Augmented triad: Difference between revisions

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The definition of P1-M3-A5 is a meantone-centrist definition. This is a chord of the augmented temp. Quick rework to reveal the implications of this chord.
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{{Wikipedia|Augmented triad}}
{{Wikipedia|Augmented triad}}


The '''augmented triad''' comprises three identical major thirds closing at the [[octave]]. 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 was simply treated as a [[triad]] comprising a root, [[major]] third, and [[interval quality|augmented]] fifth, and did not imply an equal tuning of all three intervals. 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 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]]:
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]]:
In the [[3-limit]]:
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[[Category:Augmented]]
[[Category:Augmented]]
[[Category:5-odd-limit chords]]
[[Category:5-odd-limit chords]]
[[Category:Marvel]]
[[Category:9-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Just intonation chords]]
[[Category:Just intonation chords]]

Latest revision as of 11:04, 19 April 2025

English Wikipedia has an article on:

An 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, which tempers out 128/125, the ~25/16 augmented fifth is 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 ♭II+ (16⁄15), ♭III+ (6⁄5) ♭VI+ (8⁄5), and ♭VII+ (9⁄5) in the duodene. Its 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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