So, I think it's largely safe to presume that the core of Western musical theory developed almost entirely within Europe -- Greece in its very early history, mainly Italy and also the Franco-Flemish regions during the middle ages and Renaissance, and perhaps a little after that elsewhere.
There are however notable exceptions. Spain's role in the development of classical music cannot be understated, I think. While we may not remember many famous early Spanish composers, I have heard that they were instrumental (no pun intended) in developing various forms of classical composition, perhaps bringing some in from the Middle-Eastern and North Africans musical traditions.
So here are a few of my proposals to start us off:
Some compositional forms popular in the Renaissance and Baroque, e.g. sarabande and passacaglia, are suggested to have originated in the Middle East/Arabia, in some loose stylistic sense.
The horse-travelling nomads of Central Asia are rumoured to have invented and indeed introduced the first bowed string instruments to Europe, which the Italians adopted and improved by way of the Byzantines. (Early examples of harps and lyres were of course found in Ancient Greece and indeed ancient Celtic regions for the former.)
Does anyone else have some suggestions, tentative or well-supported, to share?