Who hosts MIDI files that I can access by HTTP with no download interface?

Perhaps you are complicating things by looking for a MIDI host? Just about any "plain" web hosting should do you.

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way to embed MIDI files

If you are thinking of that, I'd suggest having a link as well. I don't think any of my current (Linux) browsers are working properly with HTML embed tags.

The only reliable way I have for me is to use a Java applet (due to a request and someone having problems playing MIDI links, I've added and am still completing writing a player for the folkinfo abc converter btw) but that depends on a user having Java and unless you can afford to pay for a proper certificate you either have an unsigned applet (one restriction is Java won't allow them to download from other sites) or you write your own certificate and probably scare your visitors away because some pop up comes up warning them your site is untrusted and may be unsafe. Can't win...