To further elaborate, here is my quick re-harmonization of it
http://www.mediafire.com/?c9bp1ttrl8r5nv8
Here is a break down of the harmony:
Dm7, then Dsus4, then Dsus2. After that, the piano plays that faster run of the ocarina melody, but harmonized in thirds. Then, it hits an A minor chord. A unison of a "C" note follows and then ends on a D minor chord. This is just one of literally limitless ways to harmonize it.
Hopefully something I've said helps with the remix and harmonizing the melody.