I have thought long and hard about the arguments surrounding whether to rephase the crankshaft in my XS650 to 83° or 90°. I have read every argument in support of Phil Irvings original theory and looked at the mathematical proof presented with interest, but if I were to believe these arguments I would have no choice but to conclude that 74° was the perfect rephase angle for the 447 engine in my XS650 and that is not what I have come to believe. I do however concede that 74° is the point where the centrelines of both crank and conrod are at right angles and as such is the point where piston speed is at its maximum and the piston has its maximum mechanical advantage over the crank, but I do not believe this is the optimum rephase angle to achieve the greatest reduction in vibration.
I believe Brian Whoolley was correct when he asserted that Phil Irving got it wrong and that to achieve a reduction in vibration then balance is the key and that can only be when the piston and conrod assemblies are in the same relative positions at the rephase angle in both halves of the stroke cycle and the crank pins are in equal and opposite positions effectively cancelling each other out. This symmetry is only achieved when the rephase angle is 90°. By Brian Whoolleys' calculations this should give a theoretical reduction in vibration of 43.5%. This holds true for all vertical twins irregardless of bore, stroke or conrod length.

As you see in the diagram, when the rephase angle is 74°, the disecting centrelines of the conrod and crank/crankpin is 90° on the downstroke and the piston is high in the bore, but on the upstroke this is only 58° and the piston is low in the bore, notice also the relative positions of the crankpins, they are some 32° apart when the left cylinder is at TDC and BDC(Blue)
With the 83° rephase the rod angle is 81° for the downstroke and 67° for the upstroke and the pistons are again high and low in the bore with the crankpins being some 14° apart.(Red)
But with the rephase angle at 90° the rod angle is 73° for both up and down strokes and the piston is in the same relative position in the bore and the crankpins are also in the same relative position 0° apart.(Black) This is symmetrical and therefore balanced!
As you can see 83° is the next best and because of its ease of construction makes it a favourite of those who are willing to settle for a compromise. As we all know this compromise is a very good one and works well and everyone who has done it sings it's praises loudly, and rightly so. But if you want the best, and for the extra cost of a custom crank pin and the sacrifice of a second crank, then nothing else but 90° will do! 
cheers
Terry Gliddon

                                                   Standard XS 650                                   Rephased XS650

Although not absolutely correct, my assumption that symmetry was the key to better balance was on the right track, see Smoothness by Degrees for a better explanation of why 90° is better than the rest.