This is about how the harmony is working in Rondo a Capriccio by Beethoven. Here is the passage I am talking about:
So starting at measure 291, there is a passage where 7th chords go up chromatically, and the majority of these 7th chords are diminished 7ths. The seventh chords only last for 2 bars until there is a break from them with a scale. So let me try to figure out the harmony here.
A, C, D#, F#, this is an enharmonic respelling of Adim7 to better fit with the key signature of G major. And then that goes to an A#dim7. What scale comes after these 2 seventh chords? Well it starts on B and then goes up and it has 5 sharps. It must be B major. C and C# are skipped.
D, F, G#, B, it is a Ddim7, again respelt to fit with the key signature better. That goes to D#dim7 which then lands on an E major scale. E and F are skipped.
G, Bb, C#, E, it is a Gdim7 though here, it doesn't really fit with the key signature better than if the C# was written as Db. But that would mean spelling the E as Fb which doesn't fit with the key signature at all. As predicted it goes upwards to G#dim7. But here, Beethoven breaks the pattern of 2 diminished 7ths followed by a major scale. If he were to continue the pattern, it would end up on an A major scale. But it doesn't. We get our first resolution. G#dim7 goes to C minor. Beethoven sure loves his C minor.
The resolution doesn't last for long though, he keeps going with the 7th chords. F, A, C, Eb, this is F7. I would expect it to resolve to Bb. So what does Beethoven give us? The unexpected is what he gives us. He gives us an Adim7. The Gb then goes back down to F giving us another F7. This time, it does resolve. Not only that but Bb is the temporary tonic note. It resolves to Bb minor.
This then goes directly into Eb7 the same way that C minor went directly into F7. I would expect it to resolve to Ab. But, of course, Beethoven has to give us the unexpected. He gives us a Gdim7 respelt with an Fb instead of an E like the previous Gdim7. This then lands back on Eb and Beethoven gives us a scale. However, it isn't a major scale like the previous ones. No, it is Bb dorian where I would expect Eb major. Then he finally resolves it to Ab without going to another 7th chord right after, ending this 20 measure long, chromatic passage. He then has to go back to G somehow but that isn't my focus here.
So, yeah, it starts off with a pattern of 2 diminished 7ths a half step apart followed by a major scale. Then it deviates from this pattern with the first resolution to C minor. This then starts another pattern of root position dominant 7th, third inversion diminished 7th with root a third above the previous chord root, same root position dominant 7th, resolution chord which then ends with Ab major. So this way, Beethoven has practically brought us all the way up the chromatic scale using these 7th chords.
But why does this work, this chromatic scale in the harmony? I think it has to do with not only the pattern and how it is deviated but also the circle of fifths. In both cases where it lands on a major scale, the second diminished 7th has the leading tone as the root and the first diminished 7th has the note 2 spaces on the circle of fifths flatter than the major scale as its root. And A to D itself is a circle of fifths motion. Likewise so is D to G. G# is where things deviate. But this still works because every note in the G#dim7 except for F is a half step away from a chord tone in the C minor chord. The C minor, as an unstable tonic sets us up for the second pattern of 7th chords. Via a motion to the major subdominant, we get F7, 1 space sharper on the circle of fifths than C minor. This then moves to the minor tonic, 3 spaces flatter on the circle of fifths. Again the tonic is unstable. This continues with another movement to the major subdominant. Then we finally get to a stable tonic with the Eb7 to Ab motion that ends the passage, a cadence in Ab. Summing this up, we move 9 spaces flatter on the circle of fifths than where we started.
Is the circle of fifths the reason why this progression of 7th chords works?