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 [[cent]]s 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.

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 [[cent]]s 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 "represented" 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 via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. 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 [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

The seventh partial ([[7/4]]) is "represented" 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 via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. 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 [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3<sup>12</sup>/2<sup>19</sup>, the Didymus comma, [[81/80]], the 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.

~~12EDO~~ is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==

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=== Commas ===

~~12EDO~~ [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"

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The two most common ~~12EDO~~ MOS scales are meantone[5] and meantone[7].

The two most common 12edo MOS scales are meantone[5] and meantone[7].

* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)

* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)

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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 [[cent]]s 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.
+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 [[cent]]s 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 "represented" 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 via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. 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 [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].
+The seventh partial ([[7/4]]) is "represented" 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 via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. 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 [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].
 In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3<sup>12</sup>/2<sup>19</sup>, the Didymus comma, [[81/80]], the 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.
-EDO is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].
+edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.
 == Intervals ==
@@ Line 307: / Line 307: @@
 === Commas ===
-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.
+edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.
 {| class="commatable wikitable center-all left-3 right-4 left-6"
@@ Line 556: / Line 556: @@
 {{main|List of MOS scales in 12edo}}
-The two most common 12EDO MOS scales are meantone[5] and meantone[7].
+The two most common 12edo MOS scales are meantone[5] and meantone[7].
 * Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
 * Pentatonic (meantone) 2L3s 22323 (generator = 7\12)