Mozart, Beethoven and others left a rich musical legacy in Europe. Why didn't China, Japan, India or the Ottoman empire write any great music in this same period of time. Or if they did, what are they?
There are a few things wrong with your question.
You refer to a "period of time" without specifying what. Classical music isn't just a niche in history -- the term "classical music" can apply to almost 750 years worth of music!
This question also assumes that symphonies and European genres are superior somehow. It's kind of like asking "why didn't Italian food take off in Japan?" (not quite but you get the idea)