Sharp was born in Topeka, Kansas in 1924. His family had settled there because Kansas was, during slavery times, a free state. Sharp’s great-grandmother was a slave and he remembers her telling him stories about lying on the floor while cannonballs blasted through the cabin. The Great Depression soon followed, and Sharp’s parents sent him to live with relatives in Los Angeles while they went to New York to seek their fortune.
Sharp’s father was a concert tenor who won several small roles in Broadway shows but never hit it big. However, their social life was rich; they lived in an apartment building in Harlem with fellow residents Duke Ellington, Walter White, Roy Wilkins and artist Aaron Douglas. It was the peak of the Harlem Renaissance, and poet Langston Hughes was a family friend; Orson Welles had recently staged his famous Black Macbeth at the nearby Lafayette Theater.