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If You Just Want to Back your Music Up
If all you want to de is back up your music, you can simply copy the iTunes Music folder to an external hard drive. To do this, go to your home directory; if you open a new Finder window, this should be in the sidebar as a picture of a house with your name next to it. If not, open your hard drive (double-click the icon on the desktop), open the Users directory, and then open the directory with your name (or short name) on it. Open the Music folder, then click-and-drag the iTunes Music folder to the external drive. Wait a while (quite a while if you're collection's large) and you're done! If you need to restore from backup, just move it back to where it was.
Otherwise, we're assuming that you want to use your library from the external drive, which is a little bit more complicated.
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By default, iTunes, Apple's music player, stores its files in the “iTunes Music” folder within the Music folder in your Home Folder (~/Music/iTunes Music), and the file containing data about all your music is stored next to them. The default behavior for iTunes is to keep your music folder organized and copy all your music there when you add it to iTunes. Whether this option is on or off, however, moving your iTunes Music folder will cause iTunes to lose track of where all your music is - and then you would have to add it all again.
The other issue is that your music may not always be in one place. This can cause problems when you try to move your iTunes library to the external drive. However, iTunes has a built-in Consolidate Library function that will do all the work for us.
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The Solution: Consolidate Your Library on the External Drive
iTunes has a built-in function to move all your music files to one location, so why not use it to move your music to the external drive?
The first thing you need to do is change the location that iTunes will save your library to. In iTunes Preferences (From the “iTunes” menu), select “Advanced.” The first option is for the iTunes Music folder location. Click the “Change” button and select where you want your music to be saved to (the external drive or a folder on it). Then, from the File menu, select “Library” and then “Consolidate Library...” iTunes will warn you. Click “Consolidate” and it will copy all your music to the new directory. This may take a while.
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Freeing up Space
Finally, you may want to free up some space on the hard drive; once you're sure everything has been copied over (check in the new iTunes Music folder to be sure), you can delete the original files from your hard drive. Make sure it's been backed up, though! You don't want to accidentally delete files. Also, keep in mind that deleting the originals will remove the benefit of having two copies - it's not a backup if the only copy you have is on the external drive.
For a more in-depth tutorial, with more explanation of why it's done this way, click the link.