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 [[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. For example~~, ~~Picardy thirds became less common, likely~~ due to ~~the stability~~ of ~~[[16:19:24]], 12edo's minor triad~~. However, it should be borne in mind that in actual performance these ~~defects~~ 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 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 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~~.

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

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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 == 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 [[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. For example, Picardy thirds became less common, likely due to the stability of [[16:19:24]], 12edo's minor triad. However, it should be borne in mind that in actual performance these defects 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 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 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.
+edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]].
 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]].