Here's an example of a moment with which every classical music lover is familiar:

There comes a point in virtually every concerto from the classical period when the orchestra comes to a sudden halt and the soloist takes flight with a couple of minutes of difficult (or at least difficult-sounding) passage-work interspersed with fragments of melody derived from what's been previously heard. This music usually culminates in an extended trill, after which the orchestra re-enters, bringing the movement to a speedy conclusion, more often than not without any further participation from the soloist.

So what's going on?

We can brush very lightly and very quickly over the technical aspects; feel free to skip this paragraph if the technical aspects don't interest you. But for the two readers remaining:

This is the most characteristic closing cadence in classical music, consisting of four chords: the subdominant, the second inversion of the tonic (called, for reasons we're not going to bother ourselves about, a six-four chord), the dominant, and finally the tonic in its root position. Never mind what any of that means, just know that the second of those chords is considered very unstable, and was traditionally the place where an opera singer might improvise some pyrotechnic embellishments before allowing his or her aria to finish.

When the classical era began experimenting with adapting vocal arias for solo instruments and marrying the da capo aria form to that of the symphony, what ultimately emerged was the concerto we now know. And when this was firmly established as a genre in its own right, that same moment and that same chord became the place where soloists could take a few minutes to display their virtuosity. Since in those days the composer and the soloist were usually the same person, the requirement to provide the latter with an opportunity to show off was no great imposition on the former.

It is believed that Mozart (generally regarded as peerless in this form of composition) improvised his own cadenzas at concerts. There's no reason to doubt it; his facility at improvisation had been legendary even when he was a child. This place in his scores was always left blank, with a fermata (pause sign) over the six-four chord, leaving the choice of what to play to the performer's discretion. An exception occurs in those of his concerti with more than one solo instrument, when the practical need for cohesion demands a written-out cadenza, for example:

We could easily have remained ignorant of what sort of thing Mozart himself did in his solo concerti when the cadenza moment hove into view. But fortunately for us, Mozart's students didn't share his improvisational gifts, and so, when a pupil of his was going to play one of his concerti, Mozart accommodatingly wrote out an improvisatory-sounding cadenza for him or her (usually it was her) to use. As a result, we have at least an approximate idea of what Mozart played and for how long he played it before arriving at the trill that welcomes the orchestra back into the fray. Here's an example:

But existing cadenzas of Mozart's own composition are the exception. For most of his concerti, soloists have either had to fend for themselves or adopt the cadenzas that were composed by other pianists or composers. In all but his final piano concerto, Beethoven followed Mozart's example and left the cadenzas unwritten. But later in his career, he thought better of the practice, and published cadenzas for his first four piano concerti, in some cases writing several alternate versions, giving the performer a choice of which to play. (This development coincided with two significant changes in the composer's life, and it's impossible to say whether either--or both --might have played a determinative role: his deafness had become sufficiently acute so that he could no longer perform comfortably in public, and he was beginning to devote himself as a teacher to the not untalented Archduke Rudolf; in either case, he might have felt an urge to be more precise about what interpreters other than himself should play.) In his final, fifth piano concerto, he still includes the six-four chord and the fermata, but this time he inserts an NB, and a note in Italian at the bottom of the page which reads, in rough translation: "One should not play a cadenza here, but go immediately to what follows." What follows is a very short, quite minimal, but fully written-out cadenza:

Unusually, it doesn't end with the orchestra's immediate return, but rather, with a continuation of the piano solo, under which the orchestra gently sidles in. As far as I'm aware, no performer of this concerto, regardless of how subversive or willful, has ever had the, I believe the technical term is chutzpah, to disregard Beethoven's instructions and to--I believe the technical term is potchkey--with what Beethoven specified. That NB is just too stern to flout.

But otherwise? Well, in cases where no original cadenza exists, the only choice is to invent one's own or play something written by someone not the composer. In other cases, performers and composers have abandoned what might seem like sensible scruples and chosen to ignore the composer's explicit intentions, replacing the original cadenza with one of someone else's devising. Most of these are unexceptionable, if not obviously improvements; Brahms, for example, composed cadenzas to some Mozart and Beethoven concerti, and although they're almost never played, they're perfectly plausible. Mendelssohn wrote cadenzas for a number of Mozart concerti, including the one for two pianos, where Mozart's splendid specimen certainly demanded no replacement, and, one might say, barely permitted it. For some reason, Robert Casadesus used to play the composer-written cadenza to Mozart's 16th piano concerto in his performances of the 26th. This was a dumbfounding choice; Casadesus was too knowledgeable a Mozartian to have been unaware of what he was doing, but he went ahead and did it anyway. It didn't cause any grotesque harm; the candenza was in the right key, and even though none of the music in the cadenza related to anything in the concerto being performed, the effect wasn't horribly grating and the final trill delivered everyone home safely.

There are cases, though, where the deviations have been, for good or ill, more noteworthy.
Take Robert Levin. Aside from being a superb and scholarly musician, he is arguably the best extemporizer in classical music today. He claims he improvises his cadenzas in the interest of musicological authenticity, but it's my suspicion he does it mainly because he can. But for whatever reason that he does it, he does it superbly. Here he is in Mozart:

And here, more controversially, in Beethoven:

Perhaps the most famous cadenza one composer ever wrote for a concerto by another is the one Beethoven wrote for the first movement of Mozart's 20th piano concerto (he wrote one for the last movement, too, but we're focusing on first movements here). The exact occasion for it isn't clear: it's known that Beethoven played this concerto at a concert to benefit Mozart's widow around the turn of the 19th century, but the manuscript paper on which the cadenza was written appears to date from a decade later than that. In any event, some have viewed Beethoven's essay as something akin to a desecration, an act of quasi-Oedipal aggression toward a master he revered. But for what it's worth, it has always sounded to me like genuine homage, in which the younger composer takes the elder's thematic material and invests it with his own characteristic vehemence. The cadenza starts at about 2 minutes,12 seconds from the start of the clip.

And then there was the brilliant and deeply eccentric Glenn Gould. He composed cadenzas for the first two Beethoven concerti, and very, very strange cadenzas they are. This one, to the first concerto, starts as if it were a fugue from the late-19th century, boasting some of the harmonic pungency of a Max Reger or a Ferruccio Busoni. It's absurdly anachronistic; in context it could even be regarded as a stylistic monstrosity. But I love it all the same. It begins about 4 minutes, 54 seconds into the clip.

And then there's the cadenza to Mozart's 22nd piano concerto that Benjamin Britten wrote for Sviatoslav Richter. This was in the '60s, when Soviet musical royalty seemed to be having an odd sort of group love affair with Britten; Shostakovich dedicated his 14th symphony to the English composer, who in turn conducted the work's premiere, Rostropovich was gifted with, and premiered, five or six major cello compositions, and the cellist and his wife spent some of their holidays with Britten and his partner Peter Pears, and Sviatoslav Richter became a regular guest at Britten's annual music festival at Aldeburgh. But this particular cadenza, while no doubt cherished as a gift, may also have left Richter perplexed; it has certainly puzzled many listeners since. It starts more or less normally, but grows increasingly deranged as it proceeds. Anyway, whether or not he really liked it, Richter consented to play it. Because of fair-use restrictions, we need to begin after the thing has already gotten under way, but trust me, the best is yet to come:

And then, saving the very weirdest for last, here is Alfred Schnittke's cadenza to the Beethoven violin concerto, commissioned (and played here) by Gidon Kremer. It's all over the stylistic map, and, in addition to its own idiosyncratic harmonic oddities, it contains allusions to the Brahms violin concerto, Shostakovich's first violin concerto, and the Berg violin concerto. I have no idea what Schnittke thought he was doing, but anyway, here's what he did.

Is there any lesson to be drawn from these examples? Would it be fair to say, for instance, that any section of a composition that permits these sorts of shenanigans--in a field that in general takes itself very seriously indeed and privileges textual fidelity above almost every other value--ought to be regarded as in some way less important than what surrounds it? I think the answer is clearly yes. This was historically--going back to a period anterior to the emergence of the concerto itself--an opportunity for soloists to thrill their fans and strut their chops. It was meant to elicit stomps and whistles and cries of "Bravo!" It was a display of athleticism as much as artistry. Some cadenzas are very good, some are ridiculous, most are in between, but in the classical era they are fundamentally, as the fermata above the six-four chord suggests, a hiatus in the progress of the musical argument.

But that doesn't mean we can do without them. After all, from the start, the musical argument formed itself around the cadenza's existence.

NOTE: The following CDs were sampled in this blog:
Viotti: Violin Concerto #23, Sinfonie concertanti #1 and 2, Naxos 8.553861
Mauro Ranieri, Roberto Baraldi, Alberto Martini, soloists, and the Accademia dei Filarmonici conducted by Aldo Sisillo
Beethoven: The Piano Concertos, Teldec 0630-13159-2
Andras Schiff, and the Staatskapelle Dresden conducted by Bernard Haitink
A Festival of Haydn, L'Oiseau-Lyre 465 642-2
The Academy of Ancient Music conducted by Christopher Hogwood
Mozart: Violin Concerto #2, Sinfonia Concertante, ARGO 411 613-2
Iona Brown, Joseph Suk, soloists, and the Academy of St. Martin-in-the-Fields conducted by Iona Brown
Mozart: The Piano Concertos, Archiv 463 111-2
Malcolm Bilson, fortepiano, with the English Baroque Soloists conducted by John Eliot Gardiner
Mozart, Piano Concertos #5, 14, and 16, Decca 289 458 285-2
Robert Levin, fortepiano, with the Academy of Ancient Music conducted by Christopher Hogwood
Beethoven: Piano Concertos #3 and 4, Archiv 457 608-2
Robert Levin, fortepiano, with the Orchestre Revolutionnaire et Romantique conducted by John Eliot Gardiner
Mozart: Piano Concertos # 22 and 27, and the Adagio and Fugue for String Orchestra, BBC 4206-2
Sviatoslav Richter, piano, with the English Chamber Orchestra conducted by Benjamin Britten