Transferring music from old iPhone to new iPhone?

I just purchased the iPhone 5, upgrading from my iPhone 4. However, I restored my new iPhone using the "Restore from backup" method. I backed up my iPhone 4 on my computer, but I have never synced my iPhone 4 and my computer. I thought that once restoring my my new iPhone from this backup that the songs would transfer over, but they did not. Everything else came through wonderfully. So my question is how to I transfer all of the music from my old iPhone to my new iPhone? I have all the songs on my old iPhone also in iTunes, but I want to keep my playlists, and I don't want to have to manually select all of the songs I want to transfer to my new iPhone. All I want is for the music on my new iPhone to look exactly like the music on my old iPhone.

Go into your carrier and they can help you transfer everything.

Make a backup for your old iPhone and restore from the backup with your new iPhone.

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