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| == Octave stretch or compression == | | == Octave stretch or compression == |
| Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[57ed12]] being a good option. | | Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]]. Suitable stretched 16edo tunings include [[zpi|15zpi]] and [[57ed12]]. |
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| What follows is a comparison of stretched- and compressed-octave 16edo tunings.
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| ; 16edo
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| * Step size: 75.000{{c}}, octave size: 1200.0{{c}}
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| Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.
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| {{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}
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| {{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}
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| ; [[WE|16et, 2.5.7.13 WE tuning]]
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| * Step size: 75.105{{c}}, octave size: 1201.7{{c}}
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| Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
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| {{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}
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| {{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}
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| ; [[zpi|15zpi]]
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| * Step size: 75.262{{c}}, octave size: 1204.2{{c}}
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| Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 both do this, their octaves differ from one another by less than 0.1{{c}}.
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| {{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}
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| {{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}
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| ; [[WE|16et, 13-limit WE tuning]]
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| * Step size: 75.315{{c}}, octave size: 1205.0{{c}}
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| Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning 37ed5, whose octave differs by only 0.1{{c}}.
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| {{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}
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| {{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}
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| ; [[57ed12]]
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| * Step size: 75.473{{c}}, octave size: 1207.6{{c}}
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| Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within 35.0{{c}}. The tunings 57ed12 and 55ed11 both do this, their octaves differ from one another by only 0.1{{c}}.
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| {{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}
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| {{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}
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| == Scales == | | == Scales == |