Here is how to get tarnish off silver or gold jewelry (gold can tarnish if the alloy contains metals that tarnish).
I use Arm & Hammer Super Washing Soda, a laundry booster that is much more alkaline than baking soda. (Washing soda is sodium carbonate; baking soda is sodium bicarbonate.).
Use a non-reactive container (i.e. not aluminum; I use tempered glass measuring cup (Pyrex) but a steel pot of bowl will work too).
Line it with aluminum foil.
Boil some water, 1-2 cups, depending on how many things you are going to clean. Have another bowl handy with tap water for rinsing the items afterward.
Protect your eyes--put on safety goggles.
Pour the hot water into the container and add 1-2 rounded tablespoons of washing soda. Stir to dissolve.
Add the tarnished silver or gold items. They need to be in contact with the aluminum foil for this to work.
A chemical reaction will result-- you'll hear and see fizzing, and smell a sulfurous odor, which is the tarnish coming off.
After a while, remove the items using tongs or a fork. If there is still some tarnish, put it back into the solution.
Then remove it and rinse it. If you still see a bit of tarnish, it's easy to rub it off with a bit of baking soda (a mild abrasive) and/or an old soft toothbrush.
Caveats:
1. Don't put pearls, opals, turquoise or other soft gems into this solution.
Also, don't put in any stones that could be damaged by thermal shock, e.g. if they have inclusions that could make the stone crack if heated this way.
2. Don't use this with plated metals.
3. Sometimes the patina of tarnish in crevices of vintage pieces is attractive and desirable. This cleaning method will remove it. In that case you'd be better off using a chemically treated, mildly abrasive polishing cloth, of which one example is a Sunshine Cloth. (That brand of cloth comes in different colors depending on how abrasive you need it to be (blue is the gentlest, yellow the most common, pink the roughest. Available on Amazon and elsewhere.)