Who owns the copyright to the MIDI files, then?
There can be several owners. Most people think there is only one copyright, but in reality a recorded piece of music is a subject of two separate, but layered copyrights sometimes referred to as copyright © and copyright (P) (should be a circle with a P in it).

The creation copyright of the author is most commonly understood and respected. The composer owns this right to his music. In most cases this copyright is assigned to a publisher to "sell" the song.

The other is the recording performance production copyright, which is the right of the performing recording artist to their specific interpretation (production/performance) of the song/music.

What this means is that in order for it to be legal to play a piece of music on a website, first, the performer must have received permission from the creator to create the performance, and second, both the creator and the performer must agree to make the performance freely available to the public.