Herb Alpert, the trumpet-playing leader of the Tijuana Brass and co-founder of the enormously successful A&M Records, makes his New Orleans Jazz and Heritage Festival debut on May 4.

In the 1960s, Alpert’s infectious, South of the Border sound sustained a ubiquitous presence on pop radio. His sales of 72 million albums led Billboard to rank him number 7 in the Greatest of All T...