From the Wikipedia entry:
"I'm Getting Sentimental Over You" was the theme song of the Tommy Dorsey Orchestra. The words were written by Ned Washington and the music was written by George Bassman. It was first performed in 1932.
It was first recorded in September 1935. A second recording on October 18, 1935, is the exact arrangement Dorsey would henceforth feature. Dorsey's expert breath control, command of the upper register, and phrasing added to the sweetness of the song which typified his orchestra. Frank Sinatra, who got his start in the Tommy Dorsey Orchestra, sang this song in the Dorsey Orchestra and also featured it in an album, I Remember Tommy (1961), after Tommy Dorsey's death in 1956. Sinatra achieved his first great success as a vocalist in the Dorsey band and claimed he learned breath control from watching Dorsey play trombone.
The tempo is considered a fox trot or swing. "I'm Getting Sentimental Over You" became the most immediately recognizable and favorite theme song of the Swing Era.
I love how listening to this music is like time traveling.