Can I just say, firstly, that I’m horribly excited that music is actually being composed for Ariel’s songs? Most of the productions I’ve seen have just had Ariel sort of breathily chanting his lines, which is lame. It just takes all the magic out of the songs and makes you feel like you’re at a hippie wedding.
I can’t recommend any specific recordings, but I can give the title of book that has a very short chapter on the history of the music with samples of actual sheet music and a link to a page with more links.
I almost did a paper on music in The Tempest once, but I decided I liked the women in The Merchant of Venice better. This is one of the citations that made its way onto my computer.
Kermode, Frank. “The Music of The Tempest.” The Tempest. Ed. Kermode, Frank. Cambridge : Harvard UP, 1958. 156-60.
Okay, as for the websites, there’s this one on English ballads of the sixteenth century. It has links to transcriptions of ballads. Most of the songs are just the lyrics and no music. Some of them, however, have scanned copies of the music, mostly in sixteenth century notation, which differs a little from modern. This page on Thomas Ravenscroft, though, has some transcribed into modern notation. As for which would ideally suit Ariel’s presence, the digging is for you.