Place 1/4 cup of cream of tartar in a bowl. Add enough hydrogen peroxide to make a thick, peanut-butterlike paste. Stir the ingredients until they're well-blended.
Spoon some of the cream of tartar paste onto a damp nylon scouring pad. Smear the paste over the manganese stains inside your dishwasher. Allow the paste to sit on the surface for 10 minutes.
Rinse the scouring pad with water and add more of the cream of tartar paste. Scrub the manganese stained areas of the dishwasher to remove the stains. Dip a wet toothbrush in the paste and use it to remove the black and brown stains from small grooves and crevices.
Pour 1 cup of white distilled vinegar into another bowl. Set the bowl inside the bottom rack of the dishwasher.
Run the dishwasher on a regular cycle. As the vinegar sloshes around and creates steam with the hot water, it will help loosen remaining hard water stains while washing away the manganese stains you removed.