Black tarnish on a sterling silver chain can be frustrating, but it's often easier to remove than you think. To clean your chain, try using a solution of equal parts water and white vinegar, and then add one tablespoon of baking soda for an extra gentle exfoliating effect. Soak the chain in the mixture for several hours or overnight, then scrub it gently with a soft-bristled toothbrush and rinse thoroughly with warm water. If the tarnish persists, you can try using a commercial silver polishing cloth specifically designed for cleaning sterling silver. Another alternative is to use rubbing alcohol as a natural cleaner, but be sure to dilute it first with an equal part of water, as undiluted rubbing alcohol can damage some types of metal finish or discolor certain gemstones. For tougher tarnish, mix 1 part white vinegar with 2 parts water and apply the solution directly to the tarnished area using a cotton swab or soft cloth. Gently rub the area in a circular motion before rinsing off with warm water.