An augmented fifth is an interval of eight semitones.
The interval between C and G#, for example, is an augmented fifth as it contains eight 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 fifth note on the scale is equal to seven semitones. This more common interval is called a perfect fifth. The augmented fifth is "augmented" as it is one semitone larger than this more common interval. The augmented fifth is a "fifth" if it occurs between the first and the fifth note on a scale (or more generally between two notes with three notes in between).
In principle, an augmented fifth is the same as a minor sixth. Both intervals contain eight semitones. An augmented fifth through would refer to the interval between the root on a scale and the fifth note on the scale (or more generally between two notes with three notes in between) whereas a minor sixth 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).
An example of a scale where the augmented fifth occurs is the Lydian with a raised fifth, which is a mode of the secondary heptatonic scale. An example of this scale is Eb, F, G, A, B, C, D and the interval between Eb and B is equal to eight semitones.
See also:
Intervals on the chromatic scale, Intervals on the chromatic scale (index)
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