31edo: Difference between revisions

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One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

~~=== Prime harmonics ===~~

~~{{Harmonics in equal|31|columns=9}}~~

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

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

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 more importantly a way better 11th harmonic at the expense of somewhat less accurate 5th and 7th harmonics. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.

=== Subsets and supersets ===

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 One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.
-edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L&nbsp;3s) [[mos]] accurate 13:17:19 chords.
-=== Prime harmonics ===
-{{Harmonics in equal|31|columns=9}}
-{{Harmonics in equal|31|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}
-=== Octave stretch ===
-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 more importantly a way better 11th harmonic at the expense of somewhat less accurate 5th and 7th harmonics. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.
 === Subsets and supersets ===