Richard Bridge - 10 Jun 10 - 04:36 AM suggests:

The other possible desirable feature might be a constant pressure to fret the string at any point upon its length, but I think (I am not sure) that that is mathematically impossible to achieve.

There have been numerous attempts to produce fingerboards with this – and a gajillion other – peculiarly desirable feature, and in this case it's quite easy to do the "thought experiment" that disproves the difficulty imagined.

A string supported at two points and under tension will be approximately a straight line. (Purists will say that the weight of the string will always make the "undeflected" string a catenary arc that's not quite straight, but the deviation from straight will be quite small.)

If you apply the same pressure against the string at each of many various points, it will deflect a certain distance at each point, depending on the location of each point along the length of the string. The difference in deflection from one fret to the next will be small, and if you include points "between the frets" you obviously will get a continuous curve "of deflection points." Consult M. Fourier for the writing of the equation, if desired, although fairly simple algebra can get you there if you have the patience and a basic understanding of stress/deflection fundamentals.

CONCLUSION:

If you apply the same pressure at each of the fret positions, and measure how far the string is pushed sideways to achieve the same "pressure" for each fret, the deflections away from the straight string position will form a curve.

To have your "constant finger pressure at all the frets" all you have to do is bend the neck (or at least the fingerboard) so the tops of the frets are on that curve.

A few luthiers have published mathematical(??) analyses purporting to "prove" that the "ideal neck" is slightly concave, and have claimed to deliberately build their products that way. The usual implementation relies on making a straight neck that is then bent concave by the string tensions, and then pulled back "to the ideal curve" by some or another (sometimes quite exotic) "truss rod" device.

Since an "end-loaded" uniform beam can only be bent to a parabolic curve by linear forces applied at the ends, some claim to build in a "tapered stiffness" so that a slightly different curve can be achieved.

(I'm a bit doubtful that any of these "analysts" actually achieve what their theories proclaim, although some of them – I'm told – have made some very nice instruments. If it really made all that much difference, it would likely be a "trade secret" that they wouldn't tell us about.)

John