I prefer using a hard new toothbrush dipped in water as the bristles are straight and get into the crevasses whereas old toothbrushes tend to slide over.
Once you have got it all out give up using brass polish and use cheap label tomato ketchup instead. Brush all over the brass (soft paintbrush quite thickly), leave for a suitable amount of time (do not let dry) depending on the level of tarnish, rinse with cold water and buff with a soft cloth. Repeat on tough tarnish if necessary.