Catalyzes the oxidation of galactitol to D-tagatose. Also catalyzes the oxidation of a wide range of substrates, including polyvalent aliphatic alcohols and polyols, to the corresponding ketones and ketoses. Galactitol is the preferred substrate. galactitol + NAD(+) = H(+) + keto-D-tagatose + NADH Binds one Mg(2+) per dimer. Can also use other divalent cations. kcat is 835 min(-1). Optimum pH is 9.5. Optimum temperature is 35 degrees Celsius. Homotetramer. Belongs to the short-chain dehydrogenases/reductases (SDR) family.