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Stars Fell On Alabama is from the Complete Pablo Solo Masterpieces recorded by Norman Granz in the early 1950’s rereleased on compact disc Vol. 5. This was written music by Frank Perkins lyrics by Mitchell Parrish. It was first recorded by Guy Lombardo with vocal by brother Carmen in 1934. Afterwards it was recorded and performed by everyone from Jack Teagarden to John Coltrane. What Tatum shows us in his version is that the stride bass piano can be moving in a slow, soft ballad style as well as dazzling in a speedy piece of music.
The title of the song comes from a book of the same name written by Carl Carmer. Carmer wrote about life in Depression era Alabama through the eyes of a New Yorker using as a metaphor a meteor shower observed in Alabama in November, 1833. For more, see this Wikipedia entry.