Benny Goodman learned clarinet at his synagogue, bolstered by two years of classical training. He made his professional debut at 12 and became part of the rich Chicago jazz scene, steeped in the music of New Orleans. Goodman first recorded in 1926 and spent several years as a first-call musician in New York. His 1934 band was broadcast on the popular “Let’s Dance” radio show which aired coast to coast. An otherwise unsuccessful tour in 1935 ended in a smash hit in Los Angeles where the audience went wild. Thus, it is said, swing was born and Goodman became its King.

The band included trumpeter Bunny Berigan, vocalist Helen Ward, and Gene Krupa, the explosive drummer who was seminal to the band’s swinging sound. In 1937 Harry James, Ziggy Elman, a...