An augmented sixth is an interval of ten semitones.
The interval between C and A#, for example, is an augmented sixth as it contains ten semitones.
In a lot of common scales, such as the standard major scale and the natural minor scale, the interval between the root on the scale and the sixth note on the scale is equal to either eight or nine semitones. These more common intervals are called a minor sixth and a major sixth respectively. The augmented sixth is "augmented" as it is one semitone larger than the larger of the two more common intervals (the major sixth). The augmented sixth is a "sixth" if it occurs between the first and the sixth note on a scale (or more generally between two notes with four notes in between).
In principle, an augmented sixth is the same as a minor seventh. Both intervals contain ten semitones. An augmented sixth through would refer to the interval between the root on a scale and the sixth note on the scale (or more generally between two notes with four notes in between) whereas a minor seventh would refer to the interval between the root on a scale and the seventh note on the scale (or more generally between two notes with five notes in between).
An example of a scale where the augmented sixth occurs is one of the modes of the minor gypsy scale. An example of the sixth mode of the scale is F, G, A, B, C, D#, E and the interval between F and D# is equal to ten semitones.
See also:
Intervals on the chromatic scale, Intervals on the chromatic scale (index)
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