Born in Kansas City, Counce studied violin and tuba early on before settling on the string bass. He went on the road when he was 16, playing with the Nat Towles Band in Omaha. After some freelancing, Counce moved to Los Angeles in 1945, working with Johnny Otis and making his recording debut the following year with Lester Young.
During the ‘50’s he was a key member of the West Coast jazz scene, recording as a sideman with Shelly Manne, Lyle Murphy, Teddy Charles, Clifford Brown, and many others.
In 1956, Counce organized a quintet comprised of trumpeter Jack Sheldon, tenor saxophonist Harold Land, pianist Carl Perkins, and drummer Frank Butler. During a 15-month period, they recorded enough material to document their endeavor, all of which were originally released by Contemporary and fall stylistically between West Coast cool jazz and hard bop.