Could you compose classical music when you're 100? Elliott Carter can. For Elliott Carter, turning 100 amounts to little more than a distraction from writing music. "The trouble with having a centennial is that you're here," he jokes to a visiting reporter. Over the last year, hundreds of events around the world have anticipated Carter's Dec. 11 birthday, and they've brought tides of visitors—musicians, composers, the dreaded journalists—to his Greenwich Village apartment, where he has lived since 1945. Those interruptions, combined with the more mundane chores of old age (regular hearing-aid checks, daily naps, mandatory strolls), add up to time...