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= [[7edo]] =
= Title1 =
== Octave stretch or compression ==
{{Harmonics in equal|40|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
What follows is a comparison of stretched- and compressed-octave 7edo tunings.
{{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}


; 7edo
= Title2 =
* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
== Octave compression ==
Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
What follows is a comparison of compressed-octave 17edo tunings.
{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
{{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}}


; [[WE|7et, 2.3.11.13 WE]]
; 17edo
* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
* Step size: 70.588{{c}}, octave size: 1200.0{{c}}  
Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
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 cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
{{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}}
{{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
{{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}}


; [[18ed6]]  
; [[44ed6]]  
* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
* Step size: NNN{{c}}, octave size: 1198.5{{c}}
Stretching the octave of 7edo 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 18ed6 does this.
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.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}
{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}


; [[WE|7et, 2.3.5.11.13 WE]]  
; [[27edt]]  
* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
* Step size: NNN{{c}}, octave size: 1197.5{{c}}
Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
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.
{{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
{{Harmonics in equal|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}
{{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}


; [[zpi|15zpi]]  
; [[zpi|56zpi]] / [[WE|17et, 2.3.7.11.13 WE tuning]]
* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
* Step size: 70.403{{c}}, octave size: 1296.9{{c}}
Stretching the octave of 7edo 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 15zpi does this.
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.
{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}
{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}
{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}


; [[11edt]]  
; [[WE|17et, 2.3.7.11 WE tuning]]  
* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
* Step size: 70.392{{c}}, octave size: 1296.7{{c}}
Stretching the octave of 7edo 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 11edt does this.
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.
{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
{{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|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}
{{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)}}
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