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If you wish to start your own Premium Family you'll first need to contact the owner of your current subscription and ask them to remove you from their plan. They can do this by visiting their Family page.
Once you've been removed from the family plan you can subscribe for Premium Family here.
The subscription type and your music collection aren't connected to each other, which means that starting your own plan won't affect your saved playlists in any way.
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“Music is the strongest form of magic.” Marilyn Manson