So there are lots of examples of this, particularly in things like opera, because it was rare that a composer would write the libretto (the lyrics or text for the music) of the opera. Sometimes they would hire a librettist, sometimes they would use one that was already made, and sometimes they would collaborate. Gilbert and Sullivan are an excellent example of an extensive ongoing collaboration between librettist and composer. Although they technically worked "together" they did not ever work in the same room at the same time. Gilbert would write the libretto, then hand it off to Sullivan who would write the music, so their music was definitely created by both, but depending on your definition of collaborative, it might or might not be.

As far as musical collaboration, another example is "Variations on a theme". A composer would write a piece of music, and then a different composer would take that piece and write variations on it. That is, to write different pieces using the main musical ideas as a base. One example is the Variations on a Theme by Haydn, written by Brahms. That is, Brahms wrote Variations on a theme that was originally written by Haydn (as a side note, it's now believed to have been incorrectly attributed, and that Haydn was not, in fact, the author of the original piece, but that's a point for a different question).

Here's the original "Haydn" theme: https://www.youtube.com/watch?v=0T4at_omvF4

And here are Brahms' variations: https://www.youtube.com/watch?v=BAuqxEMRapg

​

Or for another example, Variations on a Theme of Chopin, by Sergei Rachmaninoff, based on Chopin's Prelude in C minor.

Here's the Chopin prelude: https://www.youtube.com/watch?v=7zQYuhR-TUQ

And the Rachmaninoff variations: https://www.youtube.com/watch?v=XiDtLscGnn8

​

This kind of thing is fairly common, though again, might not fall under the definition of collaborative, since the musicians rarely worked together, and were often long dead when the variations were written.

​

As for true collaborations, it's really not very common, but it has happened. Felix Mendelssohn and his sister Fanny Mendelssohn wrote sets of 12 songs, where each sibling wrote some. Each sibling wrote some of the works, and they were released together as a group. Likewise Robert Schumann and his wife Clara Schumann did essentially the same thing - books of songs where each of them wrote a few.

Though one of my favourite examples is the Messa per Rossini (Mass for Rossini). When Rossini died, Verdi decided to create a Requiem mass in honour of him, and got a number of composers together to each write a part of it. These included (imma have to look this one up): Verdi, Antonio Buzzola, Antonio Bazzini, Carlo Pedrotti, Antonio Cagnoni, Federico Ricci, Alessandro Nini, Raimondo Boucheron, Carlo Coccia, Gaetano Gaspari, Pietro Platania, Lauro Rossi, Teodulo Mabellini. Each of whom wrote one movement. The whole shebang clocks in at just under 2 hours.

You can listen to the whole thing here: https://www.youtube.com/watch?v=jhBuGjZPbvc

There are more examples like this, but they're generally the exception rather than the rule, and usually involve composers working on a part of a work, rather than in direct collaboration with each other, although that did happen as well. Most composers worked (and in Western Concert "Classical" music today still work) on their own. But they also don't have a room full of studio executives with marketing budgets to answer to to ensure their next hit.

u/flotiste beat me to some of the most common kinds of "collaboration," namely where one composer takes the works of another and does something with it. Variations on themes, re-orchestrations, and arrangements are all fairly common examples of these.

As they mention, "collaboration" in classical music usually means some kind of piecemeal composition, one movement being added by each of the collaborators or something of the sort. But there does seem to have been a brief blossoming of collaboration in the mid 19th century, with the Schumann and Mendelssohn pieces mentioned earlier coming from that time along with a few more from prominent composers:

Anton Diabelli did a sort of crowd-sourcing experiment in 1819, writing a short waltz and sending it off to various composers to see what they'd do with it. By far the most famous of these is Beethoven's Diabelli Variations for piano, a set of 33 variations on that theme. Franz Schubert, Franz Liszt, and many, many other much less prominent composers each wrote a variation that was compiled into one big set of variations, which is rarely played.

Chopin wrote a Grand Duo Concertante in 1832 with a prominent cellist named Auguste Franchomme based on an opera by Giacomo Meyerbeer. This might be the closest to a collaborative composition in the sense that you're looking for, because each man more or less wrote one instrument part, with Chopin taking the piano and Franchomme taking the cello.

The next year, Felix Mendelssohn and Ignaz Moscheles, a piano virtuoso and Mendelssohn's teacher, wrote Fantasy and Variations on the "Gypsy March" from Weber's Preciosa. Again, it seems like this was fairly collaborative in your sense, with each composer contributing various bits, rather than composing separate parts on their own and adding them together.

The F-A-E Sonata (1853), written for the great violinist Joseph Joachim. It features a movement each by Johannes Brahms and Albert Dietrich, and two movements by Robert Schumann. Dietrich's kind of the odd guy out in this, since he was a more run-of-the-mill composer alongside two giants in Brahms and Schumann.

Now, you can probably sense a couple patterns here. One is that the rise of star virtuosi on various instruments meant that composers were eager to work with and write for specific players, whose expertise could come in handy while writing for their instrument. This would certainly be the case for Chopin and Franchomme, both of whom were celebrated virtuosi on their respective instruments, even if only one is known now as a composer.

The other is that the collaborations don't necessarily come out evenly in regards to star power in today's mind. Two of the composers of the F-A-E Sonata are bona fide stars, part of pretty much any "Top 20 Composer"-type list you could find out there. The third, Dietrich, is a name familiar only to musicologists, a footnote whose fame rests more with who he knew than with what he wrote. Franchomme and Diabelli suffered the same fate, as did Moscheles, although he's probably the best-known of the lesser-known bunch here.

Of these works, the Diabelli Variations is really the only one that is seen as a major work in the better-known composer's oeuvre. Classical music, and especially the music of the Romantic era, has a real "cult of the composer" going on, so it's not surprising that collaboration would not be seen in the best light. Instead, the stereotype is often of the individual composer working on a deeply personal vision, which doesn't really leave much room for working with others.