Applications :: Recovering ITunes Music Without IPod Or Hard Drive

I did a search for this problem but all the solutions seemed to involve using your iPod or old hard drive to recover. I recently had two tragedies in a short period of time. First, my iPod broke - a tragedy, but one I could get past. The second was the horrible death of my laptop's hard drive. I had some backups of important school and work related files, as well as a few other things like old tax documents, and much of my older files, but I did not have my iTunes folders backed up, because I was under the impression that I could download these songs again through iTunes should the need arise, since I had purchased them in the past.

Now, on my new computer, I have downloaded iTunes, and have been trying to recover my old songs. I authorized this computer, but when I "check for available downloads" it says everything has been downloaded, and that I should try to transfer the songs from the other computer. This is of course, impossible at this point. I can still see my purchases in my history, and if I try to purchase a song I had before, it warns me that this will create a duplicate. Will I have to purchase these songs again? I can see how I should have backed up my music, but I thought that iTunes would allow me to acquire them again.