Xenharmonic Wiki

User:BudjarnLambeth/Sandbox2: Difference between revisions

← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
Line 7: Line 7:
= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 12edo tunings.
What follows is a comparison of stretched- and compressed-octave 17edo tunings.


; [[WE|12et, 7-limit WE tuning]]  
; [[zpi|56zpi]]  
* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
* Step size: 70.403{{c}}, octave size: NNN{{c}}
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave.
_ing the octave of 17edo 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 56zpi does this.
{{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in cet|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}
{{Harmonics in cet|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}
{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}


; [[zpi|34zpi]]  
; [[27edt]]  
* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo.
_ing the octave of 17edo 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 27edt does this.
{{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}}
{{Harmonics in equal|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}
{{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}}
{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}


; [[WE|12et, 5-limit WE tuning]]  
; [[44ed6]]  
* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
_ing the octave of 17edo 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 44ed6 does this.
{{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}
{{Harmonics in cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}
{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}


; 12edo
; [[WE|17et, 2.3.7.11 WE tuning]]
* Step size: 100.000{{c}}, octave size: 1200.0{{c}}  
* Step size: 70.392{{c}}, octave size: NNN{{c}}
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.7.11 WE tuning and 2.3.7.11 [[TE]] tuning both do this.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo}}
{{Harmonics in cet|70.392|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo (continued)}}
{{Harmonics in cet|70.392|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)}}


; [[31ed6]]  
; [[WE|17et, 2.3.7.11.13 WE tuning]]  
* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
* Step size: 70.410{{c}}, octave size: NNN{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. The tuning 31ed6 does this.
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.7.11.13 WE tuning and 2.3.7.11.13 [[TE]] tuning both do this.
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}}
{{Harmonics in cet|70.410|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}}
{{Harmonics in cet|70.410|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[19edt]]
; 17edo
* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}  
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. The tuning 19edt does this.
Pure-octaves 17edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}}
{{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}}
{{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}
 
; [[7edf]]
* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}}
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Sandbox2"