 9: @@
 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]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|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 {{nowrap| 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 ===
 {{Harmonics in equal|104}}
+=== Octave stretch ===
+edo'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 2<sup>3</sup> × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.
+Since 104 factors into primes as {{nowrap| 2<sup>3</sup> × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.
 == Regular temperament properties ==