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 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, 15.6 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.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-[[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. 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 [[support]]s septimal meantone 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 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 [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|312/219]], the Didymus' comma, [[81/80]], the lesser 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.

The commas it tempers out include:

- the Pythagorean comma, [[Pythagorean comma|312/219]] (splitting the octave into twelve parts, 7 of which represent a perfect fifth 3/2),

- the Didymus' comma, [[81/80]] (resulting in the diatonic major third being equated to the just major third 5/4),

- the lesser diesis, [[128/125]] (resulting in 5/4 stacking three times to reach the octave),

- the diaschisma, [[2048/2025]] (resulting in a semitone 16/15 being exactly half of a whole tone 9/8, and in two 5-limit tritones, 45/32 and 64/45, being equated to the "true" tritone of [[sqrt(2/1)]] or 600c),

- the Archytas' comma, [[64/63]] (equating the dominant seventh chord and harmonic seventh chord),

- the septimal quartertone, [[36/35]] (resulting in the lack of a distinction in the 7-limit beyond "major" and "minor" intervals),

- the jubilisma, [[50/49]] (which equates [[10/7]] and [[7/5]], two simple 7-limit tritones, to the semioctave),

- the septimal semicomma, [[126/125]] (the difference between the previous two intervals),

- and the septimal kleisma, [[225/224]] (the difference between simple 7-limit intervals and 5-limit augmented or diminished intervals).

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 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 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, 15.6 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.
+edo is the basic example of an equidodecatonic scale, or more simply, a 12-[[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. 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 [[support]]s septimal meantone 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 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 [[zeta integral edo]].
-The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], the Didymus' comma, [[81/80]], the lesser 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.
+The commas it tempers out include:
+- the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]] (splitting the octave into twelve parts, 7 of which represent a perfect fifth 3/2),
+- the Didymus' comma, [[81/80]] (resulting in the diatonic major third being equated to the just major third 5/4),
+- the lesser diesis, [[128/125]] (resulting in 5/4 stacking three times to reach the octave),
+- the diaschisma, [[2048/2025]] (resulting in a semitone 16/15 being exactly half of a whole tone 9/8, and in two 5-limit tritones, 45/32 and 64/45, being equated to the "true" tritone of [[sqrt(2/1)]] or 600c),
+- the Archytas' comma, [[64/63]] (equating the dominant seventh chord and harmonic seventh chord),
+- the septimal quartertone, [[36/35]] (resulting in the lack of a distinction in the 7-limit beyond "major" and "minor" intervals),
+- the jubilisma, [[50/49]] (which equates [[10/7]] and [[7/5]], two simple 7-limit tritones, to the semioctave),
+- the septimal semicomma, [[126/125]] (the difference between the previous two intervals),
+- and the septimal kleisma, [[225/224]] (the difference between simple 7-limit intervals and 5-limit augmented or diminished intervals).
+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 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.