One day in 1964, a New York advertising-jingle composer in his early 30s received an unlikely job offer. The composer, Mitch Leigh, the Brooklyn-born son of a Jewish furrier from Ukraine, had no theater experience to speak of. All he had ever done was compose incidental music for a couple of short-lived Broadway comedies — “Too True to Be Good” (1963) and “Never Live Over a Pretzel Factory” (1964). Now he was being asked to write the music for a new show that was going to try out at the Goodspeed Opera House in East Haddam, Conn. A few numbers...