Someone on one of the mailing lists I subscribe to claimed that most all 20th century music is "junk". This set me to thinking. While I agree that much "classical" music composed in the 20th century was junk, there are, naturally, exceptions. These include a raft of English Romantics (Ralph Vaughn Williams, Gerald Finzi, Hubert Parry), a number of late-Romantic Spanish composers (Granados, Albeniz, Rodrigo), some great Latin American composers (Barrios, Villa-Lobos, Lauro), a strong Russian contingent (late Rachmaninoff, Prokofiev, Shostakovich), famous and not-so-famous Americans (Gershwin, Bernstein, Hanson, Beach), and so on.
However, art music is not limited to music in a classical vein, and I think the 20th century witnessed the rise of several genres from folk music into something approaching art music (for a fascinating exploration of this process, see the chapter "Folk Art and Fine Art" in Albert Murray's book Stomping the Blues). To my mind this happened twice: once with jazz and once with rock. While the pre-cursors to these genres were to a large degree unsophisticated (not that there is anything wrong with unsophisticated music! -- and there were exceptions here too, as in the ragtime music of Joplin et al.), each one developed into something musically complex and challenging. Jazz reached its compositional apex with the music of Duke Ellington, who I think is the composer of the 20th century (although Jelly Roll Morton and Thelonius Monk also made strong contributions to jazz composition). Rock, too, moved from the unsophisticated sounds of Bill Haley and Elvis Presley through the increasingly interesting work of the Beatles to the advanced harmonies, time signatures, and symphonic approach of progressive rock bands such as King Crimson, Gentle Giant, the Moody Blues, Jethro Tull, Emerson Lake & Palmer, and my personal favorite Yes. For an exploration of the meaning and structure of progressive rock, I continue to be impressed with philosopher Bill Martin's book Music of Yes: Structure and Vision in Progressive Rock, which I have read several times now (Martin's book deals directly with Yes, but also addresses a lot of the more-general issues in this space).