18edo: Difference between revisions

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== Octave stretch or compression ==

18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].

; 18edo

* Step size: 66.667{{c}}, octave size: 1200.0{{c}}

Pure-octaves 18edo approximates all harmonics up to 15 within 31.4{{c}}.

{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}

{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}

; [[WE|18et, 13-limit WE tuning]]

* Step size: 66.291{{c}}, octave size: 1193.2{{c}}

Compressing the octave of 18edo by around 7{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 25.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}

{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}

; [[zpi|61zpi]]

* Step size: 66.228{{c}}, octave size: 1192.1{{c}}

Compressing the octave of 18edo by around 8{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 28.9{{c}}. The tuning 61zpi does this.

{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}

{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}

; [[ed12|65ed12]]

* Octave size: 1191.3{{c}}

Compressing the octave of 18edo by around 9{{c}} results in improved primes 3, 5, 7 and 13, but a worse primes 2. This approximates all harmonics up to 15 within 31.4{{c}}. The tuning 65ed12 does this.

{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}

{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}

; [[ed6|47ed6]]

* Octave size: 1188.0{{c}}

Compressing the octave of 18edo by around 12{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 15 within 29.9{{c}}. The tuning 47ed6 does this.

{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}

{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}

== Scales ==

@@ Line 523: / Line 523: @@
 |}
 <references/>
+== Octave stretch or compression ==
+edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
+; 18edo
+* Step size: 66.667{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 18edo approximates all harmonics up to 15 within 31.4{{c}}.
+{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}
+{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}
+; [[WE|18et, 13-limit WE tuning]]
+* Step size: 66.291{{c}}, octave size: 1193.2{{c}}
+Compressing the octave of 18edo by around 7{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 25.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}
+{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}
+; [[zpi|61zpi]]
+* Step size: 66.228{{c}}, octave size: 1192.1{{c}}
+Compressing the octave of 18edo by around 8{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 28.9{{c}}. The tuning 61zpi does this.
+{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}
+{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}
+; [[ed12|65ed12]]
+* Octave size: 1191.3{{c}}
+Compressing the octave of 18edo by around 9{{c}} results in improved primes 3, 5, 7 and 13, but a worse primes 2. This approximates all harmonics up to 15 within 31.4{{c}}. The tuning 65ed12 does this.
+{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}
+{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}
+; [[ed6|47ed6]]
+* Octave size: 1188.0{{c}}
+Compressing the octave of 18edo by around 12{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 15 within 29.9{{c}}. The tuning 47ed6 does this.
+{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}
+{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}
 == Scales ==