User:BudjarnLambeth/Sandbox2: Difference between revisions
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== Octave stretch or compression == | == Octave stretch or compression == | ||
31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. | |||
What follows is a comparison of stretched-octave 31edo tunings. | What follows is a comparison of stretched-octave 31edo tunings. | ||
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; [[zpi|127zpi]] | ; [[zpi|127zpi]] | ||
* Step size: 38.737{{c}}, octave size: | * Step size: 38.737{{c}}, octave size: 1200.8{{c}} | ||
Stretching the octave of 31edo by | Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this. | ||
{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}} | {{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}} | ||
{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}} | {{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}} | ||
; [[WE|31et, 11-limit WE tuning]] | ; [[WE|31et, 11-limit WE tuning]] | ||
* Step size: 38.748{{c}}, octave size: | * Step size: 38.748{{c}}, octave size: 1201.2{{c}} | ||
Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3, 7 and 11, but moderately worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]]. | |||
{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}} | {{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}} | ||
{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued) | {{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)} | ||
; [[80ed6]] | ; [[80ed6]] | ||
* Step size: | * Step size: 38.774{{c}}, octave size: 1202.0{{c}} | ||
Stretching the octave of 31edo by | Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3, 7 and 11, but moderately worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]]... Slightly more than this - a stretch of 2.239{{c}} - is the absolute maximum amount of octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. | ||
{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}} | {{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}} | ||
{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued) | {{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)} | ||