A diminished sixth is an interval of seven semitones.
The interval between C and G, for example, is a diminished sixth as it contains seven 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 either eight semitones or nine semitones. These more common intervals are called a minor sixth and a major sixth respectively. The diminished sixth is "diminished" as it is one semitone smaller than the smaller of the two more common intervals (the minor sixth). The diminished 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, a diminished sixth is the same as a perfect fifth. Both intervals contain seven semitones. A diminished 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 perfect fifth 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).
The diminished sixth is not common. An example of the fourth mode of the enigmatic scale is F#, G#, A#, B, C, Db, E, and the interval between F# and Db is equal to seven semitones.
See also:
Intervals on the chromatic scale, Intervals on the chromatic scale (index)
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