----- Original Message ----- From: "James Ellis" <claviers@nxs.net> To: <caut@ptg.org> Sent: Tuesday, April 13, 2004 4:48 PM Subject: A 440 Hz Standard > I'm not sure who said what in this latest exchange. However, whoever it > was who said the piano will go flat when the lights come on, is right. The > strings of the piano will quickly go slightly flat when the hot light beams > hit them, because they warm up rapidly, but the plate does not. The plate > is a big heat sink. Three or four hours later, if the piano stays under > the lights that long, it will be back up to pitch when the plate warms up. > The thermal coefficients of cast iron and steel are similar, but not > identical. It will take longer for the lights to dry the board out and the > piano to go flat again. Playing against this, you have the fact that wind > instruments go about 1.7 cents sharp for every degree F temperature rise, > but the orchestra's strings go flat. So the orchestra re-tunes, but the > piano just sits there. More often than not, the piano gets rolled out on > stage either after the first number, or after the intermission. I don't > need to say more. You know what happens. [snip] > Jim Ellis Hi Jim: Having done extensive research into this phenomenon, I just thought I might tell you guys a little about what I know. You might find it interesting or useful. The effects that produce the change in pitch due to a change in temperature are all elementary physics/mechanics, but there are a number of these elementary effects taking place simultaneously, and the piano's design varies across the gamut, so it's not quite as cut and dried as you might imagine. An increase in temperature causes most materials to expand, so as you might expect, piano strings expand and get flatter in pitch as they get warmer. But the amount of expansion depends on the length of the string--longer strings will expand a longer total distance than shorter ones for the same given rise in temperature. What remains constant is the *percent* change, or inches of expansion per inch of string length. Since the strings in the piano are many different lengths, they will expand by different amounts. But hold on; it's still not that simple. Since the harp is much more rigid than a string, the length of the string will remain virtually unchanged no matter how much it would have expanded or contracted if free. Instead, it changes in tension. Let's say we have a string that thermal contraction would have made shrink by, say, .06" when free. If we hold the ends rigid and don't let the string shrink, the tension will increase just as if we pulled it .06" longer. In other words, its as if we let it contract and then pulled it back to its original size. What is actually more important than the inches of contraction is the "strain". Strain is the *percent* elongation per unit length (just like with the thermal expansion). Knowing the strain we can determine the "stress", or the tension per unit crossectional area in psi if we know the "modulus of elasticity" of the material (which we can look up in a table). If we know the diameter of a string we can calculate its crossectional area and thus determine its tension. Since both thermal expansion and strain depend on string length, it cancels out, and now we get the same change in tension for a given change in temperature regardless of the length of the string. A residual effect of this is that the segments of string between the agraffes, tuning pins, string rests and hitch pin will all change in tension by the same amount, so there will be no differences in tension across friction members to be equalized. Piece of cake, right? Well, now we must remember that the diameter of wire in a piano changes as we go down. This won't affect thermal expansion, so it won't affect the strain, so it won't affect the stress. But now when we convert from stress to actual tension, a thicker string will have more tension for a given stress since it is spread over a larger area. So temperature affects the tension more in heavier strings than in lighter ones. So now we've got it nailed, huh? The pitch produced by a given string, assuming its diameter and speaking length remain constant, is proportional to the square root of its tension. So the relation is not linear. For example, increasing the tension of a string 10% will not increase its vibrating frequency by 10%. Warmer still means "flatter", but the two are just not in constant proportion with each other. And remember that bass strings are wound with copper, which changes their mass per unit length, but not their tensile properties, further complicating matters. Now, on top of all that, recall that music is based on ratios, not differences in frequency. For example to tune a fifth (3:2) above A-110 would require 110 x 3/2 = 165 Hz, or about 55 Hz higher. But for A-440 a fifth would be 440 x 3/2 = 660 Hz, or about 220 Hz higher. That's four times as much, even though the musical interval sounds the same. So different strings require different changes in frequency and thus different changes in temperature for the same change in perceived pitch (in cents). I have found that the harp warming up has virtually no effect on the pitches. The change in tension due to the expansion/contraction of the harp is divided among over 200 strings that are under from 100 to 200 lbs of tension each, so it gets lost in the shuffle. The direct effect of heat on the strings themselves far overshadows the miniscule effects of the harp. Also remember that the effect of stage lights on the strings and harp will not "even out". It is not the same as putting a piano in a room of a given temperature. The amount that an object will heat as a result of being under radiative light depends on its absorption and the angle of incident light. That's why a dull, black, flat object will get hotter than a shiny, silver, cylindrical one under the same light. So you can see that a piano shouldn't behave according to a simple "cents per degree F" formula across the board. Having said that, through experimentation (and calculation) I have found that the thermal sensitivity of piano strings doesn't really vary that much. When all of these factors are added together the strings will all vary by about 1 or 2 cents per degree farenheit, believe it or not! Don A. Gilmore Mechanical Engineer Kansas City
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