17edo: Difference between revisions

17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. This section compares some compressed 17edo 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]]  

* Step size: 70.499{{c}}, octave size: 1198.5{{c}}

Compressing the octave of 17edo by around 1.5{{c}} results in much 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|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}

{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}

; [[27edt]]  

* Step size: 70.443{{c}}, octave size: 1197.5{{c}}

Compressing the octave of 17edo by around 2.5{{c}} results in even more improved primes 3, 7, 11 and 13 than 44ed6, but a with more error on prime 2 than 44ed6 also. 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 equal|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}

{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}

; [[zpi|56zpi]] / [[WE|17et, 2.3.7.11.13 WE tuning]]

* Step size: 70.403{{c}}, octave size: 1296.9{{c}}

* [[TE|17et, 2.3.7.11.13 TE]] * [[WE|17et, 2.3.7.11.13 WE]]  

* [[TE|17et, 2.3.7.11 TE]]

* [[WE|17et, 2.3.7.11 WE]]

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|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}

{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}