Composition types in Western classical music. Is it possible for people to recognize the types of composition without the knowledge of the name given for the piece?
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How many composition types are there?
855, by counting the entries in a contemporary list.
Less jokingly, a list of composition types will teach someone no more about western music than a list of, say, the names of Chinese herbs would teach me about Chinese cuisine. More useful is to read an introductory textbook about Western classical music, or study a "Music Appreciation 101" course.
Although a general introduction is too broad for this Q & A web site, the books cited at the end of Wikipedia's article are a useful start.
Instead of "composition types" I would call these forms, or styles or even (in more modern terminology) genres. Since these are really just classifications or categories, there are literally infinite possibilities, so it's impossible to put an exact number on it.
Here are the most common ones that I can think of: