User:BudjarnLambeth/Sandbox2: Difference between revisions

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= Title1 =

== Octave stretch or compression ==

Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and ~~gamelans~~, with [[41ed6]] and [[57ed12]] being good options.

Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[41ed6]] and [[57ed12]] being good options.

What follows is a comparison of stretched- and compressed-octave 16edo tunings.

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; [[WE|16et, 2.5.7.13 WE tuning]]

* Step size: 75.105{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 75.105{{c}}, octave size: 1201.7{{c}}

Stretching the octave of 16edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.

Stretching the octave of 16edo by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.

{{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}

{{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}

; [[zpi|15zpi]]

* Step size: 75.262{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 75.262{{c}}, octave size: 1204.2{{c}}

Stretching the octave of 16edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 15zpi does this.

Stretching the octave of 16edo by around 4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 15zpi does this.

{{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}

{{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}

; [[WE|16et, 13-limit WE tuning]]

* Step size: 75.315{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 75.315{{c}}, octave size: 1205.0{{c}}

Stretching the octave of 16edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

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

{{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}

{{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}

; [[57ed12]]

* Step size: NNN{{c}}, octave size: ~~NNN~~{{c}}

* Step size: NNN{{c}}, octave size: 1207.6{{c}}

Stretching the octave of 16edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed12 does this.

Stretching the octave of 16edo by around 7.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed12 does this.

{{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}

{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}

; [[41ed6]]

* Step size: NNN{{c}}, octave size: ~~NNN~~{{c}}

* Step size: NNN{{c}}, octave size: 1210.5{{c}}

Stretching the octave of 16edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 41ed6 does this.

Stretching the octave of 16edo by around 10.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 41ed6 does this.

{{Harmonics in equal|41|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41ed6}}

{{Harmonics in equal|41|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41ed6 (continued)}}

@@ Line 1: / Line 1: @@
 = Title1 =
 == Octave stretch or compression ==
-Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[41ed6]] and [[57ed12]] being good options.
+Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[41ed6]] and [[57ed12]] being good options.
 What follows is a comparison of stretched- and compressed-octave 16edo tunings.
@@ Line 12: / Line 12: @@
 ; [[WE|16et, 2.5.7.13 WE tuning]]
-* Step size: 75.105{{c}}, octave size: NNN{{c}}
+* Step size: 75.105{{c}}, octave size: 1201.7{{c}}
-Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
+Stretching the octave of 16edo by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
 {{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}
 {{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}
 ; [[zpi|15zpi]]
-* Step size: 75.262{{c}}, octave size: NNN{{c}}
+* Step size: 75.262{{c}}, octave size: 1204.2{{c}}
-Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 15zpi does this.
+Stretching the octave of 16edo by around 4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 15zpi does this.
 {{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}
 {{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}
 ; [[WE|16et, 13-limit WE tuning]]
-* Step size: 75.315{{c}}, octave size: NNN{{c}}
+* Step size: 75.315{{c}}, octave size: 1205.0{{c}}
-Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+Stretching the octave of 16edo by around 5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
 {{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}
 {{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}
 ; [[57ed12]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: NNN{{c}}, octave size: 1207.6{{c}}
-Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed12 does this.
+Stretching the octave of 16edo by around 7.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed12 does this.
 {{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}
 {{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}
 ; [[41ed6]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: NNN{{c}}, octave size: 1210.5{{c}}
-Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 41ed6 does this.
+Stretching the octave of 16edo by around 10.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 41ed6 does this.
 {{Harmonics in equal|41|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41ed6}}
 {{Harmonics in equal|41|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41ed6 (continued)}}