Cleaning sterling silver jewelry with gemstones

Factual Questions
1

I could find information on how best to clean one or the other, but not both. I have a gorgeous SS bracelet with multiple semi-precious stones that is badly tarnished and I would like to clean it up. (It looks just like this.) If you don’t want to bother with the link, the stones are amethyst, peridot, blue topaz, citrine, and garnet. Can I use the at-home baking soda method? Would the commercial ring cleaning liquid I got with my wedding set work?

Thanks in advance.

2

Gemstones are relatively impervious–they’re valued partly because they’ll look good for a long time, after all. If you clean the tarnished metal parts with a non-abrasive metal cleaner like Noxon or Tarn-X, you won’t damage the stones.

3

Opal is a notable exception to this general rule.