In my experience, the action has a huge impact on the timbre of the strings. At least that's what I've found on my Hanewinckels. My theory is that when you pluck the string, the string will impact slightly on the upper frets before settling into its vibration. Or, if the action is very high or your touch is light enough, it won't impact at all. By adjusting the action, you are changing where on the fretboard this impact occurs. At the point of impact, you are creating a kind of node in the string vibration. It will dampen some frequencies and enhance others. My observation is that it tends to dampen the fundamental and enhance or give the effect of enhancing the higher-order harmonics.
In college I took a differential equations course where we had to solve the problem of a vibrating string. The solution to the equations was a Fourier series which is the sum of an infinite series of sine functions (if I remember correctly). The coefficients of the terms are what determines the character of the solution. We solved the equations for plucking the string at different points and you could see how plucking near the "bridge" would cause higher-order harmonics to be more prevalent.
- Dave