Very good question David, and I had this argument (being a native pontificator) on my harp list. I couldn't understand why a string would break at the same pitch no matter the guage - and that was because I came from guitar to the harp.
The greater the mass the less the tension needed to make a certain pitch, we agree there. But the tensile strength is measured in pull per square inch, and is a matter of the thickness and the material strength. The breaking point is a matter of cross section. Yet the density (mass) is measured in cubic inches. The stretching is a factor, but such a minor one as to be not considered (although that may be different on the lute). I have a mountain dulcimer I built, the treble strings wouldn't hold when I took them up to the Mixolydian mode (a fifth up for those strings, the mountain dulcimer is designed to modal music). I dropped the guage a hundreth, but that was a mistake (they are still .011s instead of .012s, and when they break I'll go back to the .012s - they didn't break, the wrapping around the ball unwound, a drop of super glue fixed that). Again, not a problem for the lutenist, you are using nylon or gut, and tying the strings around the root bridge. But the principle stands. As to the lot difference, it is there but I don't think it is significant to the instrument unless you have a very good ear. Play with guages. I buy my harp strings in bulk from Vermont strings, costs me five dollars for a 20 foot coil of string (nylon), and 20 feet of bronze for my incipient psaltery cost me the same. Remember, density isn't a function of guage, it is a function of the material. Andy monofiliment string fits the approximate mass measures I gave you. The guage will slim down when you stretch the string, and that will change the cross section, but it won't change the density. That is a function of the material. Take a board (I've got one) and install pins and a bridge. You'll find that when the cross section thins and the tension goes up the only change is the brightness or mellowness. Not the pitch capability. Computer programs and formulae are a guide, the damned formula is complicated but I'm not bad at math - and I have a set of graphs of possible strings from a vendor whom I deal with (and they aren't always right). But there is nothing that matches trying a few strings empirically, no formula will say how your lute reacts. The formula gives you a guide as to where to start. And I'll remind you that the lute formula for strings wasn't complete, and probably because of the difference between the fixed string length of the harp and the stopped strings of the lute. Best, Jon
