12edo: 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 ==

Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[the Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

; [[WE|12et, 7-limit WE tuning]]

* Step size: 99.664{{c}}, octave size: 1196.0{{c}}

Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.

{{Harmonics in cet|99.664|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}

{{Harmonics in cet|99.664|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}

; [[zpi|34zpi]]

* Step size: 99.807{{c}}, octave size: 1197.7{{c}}

Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].

{{Harmonics in cet|99.807|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34zpi}}

{{Harmonics in cet|99.807|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34zpi (continued)}}

; [[WE|12et, 5-limit WE tuning]]

* Step size: 99.868{{c}}, octave size: 1198.4{{c}}

Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.

{{Harmonics in cet|99.868|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}

{{Harmonics in cet|99.868|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}

; 12edo

* Step size: 100.000{{c}}, octave size: 1200.0{{c}}

Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.

{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12edo}}

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

; [[31ed6]]

* Step size: 100.063{{c}}, octave size: 1200.8{{c}}

Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.

{{Harmonics in equal|31|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31ed6}}

{{Harmonics in equal|31|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31ed6 (continued)}}

; [[19edt]]

* Step size: 101.103{{c}}, octave size: 1201.2{{c}}

Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.

{{Harmonics in equal|19|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edt}}

{{Harmonics in equal|19|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edt (continued)}}

; [[7edf]]

* Step size: 100.3{{c}}, octave size: 1203.35{{c}}

Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.

{{Harmonics in equal|7|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 7edf}}

{{Harmonics in equal|7|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edf (continued)}}

== Scales ==

@@ Line 850: / Line 850: @@
 |}
 <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 ==
+Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[the Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.
+; [[WE|12et, 7-limit WE tuning]]
+* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
+Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
+{{Harmonics in cet|99.664|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
+{{Harmonics in cet|99.664|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}
+; [[zpi|34zpi]]
+* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
+Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
+{{Harmonics in cet|99.807|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34zpi}}
+{{Harmonics in cet|99.807|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34zpi (continued)}}
+; [[WE|12et, 5-limit WE tuning]]
+* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
+Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
+{{Harmonics in cet|99.868|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
+{{Harmonics in cet|99.868|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}
+; 12edo
+* Step size: 100.000{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
+{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12edo}}
+{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12edo (continued)}}
+; [[31ed6]]
+* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
+Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
+{{Harmonics in equal|31|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31ed6}}
+{{Harmonics in equal|31|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31ed6 (continued)}}
+; [[19edt]]
+* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
+Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
+{{Harmonics in equal|19|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edt}}
+{{Harmonics in equal|19|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edt (continued)}}
+; [[7edf]]
+* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
+Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
+{{Harmonics in equal|7|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 7edf}}
+{{Harmonics in equal|7|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edf (continued)}}
 == Scales ==