User:BudjarnLambeth/Sandbox2: Difference between revisions
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== Octave stretch or compression == | == Octave stretch or compression == | ||
What follows is a comparison of | What follows is a comparison of compressed-octave 27edo tunings. | ||
; 27edo | ; 27edo | ||
* Step size: 44.444{{c}}, octave size: 1200.0{{c}} | * Step size: 44.444{{c}}, octave size: 1200.0{{c}} | ||
Pure-octaves 27edo approximates all harmonics up to 16 within | Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}. | ||
{{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}} | {{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}} | ||
{{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}} | {{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}} | ||
; [[WE|27et, 13-limit WE tuning]] | ; [[WE|27et, 13-limit WE tuning]] | ||
* Step size: 44.375{{c}}, octave size: | * Step size: 44.375{{c}}, octave size: 1198.9{{c}} | ||
Compressing the octave of 27edo by around | Compressing the octave of 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. | ||
{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}} | {{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}} | ||
{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}} | {{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}} | ||
; [[97ed12]] | ; [[97ed12]] | ||
* Step size: | * Step size: 44.350{{c}}, octave size: 1197.5{{c}} | ||
Compressing the octave of 27edo by around 2.5{{c}} has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this. | |||
{{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}} | {{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}} | ||
{{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}} | {{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}} | ||
; [[70ed6]] | ; [[zpi|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]] | ||
* Step size: | * Step size (106zpi): 44.306{{c}} | ||
* Octave size (70ed6): 1196.5{{c}} | |||
{{Harmonics in equal| | * Octave size (7-lim WE): 1196.4{{c}} | ||
{{Harmonics in equal| | * Octave size (106zpi): 1196.2{{c}} | ||
Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6. | |||
{{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}} | |||
{{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}} | |||
; [[90ed10]] | |||
* Step size: 44.292{{c}}, octave size: 1195.9{{c}} | |||
Compressing the octave of 27edo by around 4{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this. | |||
{{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}} | |||
{{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}} | |||
; [[43edt]] | ; [[43edt]] | ||
* Step size: | * Step size: 44.232{{c}}, octave size: 1194.3{{c}} | ||
Compressing the octave of 27edo by around 5.5{{c}} results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2{{c}}. The tuning 43edt does this. | |||
{{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}} | {{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}} | ||
{{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}} | {{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}} | ||