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 [[gamelan]], 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 [[37ed5]] and [[57ed12]] being good options.

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

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; 16edo

* Step size: 75.000{{c}}, octave size: 1200.0{{c}}

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

Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.

{{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}

{{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}

Line 13:

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

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

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.

Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{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]]

; [[zpi|15zpi]] / [[equal tuning|59ed13]]

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

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.

Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do 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]]

; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]

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

* Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}

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.

Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.

{{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)}}

{{Harmonics in equal|37|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 37ed5}}

{{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}}

; [[57ed12]]

; [[57ed12]] / [[equal tuning|55ed11]]

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

* Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}

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.

Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do 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)}}

{{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}}

~~; [[41ed6]]~~

{{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}}

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

~~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)}}

= Title2 =

@@ 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 [[gamelan]], 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 [[37ed5]] and [[57ed12]] being good options.
 What follows is a comparison of stretched- and compressed-octave 16edo tunings.
@@ Line 7: / Line 7: @@
 ; 16edo
 * Step size: 75.000{{c}}, octave size: 1200.0{{c}}
-Pure-octaves 16edo approximates all harmonics up to 16 within NNN{{c}}.
+Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.
 {{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}
 {{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}
@@ Line 13: / Line 13: @@
 ; [[WE|16et, 2.5.7.13 WE tuning]]
 * Step size: 75.105{{c}}, octave size: 1201.7{{c}}
-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.
+Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{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]]
+; [[zpi|15zpi]] / [[equal tuning|59ed13]]
 * Step size: 75.262{{c}}, octave size: 1204.2{{c}}
-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.
+Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do 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]]
+; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]
-* Step size: 75.315{{c}}, octave size: 1205.0{{c}}
+* Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}
-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.
+Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.
 {{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)}}
+{{Harmonics in equal|37|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 37ed5}}
+{{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}}
-; [[57ed12]]
+; [[57ed12]] / [[equal tuning|55ed11]]
-* Step size: NNN{{c}}, octave size: 1207.6{{c}}
+* Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}
-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.
+Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do 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)}}
+{{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}}
-; [[41ed6]]
+{{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}}
-* Step size: NNN{{c}}, octave size: 1210.5{{c}}
-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)}}
 = Title2 =