With Tonepros wraparounds, I have found that it helps to have the tailpiece adjustment screws backed out as far as possible. This moves the tailpiece forward, so that you have to adjust the saddles back. It does take time to find the right spot. Start with the low E saddle all the way to the back. It's your "longest" string. On two of my guitars, it stays all the way back. Set the high E saddle a little back from the midpoint, and see if it can intonate using the tailpiece set screw. You want to avoid having to move that saddle forward.
The farther back the saddles are, the more of an angle the strings have coming out of the back. The farther forward, the shallower the angle. That could be where the "ping" is coming from. I hope I explained it well enough.
Yeah, that does make a lot of sense.
Originally Posted by mad monk
Even though I may decide to switch to a different bridge, everyone's responses have been really helpful and interesting.
Thank you all so much!