31edo: Difference between revisions

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=== 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.

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.

== Intervals ==

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|}

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==

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.

What follows is a comparison of stretched-octave 31edo tunings.

; 31edo

* Step size: 38.710{{c}}, octave size: 1200.0{{c}}

Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}}

{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}}

; [[WE|31et, 13-limit WE tuning]]

* Step size: 38.725{{c}}, octave size: 1200.5{{c}}

Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}

{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}

; [[zpi|127zpi]]

* Step size: 38.737{{c}}, octave size: 1200.8{{c}}

Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.

{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}

{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}

; [[WE|31et, 11-limit WE tuning]]

* Step size: 38.748{{c}}, octave size: 1201.2{{c}}

Stretching the octave of 31edo by slightly more than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]].

{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}

{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}

; [[80ed6]]

* Step size: 38.774{{c}}, octave size: 1202.0{{c}}

Stretching the octave of 31edo by about 2{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This is approaching 2.239{{c}} - the most octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. This approximates all harmonics up to 16 within 18.5{{c}}. The tuning 80ed6 does this.

{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}

{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}}

; [[49edt]]

* Step size: 38.815{{c}}, octave size: 1203.3{{c}}

Stretching the octave of 31edo by about 3.5{{c}} results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. The 13 is now differently mapped than - and much better than - 80ed6's (but not as good as the pure octaves 13). This approximates all harmonics up to 16 within 15.6{{c}}. The tuning 49edt does this.

{{Harmonics in equal|49|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt}}

{{Harmonics in equal|49|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt (continued)}}

== Instruments ==

@@ Line 31: / Line 31: @@
 === 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.
-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.
 == Intervals ==
@@ Line 1,331: / Line 1,326: @@
 |}
 <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Octave stretch or compression ==
+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.
+What follows is a comparison of stretched-octave 31edo tunings.
+; 31edo
+* Step size: 38.710{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
+{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}}
+{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}}
+; [[WE|31et, 13-limit WE tuning]]
+* Step size: 38.725{{c}}, octave size: 1200.5{{c}}
+Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}
+{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}
+; [[zpi|127zpi]]
+* Step size: 38.737{{c}}, octave size: 1200.8{{c}}
+Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.
+{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}
+{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}
+; [[WE|31et, 11-limit WE tuning]]
+* Step size: 38.748{{c}}, octave size: 1201.2{{c}}
+Stretching the octave of 31edo by slightly more than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]].
+{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}
+{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}
+; [[80ed6]]
+* Step size: 38.774{{c}}, octave size: 1202.0{{c}}
+Stretching the octave of 31edo by about 2{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This is approaching 2.239{{c}} - the most octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. This approximates all harmonics up to 16 within 18.5{{c}}. The tuning 80ed6 does this.
+{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}
+{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}}
+; [[49edt]]
+* Step size: 38.815{{c}}, octave size: 1203.3{{c}}
+Stretching the octave of 31edo by about 3.5{{c}} results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. The 13 is now differently mapped than - and much better than - 80ed6's (but not as good as the pure octaves 13). This approximates all harmonics up to 16 within 15.6{{c}}. The tuning 49edt does this.
+{{Harmonics in equal|49|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt}}
+{{Harmonics in equal|49|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt (continued)}}
 == Instruments ==