Beethoven wrote five piano concertos, and only one of these is in a minor key. I don't want to reduce these things to the easily-refuted notion that "major key equals happy music, minor key equals sad music", but there does often seem to be a degree to which a minor key brings out Beethoven's brooding side more than the major keys do. This concerto is certainly the most inward-looking, the most introspective, off the five.
Where the previous two concertos opened with genial major themes, Beethoven's Third opens with a generally low-key statement of the main march-like theme in the strings, and we actually spend what feels like a surprisingly long time awaiting the arrival of our piano soloist; if one listens to a recording without video, one might trick oneself into thinking they are listening to a symphony instead of a concerto at all. But once the soloist arrives, Beethoven's great way of integrating the soloist and the orchestra together into a cohesive whole takes over.
The real magic of this concerto, for me, comes in the second movement, which opens with the solo piano playing a major-key theme that is romance-like. The transition from the first movement to the second always puts me in mind of Mozart's Piano Concerto No. 20 in D minor, with its famous Romanze in its second movement. Then, the third movement arrives in one of Beethoven's wonderful rondos, propelling us toward the end of the concerto with typical energy and verve, after starting on a quick motif, played softly and climbing upward throughout. This mysterious opening gives way to a sequence of chords that feel like Beethoven is calling a halt to the darker mood as he starts to segue into a cheerier one, though he isn't quite done with the brooding introduction just yet.
Apparently the Third Concerto wasn't even finished when Beethoven premiered it; according to the student who served as Beethoven's page-turned at the concert (which had only had a single rehearsal for a concert which premiered not only the Third Concerto but the Second Symphony and an oratorio), the score from which Beethoven performed sometimes had blank pages and weird notations that the pupil later described as "hieroglyphics"--presumably some sort of shorthand meant by Beethoven to jog his own memory. Here is how author John Clubbe, author of Beethoven: The Relentless Revolutionary, describes that primiere:
In his Akademies [concerts Beethoven arranged for his own benefit], Beethoven would premiere his first four concertos with himself as soloist. Only the fifth and last, the so-called "Emperor" Concerto, dating from 1809, did he not play in public. By then his fading hearing did not permit a nuanced performance. Once Beethoven realized fully that his time as a piano virtuoso was over, he wrote no more piano concertos.
Although scholars usually date the Third Piano Concerto to 1800, Beethoven had likely worked on it already for several years. He employed his favored key of C minor. As was his wont, he continued to revise it right up to the hour of its first performance. Even then he still had not finished it. For the piano part he had only his erratically pencil-scrawled manuscript. Ignaz von Seyfried, who turned pages for him, recalled how the first performance went:
I saw almost nothing but empty leaves; at the most, on one page or the other a few Egyptian hieroglyphics wholly unintelligible to me scribbled down to serve as clues for him, for he played nearly all of the solo part from memory, since, as so often the case, he had not had time to put it all down on paper. He gave me a secret glance whenever he was at the end of one of the invisible passages, and my scarcely concealable anxiety not to miss the decisive moment amused him greatly.
Not finishing a work on time was not unusual for Beethoven. Nor did it overly bother him. He had lodged the concerto's piano part deep in memory. He wasted nothing. Soloists then, unlike most today, did not play from memory. Audiences would have regarded doing so as eccentric. Only decades later did Franz Liszt begin the practice of playing from memory.
I often think that I wish I could go back in history and hear inaugural performances of some of the great works of the classical canon, but it's worth remembering that performance standards now are honestly the highest in history. I suspect that could we truly hear what music sounded like in the early 1800s, we might well be shocked--and not in a good way. But still--to go back and hear Beethoven? Knowing that the score on his piano was simply for show?
Here is Beethoven's Piano Concerto No. 3.