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 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, while reasonable for its size, is unsatisfactory for some, 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. 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 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 [[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. 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. 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 maps [[3/2]] to 700c, meaning that it tempers out the Pythagorean comma [[531441/524288]] (equating Pythagorean enharmonic equivalents), and 5/4 to 400c, meaning that it tempers out the augmented comma [[128/125]] (thus equating classical enharmonic equivalents).

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth.

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~~]]. This interval is the same as the diatonic [[minor seventh]], and as such 12edo tempers out [[64/63~~]]. 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 via its ~~patent~~ [[val]] of {{val| 12 19 28 34}} ~~(tempering out [[225/224]] and [[126/125]])~~, 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]].

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]].

Other [[comma]]s ~~12edo~~ [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each ~~comma tempered out by 12edo~~ affects ~~its~~ structure in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways. For example, other diaschismic edos like 34edo separate 16/15 and 3/2 by the semioctave, and other augmented edos like 15edo conflate classical enharmonic equivalents (for example, the augmented third to the perfect fourth).

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.

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.

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 == 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, while reasonable for its size, is unsatisfactory for some, 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. 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 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 [[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. 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. 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 maps [[3/2]] to 700c, meaning that it tempers out the Pythagorean comma [[531441/524288]] (equating Pythagorean enharmonic equivalents), and 5/4 to 400c, meaning that it tempers out the augmented comma [[128/125]] (thus equating classical enharmonic equivalents).
 edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth.
-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]]. This interval is the same as the diatonic [[minor seventh]], and as such 12edo tempers out [[64/63]]. 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 via its patent [[val]] of {{val| 12 19 28 34}} (tempering out [[225/224]] and [[126/125]]), 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]].
+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]].
-Other [[comma]]s 12edo [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each comma tempered out by 12edo affects its structure in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways. For example, other diaschismic edos like 34edo separate 16/15 and 3/2 by the semioctave, and other augmented edos like 15edo conflate classical enharmonic equivalents (for example, the augmented third to the perfect fourth).
+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.