Augmented triad: Difference between revisions

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Rework to further explain what's going on in contemporary 12edo music
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reduce 12edo-centrism in introduction; group temperaments into one section and explain relevance to common practice; reduce reliance on octave closure (which would make this chord strictly a tetrad, not a triad); include more JI interpretations of essentially-tempered chords
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{{Wikipedia|Augmented triad}}
{{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 – 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 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]]:
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]]:
In the [[3-limit]]:

Revision as of 03:07, 22 August 2024

English Wikipedia has an article on:

The augmented triad is a triad comprising a root, a major third, and an augmented fifth formed by stacking another major third.

In temperaments

In augmented temperament, the ~25/16 augmented fifth is enharmonically equivalent to a minor sixth (~8/5), the difference of 128/125 being 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 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) 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 ♭II+ (16⁄15), ♭III+ (6⁄5) ♭VI+ (8⁄5), and ♭VII+ (9⁄5) in the duodene.
  • 20:25:32 is found on I+ (1⁄1), II+ (9⁄8), IV+ (4⁄3), and V+ (1⁄3/2) 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+ (5⁄4), ♯IV+ (45⁄32), VI+ (5⁄3), VII+ (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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