Hey all. My ex-husband was a jewlery maker, primarily in silver. Here's the scoop:
Although lots of chemicals will clean silver, what they are in essence doing is removing the top layer of silver along with the tarnish - even baking soda. Certainly, doing that once for a deep cleaning isn't going to make much difference, but over the long haul, it will damage your jewelry (so, everyone - Tarnex= very bad). For instance, if you have delicate designs etched into your silver, it will be dulled and could even eventually disappear with repeated chemical cleanings. If you have a very delicate, thin layer of silver, it will be worn away.
The best cleaner for silver is a thing called "jeweler's rouge." You can get it at jeweler's supply stores and possibly some craft stores. Anywhere you would find supplies for making jewelry. I Googled it and even Amazon has it -
http://www.amazon.com/gp/product/B0000BYDFB/qid=1153590156/sr=8-1/ref=pd_bbs_1/104-6304929-4412747?%5Fencoding=UTF8&v=glance&n=228013.
It comes in a stick, which you then rub onto a soft cloth (that will never be the same again, so make sure it's something you're willing to designate as your "rouge cloth"). It's kind of a dark red powdery substance. Use that spot to rub at the tarnished jewelry. If it is REALLY tarnished, it may take a little bit of work, but you'll be amazed how well it really works. For light tarnish, it'll come right off. One stick of rouge will last you for a long, long time.
Once you have cleaned the tarnish, then take a clean portion of the rag, without rouge on it, and polish the silver. You will get a gorgeous, high sheen. Much better than I've ever seen with chemical cleaners. I've done this many, many times myself and it has always worked.
It will work with other materials, too - brass, nickel, etc. But you often have to work a lot harder, especially with brass. But with silver, it works like a charm.
One note - your hands will get very dirty with red rouge and black tarnish, so either wear disposable gloves, or be prepared to wash your hands thoroughly afterwards. Also, be careful not to get it onto your clothes. I think it'll wash out, but it will make you dirty.
Cheers!