Thanks for the great comments from the last blog, with such passionate agreement and disagreement about how well the four movements of the Beethoven 4th symphony "hang together." I think the question is worth exploring. I have a hard time holding these four movements in my head as one symphony, and I've come to realize the reason is not only the different characters, but that the controlling musical idea in the 4th is far more subtle than in any other Beethoven symphony.
For instance, the 1st symphony is about the energy of gradually building a scale. One of its "secrets" is that the opening two notes of the slow introduction in the woodwinds initiates this scale process: E to F, F# to G, then the strings take over G to G# to A. When the fast allegro begins, the motive B-C in the violins is a continuation of this scale process from the slow introduction, and that scale continues ascending as the movement develops. Beethoven obsesses over ascending scales in each other movement of the symphony. The finale explicitly begins by building a G major scale note by note, just to make the accumulation of this process so obvious that it becomes a joke.
So there's no mystery behind the compositional idea of the 1st symphony. And so it goes for the other symphonies, the 5th being the most obvious, with its idea of 3 similar events followed by something different creating a fractal world where this motive occurs in many different layers of time.
But not so readily the 4th symphony. It takes more work to discover the common idea in all four movements. And while I'm convinced this motive intentionally recurs in all four movements of the symphony, it is subtle, not controlling the structure as obsessively as Beethoven does in all the others.
The clue is in the mysterious introduction that begins with a descending chain of 3rds:
These descending 3rds are themselves decorations of a descending scale, which Beethoven makes evident in the tail of the Allegro vivace theme:
The second group suggests the same chain of 3rds, both in the beginning of the 2nd theme and later in its continuation (rising 3rds):
Below are examples of the chains of 3rds and the descending scale in the 2nd, 3rd, and 4th movements.
Certainly this seems to be the large compositional "idea" of Beethoven's 4th symphony. But it doesn't "stick out" or dominate the musical journey. Rather it unifies "secretly," in the manner more of Haydn and Mozart. Maybe this is why the different movements of the 4th symphony don't seem to flow into each other as "inevitably" as in the other symphonies. Yet once attuned to this chain of 3rds, we notice it—subtly—in numerous "remote corners" of the work.
After the mystical slow introduction to the first movement, we are drawn to wonder why that music never returns. And we keep listening for it. These mysteries make the 4th symphony uniquely interesting from all the others, and both more and less than the sum of its parts.