Multiple Apple IDs

I have multiple apple IDs @ me.com, @mac.com, @ iCloud.com. My original apple ID was NOT an email address, rather it was my first initial, middle initial, and my last name. I have songs in my library that I purchased with this original account but am unable to access as my current computer is not authorized. The system keeps asking me for an email address for my original account, but this doesn't exist, never existed. You would think apple would know this. How do I gain access to songs I purchased on this original account?

I feel that I paid money and essentially got nothing in return. Did I just make a donation to apple?

Beth