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| = Title2 = | | = Title2 = |
| == Octave stretch or compression == | | == Octave stretch or compression == |
| What follows is a comparison of stretched- and compressed-octave 22edo tunings.
| | Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on. |
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| ; [[51ed5]]
| | For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]]. |
| * Step size: NNN{{c}}, octave size: nnn{{c}}
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| Stretching the octave of 22edo by around 3{{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 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning (eg for [[archy]] temperament). The tuning 57ed6 does this.
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| {{Harmonics in equal|51|5|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}
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| {{Harmonics in equal|51|5|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
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| ; 22edo
| | Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings. |
| * Step size: 54.545{{c}}, octave size: 1200.0{{c}}
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| Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.
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| {{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
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| {{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
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| ; [[WE|22et, 11-limit WE tuning]]
| | What follows is a comparison of stretched- and compressed-octave 41edo tunings. |
| * Step size: 54.494{{c}}, octave size: 1198.9{{c}}
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| Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.
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| {{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
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| {{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
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| ; [[zpi|80zpi]] | | ; [[147ed12]] / [[106ed6]] / [[65edt]] |
| * Step size: 54.483{{c}}, octave size: 1198.6{{c}} | | * 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}} |
| Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size. | | * 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}} |
| {{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}} | | * 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}} |
| {{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}} | | Compressing the octave of 41edo by around 0.2{{c}} results in just slightly improved primes 3, 11 and 13, but just slightly worse primes , 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this. |
| | {{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}} |
| | {{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}} |
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| ; [[57ed6]] | | ; 41edo |
| * Step size: NNN{{c}}, octave size: 1197.2{{c}} | | * Step size: 29.268{{c}}, octave size: 1200.00{{c}} |
| Compressing the octave of 22edo by around 3{{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 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning (eg for [[archy]] temperament). The tuning 57ed6 does this.
| | Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure. |
| {{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}} | | {{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}} |
| {{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}} | | {{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}} |
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| ; [[35edt]] | | ; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]] |
| * Step size: NNN{{c}}, octave size: 1195.5{{c}} | | * Step size: 29.277{{c}}, octave size: 1200.35{{c}} |
| 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 [[JND|just noticeably 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 tuning into a 2.3.7.13 [[subgroup]] tuning.
| | Stretching the octave of 41edo by around 0.5{{c}} results in just slightly improved primes 5 and 7, but just slightly worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}. |
| {{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt}} | | {{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}} |
| {{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}} | | {{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}} |
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| | ; [[WE|41et, 7-limit WE tuning]] |
| | * Step size: 29.288{{c}}, octave size: 1200.81{{c}} |
| | Stretching the octave of 41edo by just under 1{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. |
| | {{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}} |
| | {{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}} |