Drummer Albert “Tootie” Heath grew up in Philadelphia, but the music of New Orleans had a powerful effect on him nevertheless. Every weekend during his childhood, Heath listened in fascination to the marching band rehearsing in the American Legion Hall across the street from his house.

“New Orleans was an essential influence on the marching bands, which we didn’t know because there was no TV at the time,” says Heath, who was born in 1935. “The only thing we k...