Music in which none of the twelve notes in the Western chromatic scale is given any more emphasis than any of the others. In the key of B flat, B flat is the most important note, F is (arguably) the next most important - those notes are given more weight. In twelve-tone music (and this is a simplified explanation), none of the twelve notes can recur until all twelve have appeared. Since none of the twelve tones is emphasized, there's little or no feeling of key center; the music is atonal - which doesn't mean it's cacophonous or chaotic; it just means that there's no key center.
In jazz, besides Murphy, David Mack made a really nice album of twelve-tone music - it's pretty obscure. And John Carisi used a modified twelve-tone technique at times.
More info here.
thanks!!