I don’t know why I have to learn all these things the hard way, but apparently I do. This one took two days of unsuccessful help from my friends and one minutes with an Apple Store Genesis to resolve.
When I downloaded stuff to my new iPhone 5s I carefully marked only four albums of music that should download. I have a 16gb phone so I don’t want to use up all my space. Every song in my library downloaded. I won’t bore you with the details of everything we tried to fix this problem, and rather tell you why it happened and how to fix it.
iOS 7 has a category under the Settings called “iTunes & App Store”. I think this is a new category, but can’t swear to it. In addition to indicating what stuff you want downloaded to your device while you are in iTunes, you also need to visit this setting on your device.
The button next to Music needs to be turned off if you only want selected items of music to be listed on your phone. Mine was turned on. I turned it off and no more problem. I just downloaded my music files again after I turned it off and only my selected albums were listed on my phone. Note that contrary to my belief, aside from my selected albums the songs in my iTunes library were not actually downloading, they were just listed.