Lazy Silver Polishing
Relatives coming over and no time to polish the silver? Instructables has a cheap, easy way to polish silver without buying silver polish. Simply add baking soda and a bit of aluminum foil to a pot of water, add the silver you want to polish and boil away. —MEGHANN MARCO
Eco Silver Polish [Instructables]
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This won't work well on very heavy black tarnish, though. Not unless you have a day or so to spend on it. There are probably things I haven't tried, but so far, I've had good results with pink paste polish for that stuff. And using the glove-type with dry polish regularly will keep it from getting that bad.
This should include the caveat that you shouldn't use this method for silver items with burnished lettering that. If you do, that lettering will become basically unreadable. So if you've got your grandpa's old silver Crew trophy from Harvard, use silver polish or another method. At least that what I've read. I've use this method to clean my silver trumpet a few times. You have to have the silver piece touch the aluminium foil.

I've also heard that ketchup and toothpaste work pretty well on their own if you don't have a lot to work on.