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= Title2 =
= Title2 =
== Octave compression ==
== Octave stretch ==
What follows is a comparison of compressed-octave 17edo tunings.
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.


; 17edo
What follows is a comparison of stretched-octave 19edo tunings.
* Step size: 70.588{{c}}, octave size: 1200.0{{c}}
Pure-octaves 17edo approximates the 2.3.11 subgroup well, it arguably might approximate 7, but not well, and it doesn't really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.
{{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}}
{{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}}


; [[44ed6]]
; 19edo
* Step size: NNN{{c}}, octave size: 1198.5{{c}}
* Step size: 63.158{{c}}, octave size: 1200.0{{c}}  
Compressing the octave of 17edo by around 1.5{{c}} results in greatly improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
Pure-octaves 19edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}
{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}
{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}
{{Harmonics in equal|19|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo (continued)}}


; [[27edt]]  
; [[WE|19et, 2.3.5.11 WE tuning]]  
* Step size: NNN{{c}}, octave size: 1197.5{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}
{{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|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}
{{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)}}


; [[zpi|56zpi]] / [[WE|17et, 2.3.7.11.13 WE tuning]]
; [[WE|19et, 13-limit WE tuning]]  
* Step size: 70.403{{c}}, octave size: 1296.9{{c}}
* Step size: 63.291{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by around 3{{c}} results in improved primes NNN, but worse primes NNN. The tunings: 56zpi, [[TE|17et, 2.3.7.11.13 TE]] and [[WE|17et, 2.3.7.11.13 WE]] all do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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 cet|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}
{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}


; [[WE|17et, 2.3.7.11 WE tuning]]  
; [[49ed6]]
* Step size: 70.392{{c}}, octave size: 1296.7{{c}}
* Step size: NNN{{c}}, octave size: 1202.8{{c}}
Compressing the octave of 17edo by just over 3{{c}} results in improved primes NNN, but worse primes NNN. Its 2.3.7.11 WE tuning and 2.3.7.11 [[TE]] tuning both do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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 cet|70.392|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning}}
{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in cet|70.392|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
 
; [[zpi|65zpi]]  
* Step size: 63.331{{c}}, octave size: 1203.3{{c}}
_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 cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}
{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}
 
; [[30edt]]  
* Step size: NNN{{c}}, octave size: 1204.6{{c}}
_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|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}}
 
; [[11edf]]
* Step size: NNN{{c}}, octave size: 1212.5{{c}}
_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|11|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf}}
{{Harmonics in equal|11|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf (continued)}}
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