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When you think of classical music, there is the trinity called the "Three Bs" - Bach, Beethoven, and Brahms. The first two were dead by Edison's time, but Brahms was still alive. And so Edison sent his top men to Vienna to ask the master to play one of his songs because of the huge commercial potential of a recording by the composer himself. And you'd ask him to play one of his most popular songs, naturally.
That would be one of the 21 Hungarian Dances which were wildly popular. They were popular because they were the perfect music for most people who could afford music. Most music was performed in the home, and mostly by the daughters of the family. Music was considered a major part of a young lady's education, and so middle class gentlemen ("bourgeois" in the term of the day) families frequently purchased pianos, music lessons, and sheet music for their little Princesses. The Hungarian dances were a musical trifecta: quick tempo (in other words, challenging music that would show off the performer's mastery), upbeat tempo (fun to listen to, and therefore popular to a wide range of guests), and written by one of the undisputed greats.
Edison knew his market, as indeed did Brahms. Here is the Berlin Philharmoniker Orchestra conducted by Herbert von Karajan playing this most popular of Brahms' works:
And here is the recording of Brahms himself playing the song for Thomas Edison's wax cylinder recording device in 1889. It is the earliest recording ever performed by a major composer, and introduced by his voice: