User:BudjarnLambeth/Sandbox2: Difference between revisions
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== 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 12edo tunings. | ||
; [[WE|12et, 7-limit WE tuning]] | ; [[WE|12et, 7-limit WE tuning]] | ||
* Step size: 99.664{{c}}, octave size: | * Step size: 99.664{{c}}, octave size: 1196.0{{c}} | ||
Compressing the octave of | 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. | ||
{{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}} | {{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}} | ||
{{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|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}} | ||
; [[zpi|34zpi]] | ; [[zpi|34zpi]] | ||
* Step size: 99.807{{c}}, octave size: | * Step size: 99.807{{c}}, octave size: 1197.7{{c}} | ||
Compressing the octave of 12edo by around | 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. | ||
{{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}} | {{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}} | ||
{{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}} | {{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}} | ||
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; [[WE|12et, 5-limit WE tuning]] | ; [[WE|12et, 5-limit WE tuning]] | ||
* Step size: 99.868{{c}}, octave size: NNN{{c}} | * Step size: 99.868{{c}}, octave size: NNN{{c}} | ||
Compressing the octave of | Compressing the octave of 12edo by around a fifth of a [[cent]] 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. It has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi. | ||
{{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}} | {{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}} | ||
{{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 cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}} | ||
; 12edo | ; 12edo | ||
* Step size: 100.000{{c}}, octave size: 1200.0{{c}} | * Step size: 100.000{{c}}, octave size: 1200.0{{c}} | ||
Pure-octaves | 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. | ||
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in | {{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo}} | ||
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in | {{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo (continued)}} | ||
; [[31ed6]] | ; [[31ed6]] | ||
* Step size: 100.063{{c}}, octave size: 1200.8{{c}} | * Step size: 100.063{{c}}, octave size: 1200.8{{c}} | ||
Stretching the octave of 12edo by a little less than 1{{c}} results in improved | Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. The tuning 31ed6 does this. | ||
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}} | {{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}} | ||
{{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}} | {{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}} | ||
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; [[19edt]] | ; [[19edt]] | ||
* Step size: 101.103{{c}}, octave size: 1201.2{{c}} | * Step size: 101.103{{c}}, octave size: 1201.2{{c}} | ||
Stretching the octave of 12edo by a little more than 1{{c}} results in improved | 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. | ||
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}} | {{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}} | ||
{{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}} | {{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}} | ||
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; [[7edf]] | ; [[7edf]] | ||
* Step size: 100.3{{c}}, octave size: 1203.35{{c}} | * Step size: 100.3{{c}}, octave size: 1203.35{{c}} | ||
Stretching the octave of 12edo by around 3{{c}} results in improved primes | 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=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)}} | {{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}} | ||