First recorded by Dick Jurgens & His Orchestra (1941).
Hit version by The Glenn Miller Orchestra (US #1 1941).
From the wiki: “The story goes that ‘Elmer’s Tune’ was named for it’s creator, Elmer Albrecht, an undertaker’s assistant, who used to practice the song on the piano at the Aragon Ballroom in Chicago, Illinois. His mortuary job was nearby and so he made a deal with The Aragon Ballroom owner who let Elmer practice there every day or so. Bandleader, Dick Jurgens, would often hear Elmer practicing this melody and one day decided to help him finish his song. Dick Jurgens & His Orchestra completed the first recording of ‘Elmer’s Tune’ as an instrumental. Lyrics were later written by Sammy Gallop, at which time The Glenn Miller Orchestra recorded the number featuring Ray Eberle & The Modernaires. Miller’s recording stayed on the Hit Parade for seventeen consecutive weeks.”
The Glenn Miller Orchestra, “Elmer’s Tune” (1941):