Adding a further minor third on top of the chord (if built on C, this results in a chord consisting of C, E, G, and B — the last of which, "B double-flat," may be enharmonically respelled as A) makes a fully diminished seventh chord (so called because C to B is the interval of a diminished seventh). This chord is ambiguous as to root because a diminished seventh chord built from any note of it produces that same chord. This, combined with the fact that any of its notes may be enharmonically changed, makes it a useful pivot chord for modulation. Replacing the diminished seventh with a minor seventh above the root (ex.: C/E/G/B) creates a half-diminished seventh chord (minor third, diminished fifth, minor seventh).
The diminished chord on the leading tone can thus function as a dominant seventh and resolve to the tonic chord. The diminished fifth is part of the strong sense of resolution possible in the progression from the dominant seventh to the tonic.
The diminished seventh chord comprises frequencies that are equally spaced when considered on a logarithmic axis, and thus divides the octave into four logarithmically equal portions.
The fundamental tone or root of any diminished seventh chord, being composed of three stacked minor thirds, is ambiguous. For example, Cdim7 in root position: C + E + G + B (each has one and half interval), is just as easily viewed as an Edim7 in its first inversion:
It can also be viewed as a Gdim7 in its second inversion:
Delineating this chord in its last possibility, that of Bdim7 in its third inversion, is very clumsy and not very useful as it requires the use a triple-flatted note, something that is never used in a musical score:
The point that should not be lost in this morass of musical notation is that by any of its four possible names, this chord is played and sounds exactly the same. What makes the chord so unusual is the number of ways it can be harmonically resolved and only after it is resolved does the chord reveal its true identity.