39edo: Difference between revisions

Line 925:

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Octave stretch or compression ==

39edo's approximations of harmonics 3, 5, 7 and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[62edt]].

What follows is a comparison of stretched- and compressed-octave 39edo tunings.

; [[zpi|171zpi]]

* Step size: 30.973{{c}}, octave size: 107.9{{c}}

Stretching the octave of 39edo by around 8{{c}} results in improved primes 5, 7, 11, 17, 19 and 23, but much worse primes 2 and 3. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifth tuning]] of 39edo.

{{Harmonics in cet|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}

{{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}

; 39edo

* Step size: 30.769{{c}}, octave size: 1200.00{{c}}

Pure-octaves 39edo approximates all harmonics up to 16 within 15.0{{c}}.

{{Harmonics in equal|39|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39edo}}

{{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}

; [[WE|39et, 13-limit WE tuning]]

* Step size: 30.757{{c}}, octave size: 1199.5{{c}}

Compressing the octave of 39edo by about half a cent results in improved primes 3, 5, 7 and 11, but a worse prime 13. This approximates all harmonics up to 16 within 14.4{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}

{{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}

; [[ed6|101ed6]]

* Octave size: 1197.8{{c}}

Compressing the octave of 101ed6 by around 2{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.

{{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}

{{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}

; [[WE|39et, 2.3.5.11 WE tuning]]

* Step size: 30.703{{c}}, octave size: 1197.4{{c}}

Compressing the octave of 39edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 11.4{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.

{{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}

{{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}

; [[zpi|173zpi]]

* Step size: 30.672{{c}}, octave size: 1196.2{{c}}

Compressing the octave of 39edo by around 4{{c}} results in improved primes 3, 5, 7, 13, 17, 19 and 23, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.

{{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}

{{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}

; [[ed7|110ed7]]

* Octave size: 1194.4{{c}}

Compressing the octave of 39edo by around 5.5{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 14.4{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.

{{Harmonics in equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}

{{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}

; [[ed5|91ed5]]

* Octave size: 1194.1{{c}}

Compressing the octave of 39edo by around 6{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.3{{c}}. The tuning 91ed5 does this.

{{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}

{{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}

== Instruments ==

@@ Line 925: / Line 925: @@
 Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
+== Octave stretch or compression ==
+edo's approximations of harmonics 3, 5, 7 and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[62edt]].
+What follows is a comparison of stretched- and compressed-octave 39edo tunings.
+; [[zpi|171zpi]]
+* Step size: 30.973{{c}}, octave size: 107.9{{c}}
+Stretching the octave of 39edo by around 8{{c}} results in improved primes 5, 7, 11, 17, 19 and 23, but much worse primes 2 and 3. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifth tuning]] of 39edo.
+{{Harmonics in cet|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}
+{{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}
+; 39edo
+* Step size: 30.769{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 39edo approximates all harmonics up to 16 within 15.0{{c}}.
+{{Harmonics in equal|39|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39edo}}
+{{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}
+; [[WE|39et, 13-limit WE tuning]]
+* Step size: 30.757{{c}}, octave size: 1199.5{{c}}
+Compressing the octave of 39edo by about half a cent results in improved primes 3, 5, 7 and 11, but a worse prime 13. This approximates all harmonics up to 16 within 14.4{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}
+{{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}
+; [[ed6|101ed6]]
+* Octave size: 1197.8{{c}}
+Compressing the octave of 101ed6 by around 2{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.
+{{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}
+{{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}
+; [[WE|39et, 2.3.5.11 WE tuning]]
+* Step size: 30.703{{c}}, octave size: 1197.4{{c}}
+Compressing the octave of 39edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 11.4{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
+{{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}
+{{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}
+; [[zpi|173zpi]]
+* Step size: 30.672{{c}}, octave size: 1196.2{{c}}
+Compressing the octave of 39edo by around 4{{c}} results in improved primes 3, 5, 7, 13, 17, 19 and 23, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.
+{{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}
+{{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}
+; [[ed7|110ed7]]
+* Octave size: 1194.4{{c}}
+Compressing the octave of 39edo by around 5.5{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 14.4{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.
+{{Harmonics in equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}
+{{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}
+; [[ed5|91ed5]]
+* Octave size: 1194.1{{c}}
+Compressing the octave of 39edo by around 6{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.3{{c}}. The tuning 91ed5 does this.
+{{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}
+{{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}
 == Instruments ==