Wire Size Calculations
By Craig Brougher
I will try once more to answer Doug about the doubts he expressed regarding the ability of some bass strings to be able to place the partials more equidistant along the string.
In my last letter, I said "This changes the partials and ideally puts them more equidistant along the string. This, I believe, favors the even harmonics, strengthening them over the odd harmonics." Doug took exception and "challenged" the statement. So I wrote another letter and in it I said; "As far as "evenly spaced harmonics" are concerned, that is a physical necessity but a very short description. Whether I say "evenly spaced" or "more perfectly harmonic" is immaterial. To the degree that you have inharmonicity is the same degree that your harmonics are NOT evenly distributed along the string." Then Doug said, "I was in fact suggesting that one of your statements might be _invalid_, and was in need, at the very least, of greater elucidation. As one of our great Canadian politicians remarked once, if you can't stand the heat, stay out of the kitchen."
I thought about it and decided to try just one more time. Now please don't feel like I am condescending when I ask you to consider the very basic physical laws governing vibrating strings, or pictures of the same.
A mathematical model, as you suggested, would be strictly to impress somebody because until they studied it for weeks, they wouldn't understand it unless they were in that field of theoretical math. Besides, I know what kind of work that takes and doubt that the effort is worth it, since you don't really know what you have after you get through, except that it is indefinable. Why build a model to clarify something that 1) changes with time, temperature, and contamination too rapidly, 2) Is different every time you wrap a string, even for the same scale, 3) wouldn't mean anything after you finished because the inharmonicity and spurious parasitic vibration within the string colors the tone so differently for every pair of ears listening to it and the final result is so subjective that your "model" is moot?
So the way I am going to explain it is using some physical facts and the premise that the "cello-like tones" of a Mason & Hamlin are due primarily to a very strong second partial in the bass strings.
On pg 37 of the Piano Servicing, Tuning, and Rebuilding manual by Art Reblitz, you see a string generating four partials. This represents dynamic equilibrium with 15 stationary anti-nodes. If this wire was perfectly elastic, its shear modulus would be 1. That means, S=1=F/A/o(psi) where F/A=shearing stress/shearing strain. Assume tan psi= psi. If F=A, then S=1 and the wire has divided into perfectly harmonic partials because the nodes occupy some "dl" or theoretically infinitesimal length. These perfect harmonics are equally spaced along the string because of one reason-- the nodes don't take up any real string length. Unfortunately, the modulus of steel wire is around 12 x 10 to the 6th power/sq. in. That is a measure of stiffness and requires that each node occupy a physically measurable space on the string in order to bend the wire at those points (transverse shear moment). So you can see then that the heavier the wire per unit length, the more energy it takes to bend it, and the greater becomes the shear force to do so, so likewise the longer become the little inactive "domains" along the wire which are not vibrating, but acting as weight and adding length only to the fundamental pitch. Thus its partials are more inharmonic (sharp) because they are much shorter than they would be, had the string more elasticity per unit length.
This is not my opinion. This is a mathematically proven fact, and is also so stated in at least a dozen books on piano tuning. We have established so far that the mass, length, and tension of a string can be optimized or compromised, depending on what is at stake. That is why big concert grands have more pure partials than baby grands. Their bass strings are much thinner relative to their length, so their stiffness isn't as crucial a factor, simply due to their size.
Most experts agree that Mason & Hamlin has what is known as a "cello-like" tone. That means strong second partials, and good purity of that partial, in particular. The problem is that the second partial can be muddied up by complex odd harmonics, parasitics, and overly inharmonic higher partials. The solution is to play with the combination of core diameter, copper wire wrapping, length, and tension to optimize the scale and clean up the partials. Any time you have dead domains or sectors of the string on each end of a node, and you have dozens of interactive nodes representing many odd and even harmonics on a string, then you have overlapping partials. It is inarguable. It's an ipso-facto argument.
Lets say you are to lay six lines of sticks across a road, end for end. You must exactly fit them between two curbs so that their ends touch each other, but now you have a problem. Each time you snip a stick in two, it puts a little nib on each end where you snipped it. So if you snipped the second stick in two, it is too long by two nibs. So overlap them. The next stick you snip into four pieces, but that one is really too long, this time by 6 nibs, so to place them between the curbs, you have to overlap them in three places. The next stick overlaps will be seven, the next fifteen, and so forth. Now roll all the sticks tight together and paint just the overlaps white so you can see them when you stand and sight across the sticks sideways. Do you have overlapping sticks? Yes, you do. Why? Because they are too long when the nibs are added to fit end for end. Would you say, if you named your sticks "Partials" that your partials were overlapping because of the extra unwanted unusable nibs created as a necessary result of shear? Yes, you would.
Now let's get another stick and do the same thing in the same roadway. The difference is going to be the material the stick is made from. This material is going to be less elastic, more brittle, and perhaps a bit thinner. (IN the case of piano strings, we'll just say it is a different scale string). As you snip (shear) the stick, you notice that the nib is much less, so the overlap is much less, and the amount of white paint you use is practically nil. Now when you get off and look at those sticks, the overlaps don't show very much because they are smaller. Would you say also that those sticks are more equidistant? Yes, because they do not overlap. They appear to the eye to be laid in a very precision fashion, compared to the ones with large and indiscrete nibs. When a partial overlaps another partial it really doesn't have an exact place to be because of the ambiguity of the overlapping lengths of the nodes. You could place the majority of those sticks anywhere , just as long as they were overlapping the nibs somewhere in a plus or minus range. That is also what real partials do on a thick string. Their "domains" include a front and back porch, so to speak. SO they wander.
Now has anyone ever seen this effect? I believe they have. If I recall, this was one of the interesting high-speed color photographs made back in the 60's. But we don't have to watch it. We can hear it. We know that the second partial is pure when we hear it clearly and strongly. And when you get a piano that can solo clear down to the last bass string, and its tone is still fathomable and discrete, then you know you are working with a scale with a stronger second partial. When you don't hear so much "mud" then you know the partials on that string are not interfering as much with one-another.
I realize this doesn't begin to describe everything that's going on, and when a string is struck, you no longer have dynamic equilibrium. Longitudinal and transverse waves are reflecting back and forth in both the copper winding and the steel core, damping and reinforcing each other and hundreds of partials and even sub-partials. Young's modulus of copper is vastly different from steel and wavefronts are clashing and creating new energy waves momentarily that are entirely different and irrespective of the fundamental harmonic. For example, consider just one very simple relationship: u=(Y/p) (sq. Rt). That means, the velocity of a wave front is directly proportional to Young's modulus divided by the wavelength. So given the same wavelength and the difference between the "Y" of copper and steel (16 as compared to 29), you should be able to see how the velocity of a longitudinal wave will quickly be spit up as soon as it travels through the steel and the copper at different speeds and then reflects back. Then that sub harmonic which is split up has to go somewhere. It is vectored energy. It must then either resonate and reinforce something, or dampen and oppose something, and it does both. Each time you hit the string with a hammer, you do it a little different, and the different ratio of power in the partials quickly changes the entire coloration of the string. It is basically like awakening New York City momentarily, then turning out the lights randomly one by one.
I hope this answers your question, because I don't know any other way to put it.
Craig B.
craig_brougher@msn.com
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(Message sent Thu 2 May 1996, 04:29:18 GMT, from time zone GMT.) |
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