The very first transmission of music by radio is credited to one Dr. Nussbaumer of the University of Graz in 1904, however it was not to the public. He yodeled an Austrian folk song into an experimental transmitter which was received in the next room at the university where he worked. He does not show in any standard scientific reference works.
Lee De Forest produced a program broadcasting opera phonograph records from the Eiffel Tower in Paris in 1908. This was just an experimental stunt for other nearby hobbyists and not considered a public broadcast as the public had no access to receivers at the time - geocities.com 
At one point, when testing the radiotelephone for the Navy, Lee de Forest played patriotic phonograph music as the ships entered the harbor.