A stern black-and-white portrait of “Papa John” Joseph, bass poised, hangs over the mantle in Dr. Michael White’s home office. White, a native New Orleans traditional jazz clarinetist, inspects the photograph, admiring it. Papa John’s gaze is caught in such a way that it seems to follow you around the room. “Look at those eyes,” White says. “You’re looking at truth.”

Papa John was a cousin on White’s mother’s side, and for a long time the oldest active jazz musician in town. A contemporary...