12edo: Difference between revisions

Line 17:

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 commas it tempers out include the Pythagorean comma, [[Pythagorean comma|312/219]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal 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 Pythagorean comma, [[Pythagorean comma|312/219]] (splitting the octave into twelve parts, 7 of which represent a perfect fifth 3/2),~~

12edo also 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.

~~- 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.

=== Prime harmonics ===

@@ Line 17: / Line 17: @@
 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 commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal 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 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),
+edo also 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.
-- 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.
 === Prime harmonics ===