7edo: Difference between revisions

Line 481:

3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.

== Octave stretch ==

What follows is a comparison of stretched-octave 7edo tunings.

; 7edo

* Step size: 171.429{{c}}, octave size: 1200.0{{c}}

Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.

{{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]]

* Step size: 171.993{{c}}, octave size: 1204.0{{c}}

Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 75.0{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.

{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}

{{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)}}

; [[18ed6]]

* Step size: 172.331{{c}}, octave size: 1206.3{{c}}

Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 14. This approximates all harmonics up to 16 within 48.7{{c}}. The tuning 18ed6 does this.

{{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}

{{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}

; [[WE|7et, 2.3.5.11.13 WE]]

* Step size: 172.390{{c}}, octave size: 1206.7{{c}}

Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 85.7{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.

{{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 cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}

; [[zpi|15zpi]]

* Step size: 172.495{{c}}, octave size: 1207.5{{c}}

Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 84.0{{c}}. The tuning 15zpi does this.

{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}

{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}

; [[11edt]]

* Step size: 172.905{{c}}, octave size: 1210.3{{c}}

Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 83.6{{c}}. The tuning 11edt does this.

{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}

{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}

== Instruments ==

@@ Line 481: / Line 481: @@
 /7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.
+== Octave stretch ==
+What follows is a comparison of stretched-octave 7edo tunings.
+; 7edo
+* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
+{{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]]
+* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
+Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 75.0{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
+{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
+{{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)}}
+; [[18ed6]]
+* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
+Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 14. This approximates all harmonics up to 16 within 48.7{{c}}. The tuning 18ed6 does this.
+{{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
+{{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}
+; [[WE|7et, 2.3.5.11.13 WE]]
+* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
+Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 85.7{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
+{{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 cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
+; [[zpi|15zpi]]
+* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
+Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 84.0{{c}}. The tuning 15zpi does this.
+{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
+{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}
+; [[11edt]]
+* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
+Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 83.6{{c}}. The tuning 11edt does this.
+{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
+{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}
 == Instruments ==