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| = Title1 = | | == Approximations of odd harmonics == |
| {{Harmonics in equal|40|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| {{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|2|intervals=odd|columns=7}} |
| {{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|6|intervals=odd|columns=7}} |
| == Octave stretch ==
| | {{harmonics in equal|7|intervals=odd|columns=7}} |
| Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| What follows is a comparison of stretched-octave 19edo tunings.
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| ; 19edo
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| * Step size: 63.158{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|13|intervals=odd|columns=7}} |
| Pure-octaves 19edo approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| {{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}} | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| {{Harmonics in equal|190|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME (continued)}} | | {{harmonics in equal|16|intervals=odd|columns=7}} |
| | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| ; [[WE|19et, 2.3.5.11 WE tuning]]
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| _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 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.192|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.192|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; [[WE|19et, 13-limit WE tuning]]
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size: 63.291{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|25|intervals=odd|columns=7}} |
| _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 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| ; [[49ed6]]
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1202.8{{c}}
| | {{harmonics in equal|31|intervals=odd|columns=7}} |
| _ing the octave of 19edo 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 49ed6 does this.
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| {{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}} | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| {{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}} | | {{harmonics in equal|34|intervals=odd|columns=7}} |
| | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| ; [[zpi|65zpi]]
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * Step size: 63.331{{c}}, octave size: 1203.3{{c}}
| | {{harmonics in equal|37|intervals=odd|columns=7}} |
| _ing the octave of 19edo 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 65zpi does this.
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}} | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| {{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}} | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | | {{harmonics in equal|41|intervals=odd|columns=7}} |
| ; [[30edt]]
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1204.6{{c}}
| | {{harmonics in equal|43|intervals=odd|columns=7}} |
| _ing the octave of 19edo 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 30edt does this.
| | {{harmonics in equal|44|intervals=odd|columns=7}} |
| {{Harmonics in equal|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}} | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| {{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}} | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| | | {{harmonics in equal|47|intervals=odd|columns=7}} |
| ; [[11edf]]
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: 1212.5{{c}}
| | {{harmonics in equal|49|intervals=odd|columns=7}} |
| _ing the octave of 19edo 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 11edf does this.
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| {{Harmonics in equal|11|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf}} | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| {{Harmonics in equal|11|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf (continued)}} | | {{harmonics in equal|52|intervals=odd|columns=7}} |
| | {{harmonics in equal|53|intervals=odd|columns=7}} |