It having been said that this question admits of an answer, it'd better have one. That said, questions which ask for lists of offsite resources are off-topic, so I'll answer accordingly. You ask:
Can music be open sourced the same way code is open sourced, as in give it a MIT license or similar so that others can use, improve and build on your music?
Yes, it can. Openscore is a project which releases classical music in freely-reusable editable score form, in the free musescore format. Although the underlying music here is out of copyright, the scores from which it is usually conducted, and the orchestral parts from which it's then played, generally aren't.
The musescore format allows a number of reuses which are only possible for third-parties because of the free licensing of the openscore scores. These include printing off a full set of orchestral parts, as well as more-transformative uses such as generation of synthesised audio recordings via MIDI, the generation of the cover images (which are images processed directly from the score), and the generation of freely-reusable Braille scores.
There is no reason why in-copyright music could not be similarly made available, given the permission of the rightsholder; but things had to start somewhere, so they started with the classics.
I have no doubt there are many other examples of open-source music on the internet, but you asked if it was possible, and indicated that this example met your criteria, so here it is.