Augmented triad

English Wikipedia has an article on:

Augmented triad

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.