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A type of diatonic chord that contains an interval of an augmented 6th. Augmented 6th chords are often used in classical and romantic music, usually in minor keys. These chords come in a few different varieties: the Italian 6th, French 6th, and German 6th (or It6, Fr6, and Gr6 respectively). The geographic names don't appear to have any actual relation to where the chords are used, or where they came from.

The root of an augmented 6th chord is generally a major 3rd below the tonic in a key, and is made of a stacked major 3rd and an augmented 6th.

For example: Ab-C -F# or F-A-D#

Depending on the variety of the chord, other notes might be used. The Italian 6th uses only the basic 3 notes, while the French 6th contains an augmented 4th and the German 6th contains a perfect 5th.

Italian 6th: Ab-C-F# or F-A-D#
French 6th: Ab-C-D-F# or F-A-B-D#
German 6th: Ab-C-Eb-F# or F-A-C-D#

Notice that the German 6th is enharmonically equivalent to a dominant 7th (is made of the same notes but is spelled differently). This is because the function of the two chords is completely different. The dominant chord wants to resolve to the tonic, but the augmented 6th chord wants to resolve to the dominant or to the tonic in second inversion. This duality between the German 6th and the dominant 7th will be important later.

A typical harmonic progression involving an augmented 6th:
i - VI - It6 - V - i

In c minor, this would be:
c - Ab - It6 (Ab-C-F#) - G - c

Notice that the root of the Italian 6th chord is a major 3rd below the tonic (in this case, the Ab below C).

Now for that German 6th. A common practice in the romantic and post-romantic periods was to use the German 6th as a pivot chord in a modulation. A composer could insert a German 6th chord or a dominant chord, then alter the function of the chord. The German 6th could function as the enharmonically equivalent dominant 7th and lead us into a new key and vice-versa. So, instead of going where we, the listeners, might expect, the composer would surprise us and move us into a completely new tonal arena.

Using the above example:
c - Ab - Gr6 - Db (surprise!)

The Gr6 made us want to hear a G major chord, but instead we were given Db! The Gr6 functions as a dominant chord in the key of Db.

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