|
|
| (4 intermediate revisions by the same user not shown) |
| Line 7: |
Line 7: |
| = Title2 = | | = Title2 = |
| == 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 compressed-octave 27edo tunings. |
|
| |
|
| What follows is a comparison of stretched-octave 31edo tunings.
| | ; 27edo |
| | * Step size: 44.444{{c}}, octave size: 1200.0{{c}} |
| | 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=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}} |
|
| |
|
| ; 31edo | | ; [[WE|27et, 13-limit WE tuning]] |
| * Step size: 38.710{{c}}, octave size: 1200.0{{c}} | | * Step size: 44.375{{c}}, octave size: 1198.9{{c}} |
| Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
| | 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 equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}} | | {{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}} |
| {{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (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)}} |
|
| |
|
| ; [[WE|31et, 13-limit WE tuning]] | | ; [[97ed12]] |
| * Step size: 38.725{{c}}, octave size: 1200.5{{c}} | | * Step size: 44.350{{c}}, octave size: 1197.5{{c}} |
| Stretching the octave of 31edo by around 0.5{{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 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | 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 cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}} | | {{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}} |
| {{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}} | | {{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}} |
|
| |
|
| ; [[zpi|127zpi]] | | ; [[zpi|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]] |
| * Step size: 38.737{{c}}, octave size: 1200.8{{c}} | | * Step size (106zpi): 44.306{{c}} |
| Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.
| | * Octave size (70ed6): 1196.5{{c}} |
| {{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}} | | * Octave size (7-lim WE): 1196.4{{c}} |
| {{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}} | | * 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)}} |
|
| |
|
| ; [[WE|31et, 11-limit WE tuning]] | | ; [[90ed10]] |
| * Step size: 38.748{{c}}, octave size: 1201.2{{c}} | | * Step size: 44.292{{c}}, octave size: 1195.9{{c}} |
| Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3 and 11, but moderately worse primes 2, 5, 7 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]].
| | 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 cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}} | | {{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}} |
| {{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 equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}} |
|
| |
|
| ; [[80ed6]] | | ; [[43edt]] |
| * Step size: 38.774{{c}}, octave size: 1202.0{{c}} | | * Step size: 44.232{{c}}, octave size: 1194.3{{c}} |
| Stretching the octave of 31edo by about 2{{c}} results in moderately improved primes 3 and 11, but moderately worse primes 2, 5, 7 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]].
| | 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|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}
| | {{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}} |
| {{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}}
| | {{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}} |
| | |
| ; [[18edf]]
| |
| * Step size: nnn{{c}}, octave size: nnn{{c}}
| |
| Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3 and 11, but moderately worse primes 2, 5, 7and 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|18|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18edf}}
| |
| {{Harmonics in equal|18|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18edf (continued)}}
| |
| | |
| ; [[49edt]]
| |
| * Step size: nnn{{c}}, octave size: nnn{{c}}
| |
| Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3 and 11, but moderately worse primes 2, 5, 7and 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|49|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt}} | |
| {{Harmonics in equal|49|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt (continued)}} | |