My recall of the formulae is as rusty as the piano strings I am experimenting with. I want to replace the strings in an old grand piano. I have miced the measurements of the old strings. There are only six changes in diameter of the trebled strings. I would like to recalculate and try for a better distribution. I wondered if anyone knew what the known variables are. The friction coefficient of a solid maple pin block to a modern laminated pin block? The range of acceptable tensions of steel strings made by the best modern suppliers in 2007? The torque on a 7mm wrench pin? So question 1: Is there some data on these variables anywhere. I would expect that the trick with piano manufactures is to balance these factors and many others in the choice of string size. I would have thought that most of the choices can be accomplished mathematically given a range for the various material stresses and strains. Question 2: What would the average or the range of string tensions be over the treble strings. I have noticed that the thicker strings give a more mellow sound, but the jump to the next string size becomes a problem. So it occurred to me that if one calculated the range using torque on the wrench pin against frequency from the safe to the dangerous. Question 3: How would the sound be affected? It seems to me that the range of mathematical variations cannot be great given the length of just one string. eg A4’s length would determine the entire range of possibilities for the string within the treble range. To that you would add other characteristic such as soundboard etc. But for the frequency itself balanced against the friction coefficient of the pin block ( and therefore the torque on the wrench pins) there cannot be a huge variation.