16edo: Difference between revisions

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| [[Semidim]]

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== 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 [[57ed12]] being a good option.

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

; 16edo

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

; [[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 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]]

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

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 both do this, their octaves differ from one another by less than 0.1{{c}}.

{{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: 1205.0{{c}}

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, whose octave differs by only 0.1{{c}}.

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

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 both do this, their octaves differ from one another by only 0.1{{c}}.

{{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)}}

== Scales ==

@@ Line 636: / Line 636: @@
 | [[Semidim]]
 |}
+== 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 [[57ed12]] being a good option.
+What follows is a comparison of stretched- and compressed-octave 16edo tunings.
+; 16edo
+* Step size: 75.000{{c}}, octave size: 1200.0{{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)}}
+; [[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 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]]
+* Step size: 75.262{{c}}, octave size: 1204.2{{c}}
+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 both do this, their octaves differ from one another by less than 0.1{{c}}.
+{{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: 1205.0{{c}}
+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, whose octave differs by only 0.1{{c}}.
+{{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: 75.473{{c}}, octave size (57ed12): 1207.6{{c}}
+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 both do this, their octaves differ from one another by only 0.1{{c}}.
+{{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)}}
 == Scales ==