Augmented triad: Difference between revisions

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~~The~~ '''augmented triad''' is a [[triad]] ~~traditionally described as~~ comprising a root, [[major]] third, and [[interval quality|augmented]] fifth~~. In contemporary [[12edo]] music, the augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth, approximating [[8/5]], with the~~ major third approximating [[5/4]], so it is a stack of two [[~]]5/4's making up ~8/5, [[tempering out]] [[128/125]]. In other words, three identical ~5/4's close at the [[octave]], which implies it is a [[5-odd-limit]] [[essentially tempered chord]] of the [[augmented (temperament)|augmented]] temperament, the temperament named after this chord.

An '''augmented triad''' is a [[triad]] comprising a root, a [[major]] third, and an [[interval quality|augmented]] fifth formed by stacking another major third.

* (Augmented) 1 – 5/4 – 8/5, with steps 5/4, 5/4, 5/4.

== In ~~meantone~~ ==

== In temperaments ==

~~Before~~ the ~~ubiquity of~~ [[~~12edo~~]]~~, an~~ augmented ~~triad did not imply an equal tuning of all three thirds~~ (~~technically~~, ~~two major thirds~~ and a ~~diminished fourth~~). ~~It can be viewed~~ as a [[9-odd-limit]] essentially tempered chord of [[~~marvel~~]]. ~~See~~ [[~~marvel chords~~]]. ~~This~~ is ~~still compatible with 12edo~~ and is ~~perhaps as authentic as the 5~~-~~odd~~-~~limit interpretation above~~.

In [[augmented (temperament)|augmented temperament]], which [[tempering out|tempers out]] [[128/125]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth (~[[8/5]]). Since the resulting internal intervals are ~5/4, ~5/4, and ~8/5, the augmented triad in augmented temperament is a [[5-odd-limit]] [[essentially tempered chord]]:

* (Augmented) 1 – 5/4 – 8/5

The gap remaining to close the octave is another identical third (~5/4), giving rise to the temperament's name after this chord.

If [[225/224]] is tempered out, as in [[marvel]] and [[septimal meantone]], the ~25/16 augmented fifth is tempered together with ~[[14/9]]. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).

* (Marvel) 1 – 5/4 – 14/9

In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.

Since [[12edo]] is a good tuning of augmented temperament and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.

== In just intonation ==

In the [[7-limit]]:

* [[36:45:56]] (1/1 – 5/4 – 14/9) is the simplest JI tuning of the 7-limit marvel augmented chord.

* [[35:45:56]] (1/1 –  9/7 – 8/5)

* [[28:35:45]] (1/1 – 5/4 – [[45/28]])

* [[28:36:45]] (1/1 – 9/7 – [[45/28]])

In the [[5-limit]]:

* [[16:20:25]] is found on ~~the~~ ♭II+ ({{Frac|16|15}}), ♭III+ ({{Frac|6|5}}) ♭VI+ ({{Frac|8|5}}), and ♭VII+ ({{Frac|9|5}}) of the [[duodene]].

* [[16:20:25]] is found on ♭II+ ({{Frac|16|15}}), ♭III+ ({{Frac|6|5}}) ♭VI+ ({{Frac|8|5}}), and ♭VII+ ({{Frac|9|5}}) in the [[duodene]]. Its [[rotation|rotations]], [[20:25:32]] and [[25:32:40]], represent chords that would be enharmonically equivalent to itself in augmented temperament, substituting a minor sixth and a diminished fourth successively in place of the augmented fifth and major third.

In the [[3-limit]]:

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[[Category:Augmented]]

[[Category:5-odd-limit chords]]

[[Category:Marvel]]

[[Category:9-odd-limit chords]]

[[Category:Essentially tempered chords]]

[[Category:Just intonation chords]]

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 {{Wikipedia|Augmented triad}}
-The '''augmented triad''' is a [[triad]] traditionally described as comprising a root, [[major]] third, and [[interval quality|augmented]] fifth. In contemporary [[12edo]] music, the augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth, approximating [[8/5]], with the major third approximating [[5/4]], so it is a stack of two [[~]]5/4's making up ~8/5, [[tempering out]] [[128/125]]. In other words, three identical ~5/4's close at the [[octave]], which implies it is a [[5-odd-limit]] [[essentially tempered chord]] of the [[augmented (temperament)|augmented]] temperament, the temperament named after this chord.
+An '''augmented triad''' is a [[triad]] comprising a root, a [[major]] third, and an [[interval quality|augmented]] fifth formed by stacking another major third.
-* (Augmented) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;8/5, with steps 5/4, 5/4, 5/4.
-== In meantone ==
+== In temperaments ==
-Before the ubiquity of [[12edo]], an augmented triad did not imply an equal tuning of all three thirds (technically, two major thirds and a diminished fourth). It can be viewed as a [[9-odd-limit]] essentially tempered chord of [[marvel]]. See [[marvel chords]]. This is still compatible with 12edo and is perhaps as authentic as the 5-odd-limit interpretation above.
+In [[augmented (temperament)|augmented temperament]], which [[tempering out|tempers out]] [[128/125]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth (~[[8/5]]). Since the resulting internal intervals are ~5/4, ~5/4, and ~8/5, the augmented triad in augmented temperament is a [[5-odd-limit]] [[essentially tempered chord]]:
+* (Augmented) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;8/5
+The gap remaining to close the octave is another identical third (~5/4), giving rise to the temperament's name after this chord.
+If [[225/224]] is tempered out, as in [[marvel]] and [[septimal meantone]], the ~25/16 augmented fifth is tempered together with ~[[14/9]]. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).
+* (Marvel) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;14/9
+In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.
+Since [[12edo]] is a good tuning of augmented temperament and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.
 == In just intonation ==
+In the [[7-limit]]:
+* [[36:45:56]] (1/1 – 5/4&thinsp;– 14/9) is the simplest JI tuning of the 7-limit marvel augmented chord.
+* [[35:45:56]] (1/1&thinsp;–&thinsp; 9/7&thinsp;– 8/5)
+* [[28:35:45]] (1/1 – 5/4 – [[45/28]])
+* [[28:36:45]] (1/1 – 9/7 –  [[45/28]])
 In the [[5-limit]]:
-* [[16:20:25]] is found on the ♭II+ ({{Frac|16|15}}), ♭III+ ({{Frac|6|5}}) ♭VI+ ({{Frac|8|5}}), and ♭VII+ ({{Frac|9|5}}) of the [[duodene]].
+* [[16:20:25]] is found on ♭II+ ({{Frac|16|15}}), ♭III+ ({{Frac|6|5}}) ♭VI+ ({{Frac|8|5}}), and ♭VII+ ({{Frac|9|5}}) in the [[duodene]]. Its [[rotation|rotations]], [[20:25:32]] and [[25:32:40]], represent chords that would be enharmonically equivalent to itself in augmented temperament, substituting a minor sixth and a diminished fourth successively in place of the augmented fifth and major third.
 In the [[3-limit]]:
@@ Line 18: / Line 36: @@
 [[Category:Augmented]]
 [[Category:5-odd-limit chords]]
+[[Category:Marvel]]
+[[Category:9-odd-limit chords]]
 [[Category:Essentially tempered chords]]
 [[Category:Just intonation chords]]