33edo: Difference between revisions

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<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==

33edo is nearby to many other [[equal tuning]]s which can act as stretched or compressed versions of 33edo, improving some of its harmonics at the expense of others.

[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]

What follows is a comparison of stretched- and compressed-octave 33edo tunings.

; [[ed5|76ed5]]

* Octave size: 1209.8{{c}}

Stretching the octave of 33edo by around 10{{c}} results in improved primes 3 and 7, but worse primes 2 and 11. This approximates all harmonics up to 16 within 17.0{{c}}. The tuning 76ed5 does this.

{{Harmonics in equal|76|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 76ed5}}

{{Harmonics in equal|76|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 76ed5 (continued)}}

; [[ed7|92ed7]]

* Octave size: 1208.4{{c}}

Stretching the octave of 33edo by around 8.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 17.7{{c}}. The tuning 92ed7 does this. So does the tuning [[zpi|137zpi]] whose octave differs by only 0.3{{c}}.

{{Harmonics in equal|92|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92ed7}}

{{Harmonics in equal|92|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92ed7 (continued)}}

; [[equal tuning|114ed11]]

* Octave size: 1201.7{{c}}

Stretching the octave of 33edo by around 2{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.8{{c}}. The tuning 114ed11 does this.

{{Harmonics in equal|114|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114ed11}}

{{Harmonics in equal|114|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114ed11 (continued)}}

; [[zpi|138zpi]]

* Step size: 36.394{{c}}, octave size: 1201.0{{c}}

Stretching the octave of 33edo by around 1{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 138zpi does this. So does the tuning [[equal tuning|122ed13]] whose octave differs by only 0.1{{c}}.

{{Harmonics in cet|36.394|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 138zpi}}

{{Harmonics in cet|36.394|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 138zpi (continued)}}

; 33edo

* Step size: 36.363{{c}}, octave size: 1200.0{{c}}

Pure-octaves 33edo approximates all harmonics up to 16 within 14.3{{c}}.

{{Harmonics in equal|33|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33edo}}

{{Harmonics in equal|33|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33edo (continued)}}

; [[WE|33et, 13-limit WE tuning]]

* Step size: 36.357{{c}}, octave size: 1199.8{{c}}

Compressing the octave of 33edo by a fifth of a cent results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 13.6{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|36.357|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning}}

{{Harmonics in cet|36.357|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning (continued)}}

; [[ed7|93ed7]]

* Octave size: 1196.4{{c}}

Compressing the octave of 33edo by around 4.5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.9{{c}}. If one wishes to use both 33edo's sharp and flat fifths simultaneously (see [[dual-fifth tuning]]), then this amount of stretch is ideal, because it evenly shares error between the two fifths. The tuning 93ed7 does this. So does the tuning [[equal tuning|52ed13]] whose octave differs by only 0.1{{c}}.

{{Harmonics in equal|93|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 93ed7}}

{{Harmonics in equal|93|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93ed7 (continued)}}

; [[ed5|77ed5]]

* Octave size: 1194.1{{c}}

Compressing the octave of 33edo by around 6{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 77ed5 does this. So does the tuning [[zpi|139zpi]] whose octave differs by only 0.2{{c}}.

{{Harmonics in equal|77|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 77ed5}}

{{Harmonics in equal|77|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 77ed5 (continued)}}

; [[equal tuning|115ed11]]

* Octave size: 1191.2{{c}}

Compressing the octave of 33edo by around 9{{c}} results in improved primes 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 115ed11 does this. So do the tunings [[equal tuning|123ed13]] and [[AS|1ed47/46]] whose octaves are within 0.3{{c}} of 115ed11.

{{Harmonics in equal|115|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 115ed11}}

{{Harmonics in equal|115|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 115ed11 (continued)}}

== Scales ==

@@ Line 483: / Line 483: @@
 |}
 <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Octave stretch or compression ==
+edo is nearby to many other [[equal tuning]]s which can act as stretched or compressed versions of 33edo, improving some of its harmonics at the expense of others.
+[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]
+What follows is a comparison of stretched- and compressed-octave 33edo tunings.
+; [[ed5|76ed5]]
+* Octave size: 1209.8{{c}}
+Stretching the octave of 33edo by around 10{{c}} results in improved primes 3 and 7, but worse primes 2 and 11. This approximates all harmonics up to 16 within 17.0{{c}}. The tuning 76ed5 does this.
+{{Harmonics in equal|76|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 76ed5}}
+{{Harmonics in equal|76|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 76ed5 (continued)}}
+; [[ed7|92ed7]]
+* Octave size: 1208.4{{c}}
+Stretching the octave of 33edo by around 8.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 17.7{{c}}. The tuning 92ed7 does this. So does the tuning [[zpi|137zpi]] whose octave differs by only 0.3{{c}}.
+{{Harmonics in equal|92|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92ed7}}
+{{Harmonics in equal|92|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92ed7 (continued)}}
+; [[equal tuning|114ed11]]
+* Octave size: 1201.7{{c}}
+Stretching the octave of 33edo by around 2{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.8{{c}}. The tuning 114ed11 does this.
+{{Harmonics in equal|114|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114ed11}}
+{{Harmonics in equal|114|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114ed11 (continued)}}
+; [[zpi|138zpi]]
+* Step size: 36.394{{c}}, octave size: 1201.0{{c}}
+Stretching the octave of 33edo by around 1{{c}} results in improved primes 3, 11 and 13, but worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 138zpi does this. So does the tuning [[equal tuning|122ed13]] whose octave differs by only 0.1{{c}}.
+{{Harmonics in cet|36.394|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 138zpi}}
+{{Harmonics in cet|36.394|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 138zpi (continued)}}
+; 33edo
+* Step size: 36.363{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 33edo approximates all harmonics up to 16 within 14.3{{c}}.
+{{Harmonics in equal|33|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33edo}}
+{{Harmonics in equal|33|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33edo (continued)}}
+; [[WE|33et, 13-limit WE tuning]]
+* Step size: 36.357{{c}}, octave size: 1199.8{{c}}
+Compressing the octave of 33edo by a fifth of a cent results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 13.6{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|36.357|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning}}
+{{Harmonics in cet|36.357|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 33et, 13-limit WE tuning (continued)}}
+; [[ed7|93ed7]]
+* Octave size: 1196.4{{c}}
+Compressing the octave of 33edo by around 4.5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.9{{c}}. If one wishes to use both 33edo's sharp and flat fifths simultaneously (see [[dual-fifth tuning]]), then this amount of stretch is ideal, because it evenly shares error between the two fifths. The tuning 93ed7 does this. So does the tuning [[equal tuning|52ed13]] whose octave differs by only 0.1{{c}}.
+{{Harmonics in equal|93|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 93ed7}}
+{{Harmonics in equal|93|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93ed7 (continued)}}
+; [[ed5|77ed5]]
+* Octave size: 1194.1{{c}}
+Compressing the octave of 33edo by around 6{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 77ed5 does this. So does the tuning [[zpi|139zpi]] whose octave differs by only 0.2{{c}}.
+{{Harmonics in equal|77|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 77ed5}}
+{{Harmonics in equal|77|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 77ed5 (continued)}}
+; [[equal tuning|115ed11]]
+* Octave size: 1191.2{{c}}
+Compressing the octave of 33edo by around 9{{c}} results in improved primes 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 17.5{{c}}. The tuning 115ed11 does this. So do the tunings [[equal tuning|123ed13]] and [[AS|1ed47/46]] whose octaves are within 0.3{{c}} of 115ed11.
+{{Harmonics in equal|115|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 115ed11}}
+{{Harmonics in equal|115|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 115ed11 (continued)}}
 == Scales ==