The diminished-7th chord is formally a chord containing 4 notes, built upward in pitch from the root with the intervals of three successive minor-3rds, resulting in the chord-members identified as root, 3rd, 5th, and 7th. The actual intervals above the root are thus the minor-3rd, diminished-5th, and diminished-7th.
The diminished-7th chord does not arise naturally from the diatonic major scale, however, it is present as a tetrad whose root is the seventh degree (VII) of the harmonic-minor scale. A common abbreviation for the diminished chord quality is a small circle after the letter of the root-note: VIIo7. For example:
key of A-minor (harmonic-minor): F G# A B C D E F G# A VI VII I II III IV V VI VII I | | | | root 3rd 5th 7th \_____/ \___/ \___/ min-3rd min-3rd min-3rd
Because the V7 chord is so important in the harmonic-minor scale, it is easy to view the diminished-7th chord on VII as an extension of the V7 E:G#:B:D to include the 9th, which is specifically a minor-9th -- in the above example, (E):F -- but with the root omitted. In fact, a listener who hears this diminished-7th chord as the proportion 10:12:14:17 is likely to imagine the presence of the difference-tones 2, 3, 4, and 5 where 2 and 4 both provide the "missing root".
Interestingly, because of the theoretical similarity of the intervals (all minor-3rds), and especially because of the actual similarity of these same intervals as 300 cents each in 12-edo tuning, a composer may manipulate this chord in such a way that the enharmonic-equivalences are exploited, and allow the listener to imagine 3 other "missing roots". The above example may be reinterpreted thus:
root 3rd 5th 7th 9th (E) G# B D F in A-minor (G) B D F Ab in C-minor (Bb) D F Ab Cb in Eb-minor \ enharmonically equivalent (A#) Cx E# G# B in D#-minor / (C#) E# G# B D in F#-minor
This flexible reinterpretation allows the diminished-7th chord to act as a pivot between keys which are not closely related. Note that in 12-edo, there are only three diminished-7th chords available, and that the "missing roots" of this example chord form one of the other two diminished-7th chords, and that the tonics of the minor keys form the other of the two.
The growing acceptance of 12-edo during the 1800s led to a wider use of diminished-7th chords and the resultant modulations to remote key relationships. Eventually, the greater use of these diminished-7th chord interlockings led to experimentation with 12-edo as a member of the diminished family of temperaments which featured a new type of scale: the octatonic diminished scale (which see).
The first composer to make heavy use of the diminished-7th chord was Beethoven, and the first to base whole sections of a piece on it was Chopin (see, for example, the middle section of his Etude in E-major, op.10,no.3, which keeps alternating all three of the diminished-7th chords).