Removing tartar/calculus

I have a dark line at the bottom around the edges of a couple of my front teeth. A dentist told me he thought it was caculus, not a cavity (I was there for an extraction, so he didn't really dwell on it.) I was searching for how to remove it at home, and the only thing I found was a product called Plax. I can't have products with alcohol, so I can't use it, but the other ingredients were big ugly words I didn't recognise, so I was suspicious anyway. I thought surely there must be some natural thing I could do to make these ugly marks go away. Many of the things I found were talking about how to remove plaque, which isn't hard yet, so don't apply to my issue. I was looking for something that could soften it up so I could remove it with a toothbrush or my fingernail or something, but as I read, I started wondering if it would go away on its own if my nutritional status improved.
I was wondering if someone could comment on what affect hydrogen peroxide, iodine, borax, or oil pulling would have specifically on existing calculus. (I haven't tried any of these yet, they are just what occurred to me as I was writing this question.)
Has anyone tried anything else for this issue? Tweet