[Nick Zwar]

There is a lot of dissonance in tonal music… in fact, one could argue that actual “dissonance” can be found only in tonal music, because dissonance usually refers to harmonic tension, whereas atonal music lacks a harmonic tonal center to begin with, so there is neither “dissonance” nor “consonance” (its opposite) in atonal music.
There is a lot of dissonance in the music of Richard Wagner, Gustav Mahler, Richard Strauss, or Hector Berlioz, yet their music remains (mostly) tonal. On the other hand, Olivier Messiaen’s music often ventures deep into atonal territory (though it’s never strict about it, and it’s not “12-tone”… just not always within a certain harmonic tonal center), but I don’t find it particularly dissonant.