Mathematics Behind Grand Piano String Selections

[mileman]

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.

 

[rcgldr]

On the treble strings, the groups of 3 strings overlap between two notes. It's my understanding that the tension is the same, and it's the lengths of the strings that vary to produce the different pitches for each pair of notes. It may be that most of the trebel strings are around the same tension with the only variation being the lengths of the strings.

 

[AlephZero]

It would be more accurate to say the stress in the strings is roughly the same, not the tension (since the string diameter changes as well as the length).

Unless you know what you are doing, the best advice would be to leave the string diameters as they are. If you mess around, at worst you will finish up cracking the cast iron frame. The total string tension on a grand piano is of the order of 20 to 30 tons force. Don't try removing strings one at a time. Slacken them all by a small amount and repeat several times to take all the tension, off to avoid breaking or permanently twisting the frame, before you start actually removing the strings. And do the reverse for the replacement strings of course.

If you want to even out the tone where the diameters change, do it by re-voicing the hammers. If you restring with new rust-free strings, the tone will most likely change a lot anyway. You may need a new set of hammers as well, if they are well worn.

You will need to buy a set of overwound bass strings that are designed to fit the particular frame. The original maker's frame number should be enough to identify it to a string supplier. If not, the string makers will tell you how to make a full-size template from the original strings (BEFORE you take the strings off the piano) showing the string lengths, the end positions of the coiled outer winding and the details of the coils, etc.

BTW get some heavy duty gloves and eye protectors. Getting the sharp end of a piece of piano wire stuck in your finger, or in your eye, isn't fun.

 

[rcgldr]

It would be more accurate to say the stress in the strings is roughly the same, not the tension (since the string diameter changes as well as the length).

I only meant the single string that is shared between two notes, not all of the treble strings.