Augmented 1st: C up to C♯ Augmented 2nd: C up to D♯ Augmented 4th: C up to F♯ Augmented 5th: C up to G♯, with harmonic implication of major 3rd, e.g. augmented 5th chord on C = C–E–G♯ Augmented 6th: C up to A♯.
Chords of augmented 6th are chromatic. The 3 most common are as follows (e.g. in key of C):
(a) Chord of the Italian 6th: A♭–C–F♯(b) Chord of French 6th: A♭–C–D–F♯(c) Chord of the German 6th: A♭–C–E♭–F♯
The Ger. 6th is the commonest and serves as a convenient pivot for modulation, since it may be approached as based on the flattened submediant in one key, and quitted as based on the flattened supertonic in another (or vice versa); also, by enharmonic change (see Interval), it can be transformed into the chord of the dominant 7th of another key, and so quitted (e.g. the Ger. 6th in key C (A♭–C–E♭–F♯) can be treated as the chord of the dominant 7th in key D♭ (A♭–C–E♭–G♭)).
There are other possibilities.
Augmented 8th: C up to next C♯ but one.
"augmented intervals." The Concise Oxford Dictionary of Music. . Encyclopedia.com. (December 10, 2018). https://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/augmented-intervals
"augmented intervals." The Concise Oxford Dictionary of Music. . Retrieved December 10, 2018 from Encyclopedia.com: https://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/augmented-intervals
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