Augmented triad: Difference between revisions
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{{Wikipedia|Augmented triad}} | {{Wikipedia|Augmented triad}} | ||
An '''augmented triad''' is a [[triad]] comprising a root, a [[major]] third, and an [[interval quality|augmented]] fifth formed by stacking another major third. | |||
== In temperaments == | == In temperaments == | ||
In [[augmented (temperament)|augmented temperament]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence|enharmonically equivalent]] to a minor sixth (~[[8/5]]) | 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 | * (Augmented) 1 – 5/4 – 8/5 | ||
The gap remaining to close the octave is another identical | 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''. | 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 | 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]]: | In the [[7-limit]]: | ||
* [[36:45:56]] is the simplest JI tuning of the 7-limit marvel augmented chord. | * [[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 ♭II+ ({{Frac|16|15}}), ♭III+ ({{Frac|6|5}}) ♭VI+ ({{Frac|8|5}}), and ♭VII+ ({{Frac|9|5}}) in 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]]: | ||
| Line 35: | Line 36: | ||
[[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
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:
- 4096:5184:6561 may be considered a 3-limit approximation of 16:20:25.