Ring angle, how well quartered the wood is, should not make any difference in the damping, which is what determines how a piece rings when tapped. What it does change is the cross grain stiffness, particularly in soft woods, and the stability of the piece. Generally speaking wood shrinks and swell more with changes in humidity 'tangentially' (parallel to the surface of the log) than it does 'radially'. One outcome of this is to cause it to cup across the grain when the ring angle changes as you go across the piece. Well quartered wood will have the rings at 90 degrees to the surface all the way across, which produces the least amount of cross grain shrinkage and the greatest stability. A 'perfectly' flat cut piece, with the ring lines exactly parallel to the surface all the way across should also be pretty stable, in terms of cupping. The problem is that the only way to get a piece like that is to start with a tree of infinite diameter. There's only one of those in the universe, and it would take forever to cut it down. Flat cut wood also shrinks more than quartered, and it is not as resistant to splitting as quartered wood, owing to the orientation of the medullary rays.
The 'best' orientation to avoid splitting is probably skew cut; with the ring lines at a 45 degree angle to the surface. On soft woods this will have the lowest possible cross gain stiffness, but the difference is not so great with hardwoods, and may be negligible depending on the species. If I had to make a wenge guitar I'd try to find skew cut wood, with the least amount of variation in the ring angle I could find, for the back. It would be difficult to tell looking at it that it was not quartered, and I'd trust it more.
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