Hi Peter,

From my understanding of classical harmony, there are primarily two ways that the diminished chord is treated.

1. The diminished triad resolves to an inversion of a chord that has its root a perfect 4th higher (perfect 5th lower). This is the standard resolution for discords (dim, aug, 7, etc).

2. As you've already indicated, the diminished 7th acts as dominant harmony (without the dominant present and what would normally be the major 3rd up from the dominant as the chord's root note). For example, in the key of C major or C minor, Bdim7 is usually considered a variation of the dominant 7th (G7 chord) that has B as its root note rather than G. In this way (in the key of C), Bdim(7) leads to some inversion of C.

Arguably, though, this second point can be considered as a special case of point 1. For example, Bdim7 is enharmonically equivalent to Ddim7. The 4th above D is G and thus Ddim7 will resolve to a G chord. This, in turn, can be resolved to C (in C major or minor). So Ddim7-G7-C can be condensed to become Bdim7-C.


CONSIDERING POINT #1 ABOVE
In the key of A minor, chord ii (Bmb5 = Bdim triad) resolves to E (or E7) which, in turn, usually resolves to Am. (This is the standard ii-V-i progression in minor mode. This is also in keeping with the notion that chord ii is a variation of subdominant harmony.)

This principle can also be applied to secondary dominants. For example, consider the below progression in C major.

C-Dm-G7-C

When considering possible chords to lead to the Dm, the secondary dominant of D is A7 and so the progression could become ...

C-A7-Dm-G7-C

Equally, borrowing from minor mode harmony, the diminished triad a perfect 4th below D could also lead to Dm (using point #1 above). This gives C-Amb5-Dm-G7-C. Like A7, this diminished triad has the A root note. For this reason, I think of the above progression as the diminished variation of dominant harmony. Diminished triads often sound better when played as an inversion rather than root position. So putting the Amb5 in second inversion gives …

C-Amb5/Eb-Dm-G7

Extending the Am5b and Dm chord into chords of the 7th then gives …

C-Adim7/Eb-Dm7-G7

(Where the dim7 uses the principle of stacking notes that are successively a minor 3rd apart (as found in the seventh chord formed on the second note of the harmonic minor scale). Thus the musical distance from the root note to the top note is a diminished 7th: e.g. C-Eb-Gb-Bbb; and the notes C -> Bbb = a diminished 7 interval.)

Since Adim7 = Cdim7 = Ebdim7 = Gbdim7, it's often easier to name a diminished 7th chord by its bass note since its less confusing to interpret this visually. This gives …

C-Ebdim7-Dm7-G7 where Ebdim7 is really Adim7/Eb.

Also, Adim7 and Cdim7 are equivalent chords – that is, contain exactly the same notes – so the the above chord progression could also be written as …

C-Cdim7-Dm7-G7 which is C-Adim7/C-Dm7-G7

(To my mind, both Cdim7 and Ebdim7 are alternative ways of expressing Adim7 and so are a form of secondary dominant harmony when they lead to a Dm(7) chord.)


CONSIDERING C/G HARMONY
The progression C-D7-G7-C, where D7 is a secondary dominant, can be decorated to become

C-D7/F#-G7-C

In cadential position, a variation of G(7) is C/G which is essentially a G chord with two suspensions (the C and the E). In this position, C/G behaves as dominant harmony and not tonic harmony. Thus the above chord progression could be elaborated to become (melody permitting) …

C-D7/F#-C/G-G7-C

As explained in point 1, the diminished chord a 4th below D could used to lead to D7. Using 7ths, this gives …

C-Adim7-D7/F#-C/G-G7-C

However, Adim7=Gbdim7, and Gbdim7 is enharmonically equivalent to F#dim7. Thus …

C-F#dim7-D7/F#-C/G-G7-C

Because there are now two successive chords on the F# bass, either one can be eliminated. Choosing the dim7 chord then gives ...

C-F#dim7-C/G-G7-C

Alternatively, since F# is the leading note of G major (or G minor), and since the cadential C/G is simply a variation of G(7), the same principle of Bdim7 leading to C in the key of C major can be applied and F#dim7 can lead to C/G (if one considers a temporary key change of G major).

In the progression ...

C-F#dim7-C/G-A7

... C/G is what theory people call a passing 6/4 chord (6/4 = second inversion and comes about from Bach's day when figured basses were used). Since the bass is G, it is still good music to think of this as a form of G chord. To me, C/G feels more like a dominant chord than a tonic chord because of the dissonant interval of a 4th between the bass (G) and the chord tone C. To my way of thinking, given the above, F#dim7 to C/G is a less obvious variation of secondary dominant harmony.


AUDIO FILE
For comparison purposes, I've put together this 4-part harmony audio file of some of the above chord progression.

...link here (approx. 360 kB)

The progressions are ...

1. C - Am - Dm - G7 - C

2. C - A7 - Dm - G7 - C

3. C - Adim7 - Dm - G7 - C

4. C - Cdim7 - Dm - G7 - C

5. C - Ebdim7 - Dm - G7 - C

To my ear,even though the various second chords in the progressions sound slightly different, there's enough similarity to show that the dim7 chords are all acting in a similar way to the secondary dominant A7. (The reason that the Ebdim7 sounds more noticeably different than the others is because the melody needed to change to accommodate Eb in the bass.)


SUMMARY
In all instances, the progressions have come about by considering (a) that a discord (in this case the diminished chord) classically resolves to a chord whose root is a perfect 4th higher or (b) that a dim7 chord can resolved to an inversion of a chord whose root is one semitone higher that the root of the dim7.

Hope this helps with some kind of insight,
Noel