When Astor Piazzolla received a bandoneon from his father, as a present, he was disappointed. This was an instrument for that old tango music his dad liked. And then he played it, and he was hooked.

Today's casual observer might be forgiven for thinking of Astor Piazzolla simply as an avuncular accordion-wielding avatar of Argentine tango traditionalism. But he called his music "nuevo tango" for ...