7edo: Difference between revisions

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== Octave stretch ==

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

[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.

; 7edo

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

* Step size: 171.429{{c}}, octave size: 1200.000{{c}}

Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It ~~doesn't~~ approximate 7.

Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it is arguable whether it approximates 5 – if it does it does so poorly. It does not approximate 7.

{{Harmonics in equal|7|2|1|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 7edo}}

{{Harmonics in equal|7|2|1|intervals=integer|columns=11|collapsed=true|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)}}

{{Harmonics in equal|7|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edo (continued)}}

; [[WE|7et, 2.3.11.13 WE]]

* Step size: 171.993{{c}}, octave size: ~~1204~~.0{{c}}

* Step size: 171.993{{c}}, octave size: 1203.948{{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. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.

Stretching the octave of 7edo by around 3.9{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|171.992645|intervals=integer|columns=11|collapsed=true|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)}}

{{Harmonics in cet|171.992645|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}

; [[18ed6]]

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

* Step size: 172.331{{c}}, octave size: 1206.316{{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 13. The tuning 18ed6 does this.

Stretching the octave of 7edo by around 6.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|intervals=integer|columns=11|collapsed=true|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)}}

{{Harmonics in equal|18|6|1|intervals=integer|columns=12|start=12|collapsed=true|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}}

* Step size: 172.390{{c}}, octave size: 1206.728{{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. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.

Stretching the octave of 7edo by around 7.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|172.389769|intervals=integer|columns=11|collapsed=true|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)}}

{{Harmonics in cet|172.389769|intervals=integer|columns=12|start=12|collapsed=true|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}}

* Step size: 172.495{{c}}, octave size: 1207.471{{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. 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|172.495886|intervals=integer|columns=11|collapsed=true|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)}}

{{Harmonics in cet|172.495886|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}

; [[11edt]]

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

* Step size: 172.905{{c}}, octave size: 1210.335{{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. The tuning 11edt does this.

Stretching the octave of 7edo by around 10.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|intervals=integer|columns=11|collapsed=true|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)}}

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

== Instruments ==

@@ Line 483: / Line 483: @@
 == Octave stretch ==
-What follows is a comparison of stretched-octave 7edo tunings.
+[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.
 ; 7edo
-* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
+* Step size: 171.429{{c}}, octave size: 1200.000{{c}}
-Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7.
+Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it is arguable whether it approximates 5 – if it does it does so poorly. It does not approximate 7.
-{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
+{{Harmonics in equal|7|2|1|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in equal|7|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edo (continued)}}
 ; [[WE|7et, 2.3.11.13 WE]]
-* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
+* Step size: 171.993{{c}}, octave size: 1203.948{{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. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
+Stretching the octave of 7edo by around 3.9{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|171.992645|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in cet|171.992645|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
 ; [[18ed6]]
-* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
+* Step size: 172.331{{c}}, octave size: 1206.316{{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 13. The tuning 18ed6 does this.
+Stretching the octave of 7edo by around 6.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in equal|18|6|1|intervals=integer|columns=12|start=12|collapsed=true|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}}
+* Step size: 172.390{{c}}, octave size: 1206.728{{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. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
+Stretching the octave of 7edo by around 7.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|172.389769|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in cet|172.389769|intervals=integer|columns=12|start=12|collapsed=true|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}}
+* Step size: 172.495{{c}}, octave size: 1207.471{{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. 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|172.495886|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in cet|172.495886|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}
 ; [[11edt]]
-* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
+* Step size: 172.905{{c}}, octave size: 1210.335{{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. The tuning 11edt does this.
+Stretching the octave of 7edo by around 10.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. 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|intervals=integer|columns=11|collapsed=true|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)}}
+{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}
 == Instruments ==