Ahhh, the numbering of the works of the classical masters! If you're old enough, you may well remember owning an LP of Dvorak's Symphony No. 9 (From the New World), but it would have been labeled as his Symphony No. 5, before scholars renumbered it. Likewise, Beethoven's Piano Concerto No. 1 in C Major was actually the second piano concerto he wrote, after the one that would later be published as the Second Concerto. (There had also been an unpublished student work, but we don't need to consider that one here.) Beethoven chose to publish this concerto first, so the numbering reflects not the compositional order but the publishing order.
Classical music can be confusing.
So let's listen to both concertos, No. 1 in C Major, and No. 2 in B-flat Major.
Both of these works reflect the youthful classicism of Beethoven's early period. In each of these concertos you hear the strong influences of Mozart (who was already gone) and Haydn (who was not). But in each you do catch glimpses of the more expansive Beethoven to come, the Beethoven who would take classical forms and stretch them to unprecedented lengths. More important than that, though, is that in each work you hear Beethoven's humor and musical wit, which are characteristics not always called to mind when considering this composer. Of particular note for me in a Beethoven concerto is the third movement, the rondo, when Beethoven always seems to have a great deal of fun with the rhythms, almost making the listener wonder, at first, where the bar line and the beat even are. There's a feeling of syncopation even where there's no technical syncopation happening at all.
In the first performance I am featuring here, of the Piano Concerto No. 1, the Israel Philharmonic Orchestra, conducted by Zubin Mehta, is joined by pianist Khatia Buniatishvili, who is a wonderful artist I've just discovered this year. She plays with technical control that is amazing to behold, and she's also one of those pianists who does put on a bit of a show as she plays. (Her hair would be quite the distraction if she were playing from the score as opposed to memory.) But note also how attentive she is when she goes tacit and the orchestra picks up the main thrust of the music. Buniatishvili performs truly as a partner with the orchestra and with Maestro Mehta, and it's a fantastic performance.
In the second performance, the Piano Concerto No. 2 is played by soloist Martha Argerich and conductor Daniel Barenboim, leading the West-East Divan Orchestra. Some years ago I listened to the entire cycle of the Beethoven symphonies that Barenboim and this orchestra performed at the BBC Proms, and this combination of musicians has, to my ear, a particularly special touch with Beethoven. Argerich is, of course, an absolutely brilliant musician of Argentine and Swiss descent, and she brings here the weight of experience and years of musical training and insight to her performance of the Second Concerto. Which was really the first...but we won't go into that again.