Hey, there! Great question. I have a masters in ethnomusicology, so I hope I can be of some help.
The origins of music has long been a focus of inquiry in musicology/ethnomusicology, which concerns itself with all the music systems of the world and their relationship to culture and society. (cf. Carl Stumpf's "The Origins of Music":
https://global.oup.com/academic/product/the-origins-of-music-9780199695737?cc=us&lang=en&). However, the focus nowadays is on music universals. Why? Well, one big reason comes from the question: What is music? The answer seems obvious until you do cross-cultural analysis. What we in the West consider music, i.e. something akin to "sound art" for the purpose of artistic expression, is not a universal cultural phenomenon. Whereas the definition of language is already firmly established and clear-cut, this is not so for music. Thus, a "historical linguistics of music" would be more difficult than plain old historical linguistics, though some have certainly tried (an endeavor in this sort of vein was Alan Lomax's "cantometrics," which sought to classify the world's vocal music into a taxonomy of communications style:
https://en.m.wikipedia.org/wiki/Cantometrics).
Anyway, notwithstanding the difficulty of placing world music systems into a historical taxonomy, an area of study that I think you would find enlightening is organology, which catalogs the world's instruments and traces their historical relationships and development. (
https://en.m.wikipedia.org/wiki/Organology).
Lastly, if you would like to know why Chinese music, for example, sounds so different from Western music, then there's really no other way of knowing except researching the history of Chinese music and Chinese thought concerning music, and then doing the same for Western music. Probably somewhere along the way, you'll come across an author who has tried answering why they have developed in such different ways, but that is unfortunately out of my area of expertise.
Also, for what it's worth, contrary to Daniel's assumption that "[g]etting from modern rap, or country, or death metal back to Greek traditional music seems difficult," it really isn't that difficult, because those three former systems are unrelated to traditional Greek music, save for the fact that ancient Greek thought on music has been influential in the development of Western thought on music, and Western music is the overarching music-cultural matrix into which rap, country, and death metal were born. Anyway, a university "intro to music" or "world music" textbook could help provide a more detailed answer.
I hope this helps!