The augmented triad is:

much rarer than the other three triad types (major, minor, and diminished).

interesting for several reasons including

that rarity itself, and

the symmetrical construction, which creates potential flexibility and ambiguity (just like with the diminished seventh chord)

most often seen in one of two forms:

as III+ in harmonic minor (this being the only form in the major/minor system without chromatic alteration), and

as a chromatic passing chord between V and I in a major key: V, V+, I.

Aren’t we forgetting something here? We’re now well into chromatic harmony, yet we’ve hardly mentioned one of the four types of ostensibly diatonic triads: we’re up to speed with augmented sixth chords, but not the augmented triad. So what is this augmented triad all about? How do composers use it? How have we neglected it so long (and why do so many textbooks brush over it altogether)?

Recall that we have four types of triads that can be constructed with major and minor thirds alone:

Diminished triad (minor third + minor third)

Minor triad (minor third + major third)

Major triad (major third + minor third)

Augmented triad (major third + major third)

So the augmented triad is part of this set of possibilities, but apparently not an equal member, at least not in the eyes of common practice composers. Clearly major and minor triads are mission critical to tonal music, and so is the diminished triad, especially in its dominant function role (as viio and as a part of V7). The augmented triad is a slightly peripheral character in relation to those protagonists.

Always Chromatic?

Part of that rarity has to do with the structure of the major/minor system itself: III+ in harmonic minor is the only time the augmented triad appears in the major/minor system without chromatic alteration. This III+ triad is closely related to both the dominant (V) and minor tonic (i): in both cases, two scale degrees are held common and only one semitone distinguishes the chord tone which changes:

III+ and V: \hat5 and \hat7 in common, semitone between \hat2 and \hat3;

III+ and i: \hat3 and \hat5 in common, semitone between \hat1 and \hat7.

Overall, I'd say that the sound and usage of III+ typically suggests a dominant function. See what you think of the following example from the opening of Schubert's "Der Atlas" (Example 1) in which:

B♭ and D remain constant throughout (\hat3 and \hat5).

G moves to F♯ and back (\hat1 and \hat7).

This arguably gives the impression of a tonic-dominant alternation, but with very slight changes.

Example 1. Extract from the start of Schubert's "Der Atlas" (from Schwanengesang, D.957), showing the augmented triad III+ (possibly, depending on your analytical viewpoint) and certainly how closely related this chord is to both V and i.

Here is the entire song:

Schwanengesang, D.957 by OpenScore Lieder

The small steps between these chords relate to a key part of the "parsimonious voice leading" that’s so important to the "Neo-Riemannian" approach to harmony, which seeks to account for the extended tonal relations that become more common in the late 19th century. For more on that, see Neo-Riemannian Triadic Progressions. For now, let’s continue to look at some examples of the augmented triad in practice.

Rarely focal

The relatively peripheral role and the rarity of augmented triads may be thought to diminish its importance, though as the price of gold, diamond, and other rare commodities attest, that very rarity can be valuable. For Schoenberg, this makes the augmented triad "better protected against banality" than the diminished triad and seventh (1911, trans. ed. 1983, p. 239).

The idea of rarity also needs unpacking: when we speak of "rarity" in harmony, we usually mean that it is unusual to see that chord in a focal role. This speaks to Richard Cohn’s observation that "when an augmented triad appears in music before 1830, its behavior is normally well regulated and unobtrusive, tucked into the middle of a phrase rather than exposed at its boundaries, passed through quickly and lacking metric accent" (2012, p. 43).

This may be true in the general case, though that’s not to say there aren’t glorious counterexamples. Example 2 sets out such an example from the "mente cordis" section that concludes the "Fecit Potentiam" of Bach’s Magnificat (1723, 1733). This is a remarkable moment: not only is there an augmented triad at all, but it is introduced by nearly full forces, spanning the full register, and after a dramatic general pause which itself is preceded by a diminished seventh. You couldn’t hope to find a clearer, more dramatically foregrounded augmented triad in any repertoire.

Example 2. A prominent augmented chord in the "Fecit Potentiam" of Bach’s Magnificat with a possible harmonic reading of this passage that sees the augmented triad setting up a deceptive cadence in the middle of a passage in V minor enclosed by two secondary dominants, viio7/v.

Chromatic passing chord

Although there is only one diatonic form of the augmented triad (III+ in minor), there are clearly many more possibilities when we expand the remit to include chromatic alterations. Here too, however, some are more common than others. A particularly favored use sees the augmented chord as part of a chromatic passing motion from V to I in major, with the whole-tone step from \hat2 to \hat3 "filled in" with a chromatic semitone motion that gives a fleeing . This can appear in several ways:

As a straightforward V–V+–I

With another harmony on the initial \hat2: e.g., ii–V+–I

With or without sevenths: e.g., ii7–V+7–I

Example 3, from Fanny Mendelssohn Hensel's Gondellied (6 Lieder, Op. 1, no. 6), sets this out. Measures 19–20 can be seen as a V6/V–V+–I cadence in A (or, with sevenths, V\mathrm{^6_5}/V–V+7–I)​ where:

the augmented cho