A diminished seventh is an interval of nine semitones.
The interval between C and A, for example, is a diminished seventh as it contains nine 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 seventh note on the scale is either ten semitones or eleven semitones. These more common intervals are called a minor seventh and a major seventh respectively. The diminished seventh is "diminished" as it is one semitone smaller than the smaller of the two more common intervals (the minor seventh). The diminished seventh is a "seventh" if it occurs between the first and the seventh note on a scale (or more generally between two notes with five notes in between).
In principle, a diminished seventh is the same as a major sixth. Both intervals contain nine semitones. A diminished seventh through 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) whereas a major 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).
The diminished seventh occurs, for example, in the seventh mode of the harmonic minor scale. An example of the seventh mode of the scale is G#, A, B, C, D, E, F and the interval between G# and F is equal to nine semitones.
See also:
Intervals on the chromatic scale, Intervals on the chromatic scale (index)
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