I have always loved the Ainulindalë, but I have never thought of this question quite that close... So I was immediately lured into thinking.
I do not see it as that tightly tied to the "races" or "kinds of beings" in the hierarchy of being. But I'd also hate to read it straightforwardly through Christian theology (aka. creation, fall from paradise, apocalypsis).
I think it's quite easy to see the first theme meaning a first creation, the Ainur joining His mind and starting to create by giving their individual tones into it. But when Melkor started adding his own things as to lead the theme into dissonance (to dis-accord), then Ilúvatar made a counter-move into the theme (part two) which might be "the introducing of death", or "creating of mortals", or what will you have there to counter Melkor's schemes (it says Melkor had the mastery in the end of the second theme and the Ainur went silent). Heh. Sounds like the Christian theology version I just dissed...
So in a way Ainur and Edain, but also creation and conflict in it which is resolved with a new creation.
Then the third theme could be the original division of things to good and bad / evil? You see there is no mention of these terms in the themes but only of discord and accord, dissonance and consonance, disharmony and harmony etc.
Also before the third theme the discord was internal or grew within and was tied with the original theme, but with the third theme there were clearly two separate musics which had "unities of their own". Then the evil melody started trumpeting forwards and drowned the good one - and thus Ilúvatar had to whistle it off (apocalypsis)... so mortals with free will and the evil gaining the upper hand - introduce Christ / apocalypse?
Blah. So I ended up with a Christian interpretation anyway...
Okay, I need to give this another thought. But on another time as I now need to go to sleep.
Thanks
lmp for an interesting thread!