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| == Octave stretch or compression == | | == Octave stretch or compression == |
| 22edo can benefit from slightly compressing 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 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11. | | 22edo can benefit from slightly compressing the octave, especially when using it as an 11-limit equal temperament. With the right amount of compression we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11. |
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| ; 22edo
| | Good compressed-22 options include: [[ZPI|80zpi]], [[57ed6]] or [[35edt]]. |
| * Step size: 54.545{{c}}, octave size: 1200.000{{c}}
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| Pure-octaves 22edo approximates all harmonics up to 16 but 13 within 14.3{{c}}.
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| {{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}
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| {{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}
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| ; [[WE|22et, 11-limit WE tuning]]
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| * Step size: 54.494{{c}}, octave size: 1198.859{{c}}
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| Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3, 7, and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. Both 11-limit TE and WE tunings do this.
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| {{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
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| {{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
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| ; [[ZPI|80zpi]]
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| * Step size: 54.483{{c}}, octave size: 1198.630{{c}}
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| Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3, 7 and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. The tuning 80zpi does this.
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| {{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}
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| {{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}
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| ; [[57ed6]]
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| * Step size: 54.420{{c}}, octave size: 1197.246{{c}}
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| Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 but 13 within 15.4{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.
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| {{Harmonics in equal|57|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}
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| {{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}
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| ; [[35edt]]
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| * Step size: 54.342{{c}}, octave size: 1195.515{{c}}
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| Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7-subgroup tuning into a [[2.3.7.13 subgroup|2.3.7.13-subgroup]] tuning.
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| {{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}
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| {{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}
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| == Scales == | | == Scales == |