12edo: Difference between revisions

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== Theory ==

12edo achieved its position ~~because it~~ is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone). It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. 12edo became the standard Western tuning through a gradual historical process, primarily due to its practical advantages for keyboard instruments and its reasonable approximation of common intervals across all keys. However, it should be borne in mind that in actual performance these deviations from just intonation are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.

It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

12edo is the basic example of ~~an equidodecatonic scale, or more simply,~~ a 12-tone [[well temperament]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.

In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers.

Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

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=== Prime harmonics ===

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

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.

=== Subsets and supersets ===

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{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

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

~~{{ZPI~~

~~| zpi = 34~~

~~| steps = 12.0231830072926~~

~~| step size = 99.8071807833375~~

~~| tempered height = 5.193290~~

~~| pure height = 5.084467~~

~~| integral = 1.269599~~

~~| gap = 15.899282~~

~~| octave = 1197.68616940005~~

~~| consistent = 10~~

~~| distinct = 6~~

}}

== Regular temperament properties ==

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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 [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|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: 1195.971{{c}}

Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but much worse primes 2 and 3. Both 7-limit [[WE]] and [[TE]] tuning do this. [[40ed10]] does this as well. An argument could be made that such tunings enable [[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|99.664256|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}

{{Harmonics in cet|99.664256|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: 1197.686{{c}}

Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but worse primes 2 and 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: 1198.416{{c}}

Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but slightly worse primes 2 and 3. Both 5-limit WE and TE tuning do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.

{{Harmonics in cet|99.868021|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}

{{Harmonics in cet|99.868021|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: 1200.000{{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: 1200.757{{c}}

Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. 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: 1201.235{{c}}

Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. 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: 100.279{{c}}, octave size: 1203.351{{c}}

Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 2, 5, and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably will not 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 11: / Line 11: @@
 == Theory ==
-edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone). It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. 12edo became the standard Western tuning through a gradual historical process, primarily due to its practical advantages for keyboard instruments and its reasonable approximation of common intervals across all keys. However, it should be borne in mind that in actual performance these deviations from just intonation are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.
+edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
+It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).
-edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]].
+It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.
+Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
+In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers.
+Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}
+edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.
 The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].
@@ Line 23: / Line 30: @@
 === Prime harmonics ===
 {{Harmonics in equal|12|prec=2}}
-=== Octave stretch ===
-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.
 === Subsets and supersets ===
@@ Line 358: / Line 362: @@
 {{Q-odd-limit intervals|12}}
 {{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 34
-| steps = 12.0231830072926
-| step size = 99.8071807833375
-| tempered height = 5.193290
-| pure height = 5.084467
-| integral = 1.269599
-| gap = 15.899282
-| octave = 1197.68616940005
-| consistent = 10
-| distinct = 6
-}}
 == Regular temperament properties ==
@@ Line 860: / 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 [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|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: 1195.971{{c}}
+Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but much worse primes 2 and 3. Both 7-limit [[WE]] and [[TE]] tuning do this. [[40ed10]] does this as well. An argument could be made that such tunings enable [[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|99.664256|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
+{{Harmonics in cet|99.664256|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: 1197.686{{c}}
+Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but worse primes 2 and 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: 1198.416{{c}}
+Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but slightly worse primes 2 and 3. Both 5-limit WE and TE tuning do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
+{{Harmonics in cet|99.868021|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
+{{Harmonics in cet|99.868021|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: 1200.000{{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: 1200.757{{c}}
+Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. 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: 1201.235{{c}}
+Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. 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: 100.279{{c}}, octave size: 1203.351{{c}}
+Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 2, 5, and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably will not 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 ==