Diminished Seventh Chord - Inversions

Inversions

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 E♭dim7 in its third inversion:

D (enharmonic equivalent of C) + E♭ + G♭ + B.

It can also be viewed as a G♭dim7 in its second inversion:

D + F (enharmonic equivalent of E♭) + G♭ + B.

Delineating this chord in its last possibility, that of Bdim7 in its first 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:

D + F + A♭♭♭ (enharmonic equivalent of G♭) + B.

However, by enharmonically respelling the B to A, this can also be viewed as a first inversion Adim7 chord:

C + E♭ + G♭ + A (enharmonic equivalent of B).

Other possibilities present themselves by respelling the various roots; for instance:

C + E♭ + F♯ (enharmonic equivalent of G♭) + A (enharmonic equivalent of B) (second inversion F♯dim7).

C + D♯ (enharmonic equivalent of E♭) + F♯ (enharmonic equivalent of G♭) + A (enharmonic equivalent of B) (third inversion D♯dim7).

B♯ (enharmonic equivalent of C) + D♯ (enharmonic equivalent of E♭) + F♯ (enharmonic equivalent of G♭) + A (enharmonic equivalent of B) (root position B♯dim7).

All of the chord's inversions have the same sound harmonically. Because of the chord's symmetrical nature (superimposing more minor thirds on top of the dim 7 produces no new notes), there are only three different diminished seventh chords possible.

The diminished seventh chord can appear in first, second, or (least common) third inversion. Each inversion is enharmonic with another diminished seventh chord, and 19th-century composers in particular often make use of this enharmonic to use these chords for modulations. Percy Goetschius calls it the "enharmonic chord."

Using Piston's incomplete-ninth analysis, a single diminished seventh chord, without enharmonic change, is capable of the following analyses: V, V of ii, V of III (in min.), V of iii (in maj.), V of iv, V of V, V of VI (in min.), V of vi (in maj.), V of VII (in min.). Since the chord may be enharmonically written in four different ways without changing the sound, we may multiply the above by four, making a total of forty-eight possible interpretations. More conservatively, each assumed root may be used as a dominant, tonic, or supertonic, giving twelve possibilities.