12edo: Difference between revisions

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12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 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+2/3 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.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands 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 supports [[septimal meantone]] (meaning that the [[syntonic comma]] ~~is tempered out~~, and going 10 fifths up, i.e. C-A#, ~~give~~ 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands 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 supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==

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 edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 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+2/3 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.
-The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands 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 supports [[septimal meantone]] (meaning that the [[syntonic comma]] is tempered out, and going 10 fifths up, i.e. C-A#, give 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].
+The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands 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 supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].
 == Intervals ==