Cleaning a stainless-steel coffee carafe

Any tips for cleaning (heavy) coffee staining from the inside of a stainless steel carafe? I’ve tried soaking it with lots of dish soap, and scrubbing it with a nylon scrubber, but it didn’t make a dent in the staining. I don’t want to destroy the carafe with harsh chemicals …

…Did you try Google? This was the first result.

Granted, those are for glass carafes but there are several methods there you could try.

the lemon juice, ice, and salt method mentioned on that page works.

I’ve done the white vinegar bit. That works too.

Isn’t this one of those cream of Tartar things? I don’t know if this works with stainless steel, though.