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{{~~EDO~~ intro~~|12~~}} It is the predominating tuning system in the world today.

{{ED intro}} It is the predominating tuning system in the world today.

== 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 as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[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 fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

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).

~~The seventh partial (~~[[~~7/4~~]]) is "represented" by an interval which is sharp by 31 cents, which is 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 chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[~~support~~]]~~s septimal meantone 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 or~~ 13 ~~at all~~, ~~though it does a quite credible 17 and an even better 19. Nevertheless~~, its ~~relative tuning accuracy is quite high~~, ~~and 12edo~~ is ~~the fourth~~ [[~~zeta integral edo~~]].

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.

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|312/219]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and ~~the septimal kleisma~~, ~~[[225/224]]. Each of~~ these ~~affects~~ the ~~structure~~ of ~~12et~~ in ~~specific ways~~, and ~~tuning systems which share the comma in question will be similar to 12et in precisely those ways~~.

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 offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

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]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|312/219]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== 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]] 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 [[19edt]] or [[31ed6]], also makes sense.

=== Subsets and supersets ===

12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[~~The~~ Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.

[[24edo]], which doubles it, ~~provides a great correction for the approximate harmonics~~ 11 and 13. [[36edo]], which triples it, ~~provides a great correction for~~ the ~~approximate harmonic 7~~. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 ~~temperaments~~ that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

~~=== Miscellany ===~~

~~12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}~~.

== Intervals ==

{| class="wikitable center-all"

|+ Intervals of 12edo

|+ style="font-size: 105%;" | Intervals of 12edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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| Minor second

|

| [[25/24]] (+29.328) [[16/15]] (~~-11~~.731)

| [[25/24]] (+29.328) [[16/15]] (−11.731)

| [[28/27]] (+37.039) [[21/20]] (+15.533) [[15/14]] (~~-19~~.443)

| [[28/27]] (+37.039) [[21/20]] (+15.533) [[15/14]] (−19.443)

| [[18/17]] (+1.045) [[17/16]] (-4.955)

| [[18/17]] (+1.045) [[17/16]] (−4.955)

| [[File:piano_1_12edo.mp3]]

|-

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| 200

| Major second

| [[9/8]] (-3.910)

| [[9/8]] (−3.910)

| [[10/9]] (+17.596)

| [[28/25]] (+3.802) [[8/7]] (~~-31~~.174)

| [[28/25]] (+3.802) [[8/7]] (−31.174)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (-0.532) [[17/15]] (~~-16~~.687)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (−0.532) [[17/15]] (−16.687)

| [[File:piano_1_6edo.mp3]]

|-

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| Minor third

| [[32/27]] (+5.865)

| [[6/5]] (~~-15~~.641)

| [[6/5]] (−15.641)

| [[7/6]] (+33.129) [[25/21]] (-1.847)

| [[7/6]] (+33.129) [[25/21]] (−1.847)

| [[19/16]] (+2.487) [[44/37]] (+0.026)

| [[File:piano_1_4edo.mp3]]

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| 400

| Major third

| [[81/64]] (-7.820)

| [[81/64]] (−7.820)

| [[5/4]] (+13.686)

| [[63/50]] (-0.108) [[9/7]] (~~-35~~.084)

| [[63/50]] (−0.108) [[9/7]] (−35.084)

| [[34/27]] (+0.910) [[24/19]] (-4.442)

| [[34/27]] (+0.910) [[24/19]] (−4.442)

| [[File:piano_1_3edo.mp3]]

|-

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| [[Tritone]]

|

| [[45/32]] (+9.776) [[64/45]] (−9.776)

| [[7/5]] (+17.488) [[10/7]] (~~-17~~.488)

| [[7/5]] (+17.488) [[10/7]] (−17.488)

| [[24/17]] (+3.000) [[99/70]] (-0.088) [[17/12]] (-3.000)

| [[24/17]] (+3.000) [[99/70]] (−0.088) [[17/12]] (−3.000)

| [[File:piano_1_2edo.mp3]]

|-

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| 700

| Fifth

| [[3/2]] (-1.955)

| [[3/2]] (−1.955)

|

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| Minor sixth

| [[128/81]] (+7.820)

| [[8/5]] (~~-13~~.686)

| [[8/5]] (−13.686)

| [[14/9]] (+35.084) [[100/63]] (+0.108)

| [[19/12]] (+4.442) [[27/17]] (-0.910)

| [[19/12]] (+4.442) [[27/17]] (−0.910)

| [[File:piano_2_3edo.mp3]]

|-

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| 900

| Major sixth

| [[27/16]] (-5.865)

| [[27/16]] (−5.865)

| [[5/3]] (+15.641)

| [[42/25]] (+1.847) [[12/7]] (~~-33~~.129)

| [[42/25]] (+1.847) [[12/7]] (−33.129)

| [[37/22]] (-0.026) [[32/19]] (-2.487)

| [[37/22]] (−0.026) [[32/19]] (−2.487)

| [[File:piano_3_4edo.mp3]]

|-

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| Minor seventh

| [[16/9]] (+3.910)

| [[9/5]] (~~-17~~.596)

| [[9/5]] (−17.596)

| [[7/4]] (+31.174) [[25/14]] (-3.802)

| [[7/4]] (+31.174) [[25/14]] (−3.802)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (-0.020) [[34/19]] (-7.442)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (−0.020) [[34/19]] (−7.442)

| [[File:piano_5_6edo.mp3]]

|-

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| Major seventh

|

| [[15/8]] (+11.731) [[48/25]] (~~-29~~.328)

| [[15/8]] (+11.731) [[48/25]] (−29.328)

| [[28/15]] (+19.443) [[40/21]] (~~-15~~.533) [[27/14]] (~~-37~~.039)

| [[28/15]] (+19.443) [[40/21]] (−15.533) [[27/14]] (−37.039)

| [[32/17]] (+4.955) [[17/9]] (-1.045)

| [[32/17]] (+4.955) [[17/9]] (−1.045)

| [[File:piano_11_12edo.mp3]]

|-

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{| class="wikitable center-all"

|+~~Notation of 12edo~~

|+ style="font-size: 105%;" | Notation of 12edo

~~! rowspan~~="2" |~~[[Degree]]~~

~~! rowspan="2" |[[Cent]]s~~

~~! colspan="2" |[[Chain-~~of~~-fifths notation|Standard notation]]~~

|-

! Diatonic ([[5L 2s]]) interval names

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Chain-of-fifths notation|Standard notation]]

|-

! Diatonic ([[5L 2s]]) interval names

! Note names (on D)

|-

| 0

|'''Perfect unison (P1)'''

| '''Perfect unison (P1)'''

|'''D'''

| '''D'''

|-

| 1

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| 2

| 200

|'''Major second (M2)''' Diminished third (d3)

| '''Major second (M2)''' Diminished third (d3)

|'''E''' Fb

| '''E''' Fb

|-

| 3

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| 5

| 500

|'''Perfect fourth (P4)'''

| '''Perfect fourth (P4)'''

|'''G'''

| '''G'''

|-

| 6

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| 7

| 700

|'''Perfect fifth (P5)'''

| '''Perfect fifth (P5)'''

| A

|-

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| 9

| 900

|'''Major sixth (M6)''' Diminished seventh (d7)

| '''Major sixth (M6)''' Diminished seventh (d7)

|'''B''' Cb

| '''B''' Cb

|-

| 10

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| 12

| 1200

|'''Perfect octave (P8)'''

| '''Perfect octave (P8)'''

|'''D'''

| '''D'''

|}

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* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===

=== Sagittal notation ===

~~====Evo flavor====~~

==== Evo flavor ====

File:12-EDO_Evo_Sagittal.svg

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Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

~~====Revo flavor====~~

==== Revo flavor ====

File:12-EDO_Revo_Sagittal.svg

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== Solfege ==

{| class="wikitable center-all"

|+ Solfege of 12edo

|+ style="font-size: 105%;" | Solfege of 12edo

|-

! [[Degree]]

! [[Cents]]

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=== 15-odd-limit interval mappings ===

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

== Regular temperament properties ==

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| 2.64

|}

* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===

=== Commas ===

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| [[Hexe]]

|}

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

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

== ~~Zeta properties~~ ==

== 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)}}

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

~~{| class="wikitable"~~

~~! colspan="3" |Tuning~~

~~! colspan="3" |Strength~~

~~! colspan="2" |Closest EDO~~

~~! colspan="2" |Integer limit~~

|-

~~!ZPI~~

~~!Steps per octave~~

~~!Step size (cents)~~

~~!Height~~

~~!Integral~~

~~!Gap~~

~~!EDO~~

~~!Octave (cents)~~

~~!Consistent~~

~~!Distinct~~

|-

~~|[[34zpi]]~~

~~|12.0231830072926~~

~~|99.8071807833375~~

~~|5.193290~~

~~| 1.269599~~

~~|15.899282~~

~~|12edo~~

~~|1197.68616940005~~

~~|10~~

|6

|}

== Scales ==

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* [[Lumatone mapping for 12edo]]

* [[:purdal:12-EDD]]{{dead link}}

* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==

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* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]]

[[Category:Historical]]

[[Category:Meantone]]

@@ Line 8: / Line 8: @@
 {{Infobox ET}}
 {{Wikipedia|12 equal temperament}}
-{{EDO intro|12}} It is the predominating tuning system in the world today.
+{{ED intro}} It is the predominating tuning system in the world today.
 == 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 as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[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 fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
+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).
-The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is 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 chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone 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 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].
+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.
-The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
+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 offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.
+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]].
+Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
+edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.
 === 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]] 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 [[19edt]] or [[31ed6]], also makes sense.
 === Subsets and supersets ===
-edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.
+edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.
-[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].
+[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].
-=== Miscellany ===
-edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.
 == Intervals ==
 {| class="wikitable center-all"
-|+ Intervals of 12edo
+|+ style="font-size: 105%;" | Intervals of 12edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 60: / Line 64: @@
 | Minor second
 |
-| [[25/24]] (+29.328)<br>[[16/15]] (-11.731)
+| [[25/24]] (+29.328)<br>[[16/15]] (−11.731)
-| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (-19.443)
+| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (−19.443)
-| [[18/17]] (+1.045)<br>[[17/16]] (-4.955)
+| [[18/17]] (+1.045)<br>[[17/16]] (−4.955)
 | [[File:piano_1_12edo.mp3]]
 |-
@@ Line 68: / Line 72: @@
 | 200
 | Major second
-| [[9/8]] (-3.910)
+| [[9/8]] (−3.910)
 | [[10/9]] (+17.596)
-| [[28/25]] (+3.802)<br>[[8/7]] (-31.174)
+| [[28/25]] (+3.802)<br>[[8/7]] (−31.174)
-| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (-0.532)<br>[[17/15]] (-16.687)
+| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (−0.532)<br>[[17/15]] (−16.687)
 | [[File:piano_1_6edo.mp3]]
 |-
@@ Line 78: / Line 82: @@
 | Minor third
 | [[32/27]] (+5.865)
-| [[6/5]] (-15.641)
+| [[6/5]] (−15.641)
-| [[7/6]] (+33.129)<br>[[25/21]] (-1.847)
+| [[7/6]] (+33.129)<br>[[25/21]] (−1.847)
 | [[19/16]] (+2.487)<br>[[44/37]] (+0.026)
 | [[File:piano_1_4edo.mp3]]
@@ Line 86: / Line 90: @@
 | 400
 | Major third
-| [[81/64]] (-7.820)
+| [[81/64]] (−7.820)
 | [[5/4]] (+13.686)
-| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)
+| [[63/50]] (−0.108)<br>[[9/7]] (−35.084)
-| [[34/27]] (+0.910)<br>[[24/19]] (-4.442)
+| [[34/27]] (+0.910)<br>[[24/19]] (−4.442)
 | [[File:piano_1_3edo.mp3]]
 |-
@@ Line 105: / Line 109: @@
 | [[Tritone]]
 |
-|
+| [[45/32]] (+9.776)<br>[[64/45]] (−9.776)
-| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
+| [[7/5]] (+17.488)<br>[[10/7]] (−17.488)
-| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)
+| [[24/17]] (+3.000)<br>[[99/70]] (−0.088)<br>[[17/12]] (−3.000)
 | [[File:piano_1_2edo.mp3]]
 |-
@@ Line 113: / Line 117: @@
 | 700
 | Fifth
-| [[3/2]] (-1.955)
+| [[3/2]] (−1.955)
 |
 |
@@ Line 123: / Line 127: @@
 | Minor sixth
 | [[128/81]] (+7.820)
-| [[8/5]] (-13.686)
+| [[8/5]] (−13.686)
 | [[14/9]] (+35.084)<br>[[100/63]] (+0.108)
-| [[19/12]] (+4.442)<br>[[27/17]] (-0.910)
+| [[19/12]] (+4.442)<br>[[27/17]] (−0.910)
 | [[File:piano_2_3edo.mp3]]
 |-
@@ Line 131: / Line 135: @@
 | 900
 | Major sixth
-| [[27/16]] (-5.865)
+| [[27/16]] (−5.865)
 | [[5/3]] (+15.641)
-| [[42/25]] (+1.847)<br>[[12/7]] (-33.129)
+| [[42/25]] (+1.847)<br>[[12/7]] (−33.129)
-| [[37/22]] (-0.026)<br>[[32/19]] (-2.487)
+| [[37/22]] (−0.026)<br>[[32/19]] (−2.487)
 | [[File:piano_3_4edo.mp3]]
 |-
@@ Line 141: / Line 145: @@
 | Minor seventh
 | [[16/9]] (+3.910)
-| [[9/5]] (-17.596)
+| [[9/5]] (−17.596)
-| [[7/4]] (+31.174)<br>[[25/14]] (-3.802)
+| [[7/4]] (+31.174)<br>[[25/14]] (−3.802)
-| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (-0.020)<br>[[34/19]] (-7.442)
+| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (−0.020)<br>[[34/19]] (−7.442)
 | [[File:piano_5_6edo.mp3]]
 |-
@@ Line 150: / Line 154: @@
 | Major seventh
 |
-| [[15/8]] (+11.731)<br>[[48/25]] (-29.328)
+| [[15/8]] (+11.731)<br>[[48/25]] (−29.328)
-| [[28/15]] (+19.443)<br>[[40/21]] (-15.533)<br>[[27/14]] (-37.039)
+| [[28/15]] (+19.443)<br>[[40/21]] (−15.533)<br>[[27/14]] (−37.039)
-| [[32/17]] (+4.955)<br>[[17/9]] (-1.045)
+| [[32/17]] (+4.955)<br>[[17/9]] (−1.045)
 | [[File:piano_11_12edo.mp3]]
 |-
@@ Line 175: / Line 179: @@
 {| class="wikitable center-all"
-|+Notation of 12edo
+|+ style="font-size: 105%;" | Notation of 12edo
-! rowspan="2" |[[Degree]]
-! rowspan="2" |[[Cent]]s
-! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
 |-
-! Diatonic ([[5L 2s]]) interval names
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
+|-
+! Diatonic ([[5L&nbsp;2s]]) interval names
 ! Note names (on D)
 |-
 | 0
 | 0
-|'''Perfect unison (P1)'''
+| '''Perfect unison (P1)'''
-|'''D'''
+| '''D'''
 |-
 | 1
@@ Line 195: / Line 200: @@
 | 2
 | 200
-|'''Major second (M2)'''<br>Diminished third (d3)
+| '''Major second (M2)'''<br>Diminished third (d3)
-|'''E'''<br>Fb
+| '''E'''<br>Fb
 |-
 | 3
@@ Line 210: / Line 215: @@
 | 5
 | 500
-|'''Perfect fourth (P4)'''
+| '''Perfect fourth (P4)'''
-|'''G'''
+| '''G'''
 |-
 | 6
@@ Line 220: / Line 225: @@
 | 7
 | 700
-|'''Perfect fifth (P5)'''
+| '''Perfect fifth (P5)'''
 | A
 |-
@@ Line 230: / Line 235: @@
 | 9
 | 900
-|'''Major sixth (M6)'''<br>Diminished seventh (d7)
+| '''Major sixth (M6)'''<br>Diminished seventh (d7)
-|'''B'''<br>Cb
+| '''B'''<br>Cb
 |-
 | 10
@@ Line 245: / Line 250: @@
 | 12
 | 1200
-|'''Perfect octave (P8)'''
+| '''Perfect octave (P8)'''
-|'''D'''
+| '''D'''
 |}
@@ Line 253: / Line 258: @@
 * Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:12-EDO_Evo_Sagittal.svg
@@ Line 266: / Line 271: @@
 Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:12-EDO_Revo_Sagittal.svg
@@ Line 278: / Line 283: @@
 == Solfege ==
 {| class="wikitable center-all"
-|+ Solfege of 12edo
+|+ style="font-size: 105%;" | Solfege of 12edo
+|-
 ! [[Degree]]
 ! [[Cents]]
@@ Line 355: / Line 361: @@
 === 15-odd-limit interval mappings ===
 {{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}}
 == Regular temperament properties ==
@@ Line 417: / Line 424: @@
 | 2.64
 |}
-* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.
+* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.
 === Uniform maps ===
-{{Uniform map|13|11.5|12.5}}
+{{Uniform map|edo=12}}
 === Commas ===
@@ Line 842: / Line 849: @@
 | [[Hexe]]
 |}
-<nowiki>*</nowiki> [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-== Zeta properties ==
+== 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)}}
-=== Zeta peak index ===
-{| class="wikitable"
-! colspan="3" |Tuning
-! colspan="3" |Strength
-! colspan="2" |Closest EDO
-! colspan="2" |Integer limit
-|-
-!ZPI
-!Steps per octave
-!Step size (cents)
-!Height
-!Integral
-!Gap
-!EDO
-!Octave (cents)
-!Consistent
-!Distinct
-|-
-|[[34zpi]]
-|12.0231830072926
-|99.8071807833375
-|5.193290
-| 1.269599
-|15.899282
-|12edo
-|1197.68616940005
-|10
-|6
-|}
 == Scales ==
 {{Main| List of MOS scales in 12edo }}
@@ Line 910: / Line 931: @@
 * [[Lumatone mapping for 12edo]]
 * [[:purdal:12-EDD]]{{dead link}}
-* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step
+* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
 == Notes ==
@@ Line 918: / Line 939: @@
 * [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
+[[Category:3-limit record edos|##]] <!-- 2-digit number -->
 [[Category:Historical]]
 [[Category:Meantone]]