If you add Sodium Carbonate (typical pH Up or Soda Ash or washing soda) you can raise the TA and pH which will add carbonate to the water and (along with the higher pH) will precipitate out calcium carbonate. With enough carbonate you can even precipitate magnesium carbonate though it is about 10 times more soluble.

Adding just 1 ounce (weight) of Sodium Carbonate to 100 gallons of water will make the equilibrium of calcium carbonate out of balance by a factor of 45 which should precipitate over 90% of the calcium. You might also precipitate half of the magnesium. This assumes you get the pH over 9 (which will happen unless your water has over 100 TA to start with).

Of course, you'll then be left with water with high TA, but you know how to handle that. Using Ben's procedure of adding acid and aerating, you can get rid of the excess TA. Adding 1 ounce of Muriatic Acid will bring you back down to 7.5 pH, but your TA will be higher by about 35.

Then you'll be left with salty water (Sodium from the Sodium Carbonate and Chloride from the Muriatic Acid). Just with adding what I described above, your TDS in this make-up water goes up by 100. It will go up even more with the acid/aeration process, but I can't tell how much since it depends on how efficiently you aerate.

Well, you asked! Sorry I don't know of a cleaner method.

Richard