I used to get a nice white (but not stark white like store bought) soap without td; used lots of lard, coconut, castor and whatever high oleic oil I had on hand. Then I found Big Lots coconut oil - 12 bucks per gallon which is a great price for a hobbyist. It's food grade, but more yellow than the coconut oil I used to use so no more white soap. I have to use td now.
I love Brambleberry's white soluble td best - never need to heat the water, just plop some td into a cup, add water, swish and good to go.
I now have Nurture's td and I hate it. It's a bear to mix properly. It absolutely has to be mixed for several minutes OR you need a mini mixer, plus it should sit in solution for a few minutes or you'll end up with spots in your soap.