The problem I have with "modern" music is all the confusion arising from the many different uses of the word "modern." To some people, Bartok is old music and Frank Zappa or Lady Gaga is modern music...... Boy o' boy, it's a Babel Tower of missunderstandings.
So, the question cannot be answered until the term "modern" is clearly and totally defined. First task at hand, it seems to me, would be to distinguish between "modern" music and "contemporary" music, explaining substantively or logically the basis for such a distinction. At the other hand of the aesthetic spectrum, and equally required in this discussion, would be to distinguish between "late Romantic," "post Romantic," "neo-Classical," and "modern" music. For instance, were Debussy, Ravel, Satie, Honegger, Berg, Prokofiev, Shostakovich, and Sibelius "modern" classical music composers? If not, what were they? Clearly not Romantics like Schumann, Chopin, and Brahms!! Not even late Romantics like Mahler, Rachmaninov, Villa-Lobos or Myaskovsky!!
If these composers are said to be "modern" music composers then how to characterize the vast, vast cultural world separating them from Messiaen, Carter, Cage, Schnittke, Gubaidulina and so many others? If you say that the latter are contemporary composers, not modern, then how to distinguish them, most of them dead, with today's composers -- i.e., contemporary to us?
Please do give it a shot! Otherwise this question cannot be answered.