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The short answer is this: you can't transfer songs from your iPod to your computer using iTunes alone. It's a built-in iTunes feature that prevents oversharing music files. Chances are, the songs recognized by the computer were songs you purchased through the iTunes store, and those missing came from importing music files from a CD or were sent to you by a friend. The good news is that there are plenty of programs that allow you to transfer music from your iPod to an authorized computer. If you're on a PC, try Pod To PC, for Mac, try Pod To Mac. These programs will pull files off of your iPod onto your new computer; just make sure none of them violate the iTunes Terms of Service.
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