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| == Octave stretch or compression == | | == Octave stretch or compression == |
| Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[the Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. | | Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. |
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| ; [[WE|12et, 7-limit WE tuning]]
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| * Step size: 99.664{{c}}, octave size: 1195.971{{c}}
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| Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but much worse primes 2 and 3. Both 7-limit [[WE]] and [[TE]] tuning do this. [[40ed10]] does this as well. An argument could be made that such tunings enable [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
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| {{Harmonics in cet|99.664256|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
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| {{Harmonics in cet|99.664256|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}
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| ; [[ZPI|34zpi]]
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| * Step size: 99.807{{c}}, octave size: 1197.686{{c}}
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| Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but worse primes 2 and 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. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
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| {{Harmonics in cet|99.807|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34zpi}}
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| {{Harmonics in cet|99.807|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34zpi (continued)}}
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| ; [[WE|12et, 5-limit WE tuning]]
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| * Step size: 99.868{{c}}, octave size: 1198.416{{c}}
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| Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but slightly worse primes 2 and 3. Both 5-limit WE and TE tuning do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
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| {{Harmonics in cet|99.868021|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
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| {{Harmonics in cet|99.868021|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}
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| ; 12edo
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| * Step size: 100.000{{c}}, octave size: 1200.000{{c}}
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| 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.
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| {{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12edo}}
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| {{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12edo (continued)}}
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| ; [[31ed6]]
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| * Step size: 100.063{{c}}, octave size: 1200.757{{c}}
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| Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
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| {{Harmonics in equal|31|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31ed6}}
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| {{Harmonics in equal|31|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31ed6 (continued)}}
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| ; [[19edt]]
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| * Step size: 101.103{{c}}, octave size: 1201.235{{c}}
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| Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
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| {{Harmonics in equal|19|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edt}}
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| {{Harmonics in equal|19|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edt (continued)}}
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| ; [[7edf]]
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| * Step size: 100.279{{c}}, octave size: 1203.351{{c}}
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| Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 2, 5, and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably will not sound very good here because of the off 5th harmonic. The tuning 7edf does this.
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| {{Harmonics in equal|7|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 7edf}}
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| {{Harmonics in equal|7|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edf (continued)}}
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| == Scales == | | == Scales == |