Augmented triad
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:
- 4096:5184:6561 may be considered a 3-limit approximation of 16:20:25.