31edo: Difference between revisions

Line 24:

{{Harmonics in equal|31|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}

=== ~~Octave stretch~~ ===

=== Subsets and supersets ===

31edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to the 13- and 17- and 19-limit.

=== Stretched and compressed tunings ===

31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.

~~=== Subsets~~ and ~~supersets ===~~

229ed169 has an octave stretched by 2.23893{{c}}. Since the 13th harmonic is exactly halfway between 114 and 115 steps, this difference is the absolute maximum amount of octave stretch 31edo can tolerate before a discrepancy for the 13th harmonic occurs.

~~31edo~~ is the ~~11th [[prime edo]], following [[29edo]] and coming~~ before ~~[[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to~~ the ~~13- and 17- and 19-limit~~.

== Intervals ==

@@ Line 24: / Line 24: @@
 {{Harmonics in equal|31|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}
-=== Octave stretch ===
+=== Subsets and supersets ===
+edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to the 13- and 17- and 19-limit.
+=== Stretched and compressed tunings ===
 edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.
-=== Subsets and supersets ===
+ed169 has an octave stretched by 2.23893{{c}}. Since the 13th harmonic is exactly halfway between 114 and 115 steps, this difference is the absolute maximum amount of octave stretch 31edo can tolerate before a discrepancy for the 13th harmonic occurs.
-edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to the 13- and 17- and 19-limit.
 == Intervals ==