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| == Octave stretch or compression == | | == Octave stretch or compression == |
| 38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]]. | | 38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]]. |
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| What follows is a comparison of stretched-octave 38edo tunings.
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| ; 38edo
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| * Step size: 31.579{{c}}, octave size: 1200.00{{c}}
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| Pure-octaves 38edo approximates all harmonics up to 16 within 14.6{{c}}.
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| {{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}}
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| {{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}}
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| ; [[WE|38et, 13-limit WE tuning]]
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| * Step size: 31.599{{c}}, octave size: 1200.77{{c}}
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| Stretching the octave of 38edo by around 1{{c}} results in improved primes 3, 5, 11, 17 and 19, but worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 14.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
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| {{Harmonics in cet|31.599|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning}}
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| {{Harmonics in cet|31.599|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning (continued)}}
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| ; [[ed5|88ed5]]
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| * Step size: 31.663{{c}}, octave size: 1203.18{{c}}
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| Stretching the octave of 38edo by around 3{{c}} results in improved primes 3, 5, 11, 13, 17 and 19 but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.7{{c}}. The tuning 88ed5 does this.
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| {{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}}
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| {{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}}
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| ; [[zpi|166zpi]]
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| * Step size: 31.671{{c}}, octave size: 1203.48{{c}}
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| Stretching the octave of 38edo by around 3.5{{c}} results in improved primes 3, 5, 11, 13, 17 and 19, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.0{{c}}. This results in the same [[mapping]] as [[wart|38df]], which is the mapping used by most of 38edo's lowest-[[badness]] temperaments. The tuning 166zpi does this.
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| {{Harmonics in cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}
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| {{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}
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| ; [[60edt]]
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| * Step size: 31.699{{c}}, octave size: 1204.57{{c}}
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| Stretching the octave of 38edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13, 17 and 19, but a much worse prime 2. This approximates all harmonics up to 16 within 13.7{{c}}. This results in the same [[mapping]] as [[wart|38df]], which is the mapping used by most of 38edo's lowest-[[badness]] temperaments. The tuning 60edt does this.
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| {{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}
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| {{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}
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| == Scales == | | == Scales == |