7edo: Difference between revisions

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It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>

~~=== Octave stretch ===~~

[[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.

~~The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.~~

=== Subsets and supersets ===

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== Approximation to JI ==

[[File:7ed2-001.svg~~|alt=alt : Your browser has no SVG support.~~]]

[[File:7ed2-001.svg]]

~~[[:File:7ed2-001.svg|7ed2-001.svg]]~~

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 15~~

~~| steps = 6.95668765658792~~

~~| step size = 172.495885863671~~

~~| tempered height = 4.166936~~

~~| pure height = 3.940993~~

~~| integral = 1.162332~~

~~| gap = 14.234171~~

~~| octave = 1207.47120104570~~

~~| consistent = 6~~

~~| distinct = 5~~

}}

== Regular temperament properties ==

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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 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.

{{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. 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 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|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. 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. 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. 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 40: / Line 40: @@
 It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>
-=== Octave stretch ===
-[[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.
-The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.
 === Subsets and supersets ===
@@ Line 265: / Line 260: @@
 == Approximation to JI ==
-[[File:7ed2-001.svg|alt=alt : Your browser has no SVG support.]]
+[[File:7ed2-001.svg]]
-[[:File:7ed2-001.svg|7ed2-001.svg]]
-=== Zeta peak index ===
-{{ZPI
-| zpi = 15
-| steps = 6.95668765658792
-| step size = 172.495885863671
-| tempered height = 4.166936
-| pure height = 3.940993
-| integral = 1.162332
-| gap = 14.234171
-| octave = 1207.47120104570
-| consistent = 6
-| distinct = 5
-}}
 == Regular temperament properties ==
@@ Line 502: / 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 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.
+{{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. 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 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|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. 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. 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. 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 ==