After reading the other responses I think that either I misread the
question or there is some ambiguity in it.  If the question was, "what
is--or what was--"modern" music? then L'Apres midi...  seems as good a
start as any.  The "modern" in art and in literature had its beginning
within a few years of that and the very practical question that the Museum
of Modern Art in New York had as to whether they should simply consider
their collection closed and representing an historical period--or whether
MOMA should continue to collect and display more recent work without
violating its original mission shows the difficulty and seriousness of
the question.  I would say that modernism in music began when I said and
ended as an historical period at the onset of minimalism and the kind of
postmodernism or neoromanticism that George Rochberg began with his Concord
Quartets.  (There had been neoromanticism before that, e.g.  in Hanson,
Barber and Thomson, to further muddy things.) Thus if the student meant to
ask what should we call the music of the period 1900-2000 then the only
adequate answer would be 20th Century.  But if the questioner meant what
should we call the music of the 21st Century, then "contemporary" is what
makes most sense.

Jim Tobin