38edo: Difference between revisions

Line 366:

| [[Divination]] || 7\38 || 2

|}

== Octave stretch or compression ==

38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].

What follows is a comparison of stretched-octave 38edo tunings.

; 38edo

* Step size: 31.579{{c}}, octave size: 1200.00{{c}}

Pure-octaves 38edo approximates all harmonics up to 16 within 14.6{{c}}.

{{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}}

{{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}}

; [[WE|38et, 13-limit WE tuning]]

* Step size: 31.599{{c}}, octave size: 1200.77{{c}}

Stretching the octave of 38edo by around 1{{c}} results in improved primes 3, 5 and 11, but worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 14.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|31.599|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning}}

{{Harmonics in cet|31.599|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning (continued)}}

; [[ed5|88ed5]]

* Step size: 31.663{{c}}, octave size: 1203.18{{c}}

Stretching the octave of 38edo by around 3{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.7{{c}}. The tuning 88ed5 does this.

{{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}}

{{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}}

; [[zpi|166zpi]]

* Step size: 31.671{{c}}, octave size: 1203.48{{c}}

Stretching the octave of 38edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.0{{c}}. The tuning 166zpi does this.

{{Harmonics in cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}

{{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}

; [[60edt]]

* Step size: 31.699{{c}}, octave size: 1204.57{{c}}

Stretching the octave of 38edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13, 17 and 19, but a worse primes 2. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 60edt does this.

{{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}

{{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}

== Instruments ==

@@ Line 366: / Line 366: @@
 | [[Divination]] || 7\38 || 2
 |}
+== Octave stretch or compression ==
+edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].
+What follows is a comparison of stretched-octave 38edo tunings.
+; 38edo
+* Step size: 31.579{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 38edo approximates all harmonics up to 16 within 14.6{{c}}.
+{{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}}
+{{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}}
+; [[WE|38et, 13-limit WE tuning]]
+* Step size: 31.599{{c}}, octave size: 1200.77{{c}}
+Stretching the octave of 38edo by around 1{{c}} results in improved primes 3, 5 and 11, but worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 14.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|31.599|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning}}
+{{Harmonics in cet|31.599|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning (continued)}}
+; [[ed5|88ed5]]
+* Step size: 31.663{{c}}, octave size: 1203.18{{c}}
+Stretching the octave of 38edo by around 3{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.7{{c}}. The tuning 88ed5 does this.
+{{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}}
+{{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}}
+; [[zpi|166zpi]]
+* Step size: 31.671{{c}}, octave size: 1203.48{{c}}
+Stretching the octave of 38edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.0{{c}}. The tuning 166zpi does this.
+{{Harmonics in cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}
+{{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}
+; [[60edt]]
+* Step size: 31.699{{c}}, octave size: 1204.57{{c}}
+Stretching the octave of 38edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13, 17 and 19, but a worse primes 2. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 60edt does this.
+{{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}
+{{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}
 == Instruments ==

38edo: Difference between revisions

BudjarnLambeth (talk | contribs)