How do I re-download previously purchased music without requiring iCloud?

Music I previously purchased from iTunes mysteriously disappeared from my iTunes library recently. I'm trying to re-download it from the "purchased" music section of the iTunes store but the website won't let me. It shows the music as having been purchased and downloaded but gives no option to re-download it. Other message boards on this subject speak to going into iCloud. I don't subsribe to that service, and I hope I'm not required to pay and subscribe to that service in order to access what I've already purchased. Thanks in advance for your help.