In an equal tempered tuning, a diminished seventh is equal to nine semitones, a ratio of 1:29/12 (approximately 1.682), or 900 cents. There is no standard just tuning of this interval, but one possibility, assuming the flat submediant is a perfect (5:4) major third below the octave, and the leading tone to be 15:16, would lead to an interval of 128:75, about 925 cents.
The most common form of the diminished seventh chord is one which includes the leading tone, as well as the second, fourth, and flatted sixth (flat submediant) scale degrees. These notes occur naturally in the harmonic minor scale, but this chord also appears in major keys, especially after the time of Bach, where it is "borrowed" from the parallel minor.
Seventh chords may also be rooted on other scale degrees, either as secondary function chords temporarily borrowed from other keys, or as appoggiatura chords: a chord rooted on the raised second scale degree (D-F-A-C in the key of C) acts as an appoggiatura to the tonic (C major) chord, and one rooted on the raised sixth scale degree (A-C-E-G in C major) acts as an appoggiatura to the dominant (G major) chord. These chords may be referred to as "secondary diminished seventh chords" or as a subclass of secondary dominants.
In jazz, the diminished seventh chord is often based on the lowered third scale degree (the flat mediant) and acts as a passing chord between the mediant triad (or first-inversion tonic triad) and the supertonic triad: in C major, this would be the chord progression E minor - E diminished - D minor.
The other method is to analyze the chord as an "incomplete dominant ninth", that is a ninth chord with its root on the dominant, whose root is missing or implied, as shown in the diagram above right. Walter Piston has long been the champion of this analysis.
The dominant ninth theory has been questioned by Heinrich Schenker. He explained that although there is a kinship between all univalent chords rising out of the fifth degree, the dominant ninth chord is not a real chord formation.
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 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 maj.). 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.