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| = Title1 = | | = Title1 = |
| == Octave stretch or compression == | | == Octave stretch or compression == |
| What follows is a comparison of stretched- and compressed-octave 60edo tunings. | | What follows is a comparison of stretched- and compressed-octave EDONAME tunings. |
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| ; [[35edf]] | | ; [[zpi|ZPINAME]] |
| * Step size: 20.056{{c}}, octave size: 1203.35{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 60edo by a little over 3{{c}} results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00{{c}}. The tuning 35edf does this.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. |
| {{Harmonics in equal|35|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edf}} | | {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} |
| {{Harmonics in equal|35|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edf (continued)}} | | {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} |
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| ; [[139ed5]] | | ; [[EDONOI]] |
| * Step size: 20.045{{c}}, octave size: 1202.73{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 60edo by a little under{{c}} results in improved primes 5, 7 and 11, but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 9.56{{c}}. The tuning 139ed5 does this.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. |
| {{Harmonics in equal|139|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed5}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} |
| {{Harmonics in equal|139|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed5 (continued)}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} |
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| ; [[zpi|301zpi]] | | ; [[WE|ETNAME, SUBGROUP WE tuning]] |
| * Step size: 20.027{{c}}, octave size: 1201.62{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 60edo by around 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48{{c}}. The tuning 301zpi does this.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. |
| {{Harmonics in cet|20.027|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 301zpi}} | | {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} |
| {{Harmonics in cet| 20.027 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 301zpi (continued)}} | | {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} |
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| ; [[95edt]] | | ; EDONAME |
| * Step size: 20.021{{c}}, octave size: 1201.23{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06{{c}}. The tuning 95edt does this.
| | Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}. |
| {{Harmonics in equal|95|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 95edt}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}} |
| {{Harmonics in equal|95|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 95edt (continued)}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}} |
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| ; [[WE|60et, 13-limit WE tuning]] / [[155ed6]] | | ; [[WE|ETNAME, SUBGROUP WE tuning]] |
| * Step size: 20.013{{c}}, octave size: 1200.78{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does 155ed6 whose octaves differ by only 0.02{{c}}.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. |
| {{Harmonics in cet|20.013|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning}} | | {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} |
| {{Harmonics in cet|20.013|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning (continued)}} | | {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} |
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| ; [[ed12|215ed12]] | | ; [[EDONOI]] |
| * Step size: 20.009{{c}}, octave size: 1200.55{{c}} | | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Stretching the octave of 215ed12 by around half a cent results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 9.44{{c}}. The tuning 215ed12 does this.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. |
| {{Harmonics in equal|215|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 215ed12}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} |
| {{Harmonics in equal|215|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 215ed12 (continued)}} | | {{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} |
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| ; 60edo
| | ; [[zpi|ZPINAME]] |
| * Step size: 20.000{{c}}, octave size: 1200.00{{c}}
| | * Step size: NNN{{c}}, octave size: NNN{{c}} |
| Pure-octaves 60edo approximates all harmonics up to 16 within 8.83{{c}}.
| | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. |
| {{Harmonics in equal|60|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edo}}
| | {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} |
| {{Harmonics in equal|60|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edo (continued)}}
| | {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} |
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| ; [[zpi|302zpi]]
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| * Step size: 19.962{{c}}, octave size: 1197.72{{c}}
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| Compressing the octave of 60edo by around 2{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84{{c}}. The tuning 202zpi does this. So does the tuning [[equal tuning|208ed11]] whose octave is identical within 0.3{{c}}.
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| {{Harmonics in cet|19.962|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 302zpi}}
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| {{Harmonics in cet|19.962|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 302zpi (continued)}}
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| 302zpi is particularly well suited to [[catnip]] temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the [[patent val]]s, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.
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| ; [[ed7|169ed7]]
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| * Step size: 19.958{{c}}, octave size: 1197.50{{c}}
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| Compressing the octave of 60edo by around 2.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94{{c}}. The tuning 169ed7 does this.
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| {{Harmonics in equal|169|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 169ed7}}
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| {{Harmonics in equal|169|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 169ed7 (continued)}}
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| ; [[zpi|303zpi]] | |
| * Step size: 19.913{{c}}, octave size: 1194.78{{c}} | |
| Compressing the octave of 60edo by around 5{{c}} results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.75{{c}}. The tuning 303zpi does this. So does [[equal tuning|223ed13]] whose octave is identical within 0.03{{c}}.
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| {{Harmonics in cet|19.913|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 303zpi}} | |
| {{Harmonics in cet|19.913|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 303zpi (continued)}} | |
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| = Title2 = | | = Title2 = |