An interval which is one chromatic semitone higher than the major-2nd of the diatonic scale. For example, in the key of A-minor:
major-2nd | | M2 t A B C D E F G = A-natural-minor diatonic scale chromatic semitone +1 (t-s) / \ G G# A B C D E F G# = A-harmonic-minor diatonic scale t + (t-s) M2 + +1 | | augmented-2nd
The augmented-2nd arose primarily out of "common-practice" harmony as part of the harmonic-minor scale, where the 7th degree ("leading-tone") is raised a half-step so as to approach more closely to the tonic, the note to which it normally resolves.
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