Help solve a musical mystery. 

It's easy to explain what a rondo is. The tough question, the one that stumps me, is why a certain kind of rondo—the kind that typically brings a concerto to a conclusion—sounds the way it does.

Rondo form, at its most basic, is very basic indeed. Descended from the rondeau, a kind of troubadour song in which varied verses are separated by a repeated refrain, it can be represented by the diagram ABACADA, wherein each letter represents an individual theme or musical idea. In other words, it's simply a string of tunes in which the first tune, the so-called ritornello, keeps recurring, separating all the succeeding tunes from one another; the same ritornello then brings the piece to a satisfying conclusion with a farewell appearance.

In the classical period, the rondo became one of the favored forms for the final movement of a multi-movement work, especially the concerto, where it was used almost invariably. (Mozart diverged from this practice only twice out of 27 piano concertos, Beethoven never.)

The great achievement of the early classical period was the creation of sonata form, a structural innovation that transformed musical discourse into a rigorous tonal dialectic, a closely reasoned and dynamic argument pitting conflicting tonalities and contrasting musical ideas in dramatic opposition to one another. It was Mozart, with his profoundly orderly mind, who rethought rondo form in order to bring it into line with sonata principles. (He did this in company with another great architectonic thinker, Haydn; but Haydn, who was not a first-class instrumentalist and did not therefore appear in public as a soloist, wrote few concertos since he did not need them for his own use.) This formal integration was accomplished by embedding the defining elements of sonata form into an overall rondo structure: After the A section, which was the initial presentation of the ritornello theme, the B section consisted of a sonata exposition, and then, following the recurrence of the ritornello, the C section was a sonata development, etc. Thus, Mozart and Haydn devised an ingenious way to merge the rigor of sonata logic with the variety and apparent spontaneity of rondo form. Their re-conception of the rondo established the template for most of the composers who followed them.

But, as I say, that's the simple part. What interests me and, frankly, has puzzled me over a lifetime of close listening to classical music is the character of rondo music, which has remained constant even when the form itself has been bent out of shape by subsequent composers. One can usually recognize a rondo finale from its very first measures, well before its formal ground plan has had a chance to reveal itself. So what exactly is the elusive quality that gives these movements their distinctive personality? This isn't intended as a rhetorical question: I'm soliciting your input in "The Fray." Let's consider a variety of concerto finales (all taken, rather arbitrarily, from pieces for piano and orchestra, although works with other solo instruments would serve equally well), pieces spanning over 150 years of music history and widely divergent in style, nationality, and harmonic complexity. To my ears, despite these differences, they all have something in common. Perhaps you can help me figure out what it is.

We'll start, naturally, with Mozart. Here are the opening bars of the last movement of a fairly early but thoroughly miraculous piano concerto, K. 449.

And here are the opening bars of the finale of K. 488, a piece from Mozart's glory days as the most celebrated composer-clavier virtuoso in Vienna. (Incidentally, don't let the apparent closeness of the Kochel numbers fool you: Fully eight piano concertos separate the two pieces.)

Are we any closer to an answer yet? Let's take stock. Both of these pieces are lively (although the second is considerably faster than the first), both are in a relatively square-cut 4/4 time, and both have a certain jollity about them. Could we be closing in on an answer?

Well, before we jump to any conclusions, let's listen to the last movement of Beethoven's second piano concerto. Although published second, this was actually the first concerto Beethoven wrote, and its debt to Mozart is patent. Like the two Mozart examples, the movement is fast and jolly all right, and it sure sounds like a rondo to me. But it's in 6/8, not 4/4.

And then there's his third piano concerto. This music may be quick, and it's in 4/4, but there's nothing terribly jolly about it.