I've been using MP3 Player to provide an on-page mp3 player, but I also need to be able to handle midi files. Right now the best I can do is allow them to be uploaded into the "mp3 file" field, but when it renders the node, it just gives a link to the file, which mean you have to navigate to the file (or open it in a new tab) for the browser to play it.
MIDI files can be embedded straight into the page with the
<object> tag, so there should be some way to then output that with the link that views gives you at the moment. If you went with a separate field then it's be easy enough to do with Views output rewriting.
After a quick look for a flash solution, it seems that flash itself doesn't have any support for MIDI, so a flash based player is unlikely.
If you actually want a module to do this, you can look at the Audio module. However, this module is currently said to be "unstable", even though over 3000 sites report having it installed. So with that many users it might be worth taking a look at anyways. The author also has an alternative suggestion using FileField and SWF Tools modules.