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, ~~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. 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 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.

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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, 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. 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 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.