why is a 13" tonearm design really superior?

Mathematical explanation:

The cartridge traces out an arc as the arm moves from the golder to the center of the record. When the arm is short this arc is very curved on the region between the holder spindle, where as if the arm is longer the arc becomes less curves. Now, when the arc is bigger and relatively less curved, it maintains tangency for a brief period just like the short arm, but the error dur to the curvature is minimzed. I suppose you can call it more tangent if that makes any sense.

HAHAHAHAHAHA, LOL. Yeah there are trade-offs when the arm length becomes to large. The change in effective weight placed on the stylus proportional to changes in up and down vertical movement increase with arm mass. So yeah don't plan on using a flag pole as a tone arm just yet. Nonetheless, a slightly longer tonearm certainly does decrease tangency error for the reasons given above, it is simple geometry. The optimization problem is slightly more complicated as Baerwald was helped by 12 guys from nasa, as I understand it. Well, I have not found it to be too difficult, nothing a little calculus of variations and free parameter sensitivity analysis of some parametrically defined functions can't solve. IE> a slight bit beyond highschool calculus if you want to do it from scratch, and 9th grade math if you would like to use Baerwald's, Loengren's, or Stevenson's optimal null points tweaked for your turntable.