The opposite of “tonality” is atonality. An atonal piece is one where there is no feeling of a home key. Playing lots of random notes will sound atonal. Schoenberg was one of the most famous composers of atonal music. Of course, his music is not just random notes (although it may sound like it to the listener at first), so he had to find another way of giving his music shape. That is why he invented the twelve-tone system.