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| == Octave stretch or compression == | | == Octave stretch or compression == |
| 58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]]. | | 58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]], [[150ed6]] or [[zpi|289zpi]]. |
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| What follows is a comparison of stretched- and compressed-octave 58edo tunings.
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| ; [[zpi|288zpi]]
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| * Step size: 20.736{{c}}, octave size: 1202.69{{c}}
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| Stretching the octave of 58edo by around 2.5{{c}} results in improved primes 11, 13, 19 and 23, but worse primes 2, 3, 5, 7 and 17. This approximates all harmonics up to 16 within 9.98{{c}}. The tuning 288zpi does this.
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| {{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}
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| {{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}
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| ; 58edo
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| * Step size: 20.690{{c}}, octave size: 1200.00{{c}}
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| Pure-octaves 58edo approximates all harmonics up to 16 within 8.28{{c}}.
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| {{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}
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| {{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}
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| ; [[150ed6]]
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| * Step size: 20.680{{c}}, octave size: 1199.42{{c}}
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| Compressing the octave of 58edo by around half a cent results in improved primes 3, 5, 7, 11 and 13 but a worse prime 2. This approximates all harmonics up to 16 within 6.02{{c}}. The tuning 150ed6 does this.
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| {{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}
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| {{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}
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| ; [[92edt]]
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| * Step size: 20.673{{c}}, octave size: 1199.06{{c}}
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| Compressing the octave of 58edo by around 1{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 4.60{{c}}. The tuning 92edt does this.
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| {{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}
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| {{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}
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| ; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]
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| * Step size: 20.666{{c}}, octave size: 1198.63{{c}}
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| Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 5.49{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.
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| {{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}
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| {{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}
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| ; [[WE|58et, 13-limit WE tuning]]
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| * Step size: 20.663{{c}}, octave size: 1198.45{{c}}
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| Compressing the octave of 58edo by just over 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 6.18{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
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| {{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}
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| {{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}
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| ; [[34edf]]
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| * Step size: 20.646{{c}}, octave size: 1197.45{{c}}
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| Compressing the octave of 58edo by around 2.5{{c}} results in much improved primes 5, 11 and 13, but worse primes 2 and 3. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 34edf does this.
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| {{Harmonics in equal|34|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34edf}}
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| {{Harmonics in equal|34|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34edf (continued)}}
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| == Scales == | | == Scales == |