111edo: Difference between revisions

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In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]].

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &~~amp;~~ 80}} temperament, and [[buzzard]], the {{nowrap|53 &~~amp;~~ 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[~~Orwellismic family #Guanyin|~~guanyin ~~temperament~~]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===

{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===

Since 111 factors into 3 × 37, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==

@@ Line 9: / Line 9: @@
 In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]].
-It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
+It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
 === Prime harmonics ===
-{{Harmonics in equal|111}}
+{{Harmonics in equal|111|columns=9}}
+{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}
 === Subsets and supersets ===
-Since 111 factors into 3 × 37, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.
+Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.
 == Intervals ==