Even though I do feel that the term "classical music" can be a little misleading (as people generally use it to describe music from the Baroque, Renaissance and Classical periods exclusively), I see that people are mentioning more modern composers as well. So here are some of my favourites across the different periods:
Baroque: Albinoni, Bach, Handel, Pachelbel
Classical: Beethoven (even though he teeters on the edge towards Romantic), Brahms, Mozart, Clementi, Haydn
Romantic: Berlioz, Chopin, Wagner (yes, he is actually rather good, in my opinion - at least his different themes)
Impressionism: Debussy (really, he's the only one)
Expressionism: Berg
Various 20-th century composers: Ligeti, Stravinskij, Philip Glass, George Gershwin
On another note, I really enjoyed reading the earlier debate on dodecaphony. Even though I don't really enjoy it that much (Alban Berg is an exception to that rule because of
Wozzek) I have composed using the technique a few times. It can be said to be both difficult and easy, I suppose... since you actually have a very strict set of rules to follow and you can't really do more than follow them once you've created the twelve note order.
Also, I must add my two cents considering the whole atonality vs. tonal music: Every piece of music has its own beauty, in my opinion. Ligeti's music may not be easy on the ear, but it can be hauntingly beautiful. John Cage may have written a piece called
4′33″ (which is one long pause - the pianist counts measures in complete silence for four minutes and thirty-three seconds) that may not exactly be what you'd expect from a composition, but the beauty instead lies in the noise the uncomfortable audience generates. Stravinskij's
The Rite of Spring is both tonal and atonal, and one of the most beautiful pieces of music I've ever heard. And Debussy's
Clair de Lune may be simplistic, but that is also the reason why it's so wonderful.
Anyway. Enough of my essay.
