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.

The '''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]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth (~[[8/5]]), the difference of [[128/125]] being [[tempering out|tempered out]]. 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 fifth (~5/4), giving rise to the temperament's name after this chord.

In [[marvel|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[224/225]). 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 marvel, and the historically prevalent [[quarter-comma meantone]] is a good tuning of marvel, 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]] is the simplest JI tuning of the 7-limit marvel augmented chord.

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]].

* [[20:25:32]] is found on I+ ({{Frac|1|1}}), II+ ({{Frac|9|8}}), IV+ ({{Frac|4|3}}), and V+ ({{Frac|3/2}}) in the duodene. It substitutes the 8/5 minor sixth found in augmented temperament in place of the 25/16 augmented fifth.

* [[25:32:40]] is found on III+ ({{Frac|5|4}}), ♯IV+ ({{Frac|45|32}}), VI+ ({{Frac|5|3}}), VII+ ({{Frac|15|8}}) in the duodene. It substitutes an [[8/5]] ''minor sixth'' and [[32/25]] ''diminished fourth'' in place of the augmented fifth and lower major third respectively.

In the [[3-limit]]:

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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.
+The '''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]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth (~[[8/5]]), the difference of [[128/125]] being [[tempering out|tempered out]]. 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 fifth (~5/4), giving rise to the temperament's name after this chord.
+In [[marvel|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[224/225]). 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 marvel, and the historically prevalent [[quarter-comma meantone]] is a good tuning of marvel, 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]] is the simplest JI tuning of the 7-limit marvel augmented chord.
 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]].
+* [[20:25:32]] is found on I+ ({{Frac|1|1}}), II+ ({{Frac|9|8}}), IV+ ({{Frac|4|3}}), and V+ ({{Frac|3/2}}) in the duodene. It substitutes the 8/5 minor sixth found in augmented temperament in place of the 25/16 augmented fifth.
+* [[25:32:40]] is found on III+ ({{Frac|5|4}}), ♯IV+ ({{Frac|45|32}}), VI+ ({{Frac|5|3}}), VII+ ({{Frac|15|8}}) in the duodene. It substitutes an [[8/5]] ''minor sixth'' and [[32/25]] ''diminished fourth'' in place of the augmented fifth and lower major third respectively.
 In the [[3-limit]]: