Dwarfed by skyscrapers and distracting glittering billboards while surrounded by throngs of international tourists in the middle of New York’s Times Square, drummer Darren King of Mutemath experienced a profound, if not surreal, moment that reflected the electronic-rock group’s burgeoning music career.

The date was July 17, 2007, and he and his band, which at the time included frontman Paul Meany, bassist Roy Mitchell-C...