1. A diminished seventh chord is a stack of minor thirds (causing the interval of an augmented second from the seventh going up to the root).
2. A diminished seventh chord thus divides the octave up into four equal parts -- four minor thirds.
3. A structure such as this, that's the same in both directions from some central pitch axis, is called a symmetrical pitch structure. (The whole tone scale is another such symmetrical structure.)
4. Because all of its intervals are equal and it divides the octave equally, it's tonally ambiguous.
5. Any transposition of it by a minor third results in the same four pitch classes. Thus, for example, a B°7, D°7, F°7, and Ab(G#)°7 chord all contain the same four pitch classes.
6. Any given diminished seventh chord can therefore potentially have four different interpretations, that is, any one of the pitches might be considered the root.
7. The actual root is determined by the chord's context and usage (how it functions in the music).
8. Usually (ideally), but not always, the function is indicated by the way the chord is spelled (e.g., G# vs. Ab).
9. The ambiguity, the four different possible interpretations, make it a very useful pivot chord for modulation.
10. Most commonly its function is as a leading tone chord in minor, or as an altered (borrowed) leading tone chord in major.
12. The leading tone diminished seventh chord is nearly identical to the dominant seventh chord, differing by only one pitch. (For example, in c minor, the dominant seventh chord is GBDF, and the leading tone diminished seventh chord is BDFAb. The only difference is between the G and the Ab.) Note that the entire leading tone triad (e.g. BDF) is contained within the dominant seventh chord.
13. Because of that similarity, Arnold Schoenberg posited that the diminished seventh chord is best interpreted as being, or at least has the same function as, the upper notes of a V7 chord with an added 9th (e.g., GBDFAb).
14. Even if the V root is not present, he contends, the chords functions in the same way, as if it were there by implication. In other words, the V7 and the vii°7 have the same function.
15. In practice the V7 and the vii°7 might be merged, with the 7th of the vii°7 chord functioning as a non-chord tone (appoggiatura or neighbor tone) to the V7 chord, as in the Bach minuet in D minor.
16. In jazz, that flat 9th might not be resolved as a non-chord tone would be, but might simply be treated as a legitimate member of an extended (b9) chord. (Most often it will resolve to the expected note in the next chord, but not always.)
17. The diminished seventh chord's next most common function is as a secondary leading tone (implied secondary dominant) seventh chord to tonicize some note other than the tonic, either as a fleeting tonicization without actually leaving the home key or as an important force in modulating to a new key.
18. Next most common is for the leading tone note to resolve as expected, even though that resolution note is not the root of the resolution chord. Any chord that contains that expected resolution note might possibly work as a resolution chord.
19. Equally commonly, a diminished seventh chord might occur as a result of multiple simultaneous chromatic non-chord tones, especially neighbor tones, as in the song In stiller Nacht by Brahms. (In that song, a the bass note of Bb stays constant as the upper notes of a Bb7 chord -- DFAb -- move to their chromatic neighbors C#EG -- momentarily creating a diminished seventh chord because of the three simultaneous non-chord tones against the sustained Bb. The chord then resolves back to Bb7 as all those non-chord tones return to their chord tones.)
20. In the textbook, p. 587, Gauldin calls this an embellishing diminished seventh chord, specifically a common-tone diminished seventh chord or a neighboring diminished seventh chord, which he writes as N°7 in his Roman numeral analyses.