Formulæ for Piano String Calculation
Introduction—The purpose of this paper is to provide simple and very exact formulæ for calculating properties of both plain wire and covered strings, for the use of piano technicians. Given any three of the four properties diameter, frequency, length and tension, they provide a means to calculate the fourth. The value of a constant K is determined by the relative density of the string, and a table of values for a full range of densities is given. These equations are not only easy to memorize but are also most suitable for use in spreadsheets, scripts etc. The tension equation can be used on the simplest of pocket calculators and the others on any calculator with a √ (square root) button. No special scientific calculator is required. In addition to the four equations above, which will be explained below, formulæ will also be given showing how these are derived from the basic Mersenne-Taylor formula as well as formulæ for calculating the relative density, and hence the constant K, for covered strings. The French monk and mathematician Marin Mersenne (1588-1648) discovered the following relationships:
Given quantities
The Mersenne-Taylor formula, named after the English mathematician Brook Taylor (1685-1731), to be found in most physics textbooks, to calculate the frequency of a vibrating string is given on the right and can be expressed thus : “The frequency equals 1 divided by twice the length all multiplied by the square root of the tension divided by the mass per unit length”. In this formula t, the tension, is measured in units of force; f, the frequency, in cycles per second and m, the mass per unit length, in units of mass. For practical purposes, and in order to enable calculations to be done using an ordinary electronic calculator, we shall use the CGS (centimetre, gram, second) system of units. As it stands, the formula has no practical use; it is simply the statement of a relationship between four quantities. We shall now proceed to produce from this formula a set of practical formulæ for the piano technician correlating length, diameter and frequency. In what follows the symbol ϱ (the Greek letter rho) symbolizes the relative density of the string considered as a cylinder comprising a cast steel core and, in the case of covered strings, air and the material of the covering wire. The relative density of piano strings ranges from ca. 6.9 for the thickest double-covered strings to ca. 7.4 for the thinnest single-covered string, and for the plain steel strings ca. 7.85. If this seems topsy-turvy in view of the greater density of copper, consider how much air is included in the cylinder you measure. Mass — Let us deal first with m, the mass of the string. This is calculated by the formula on the right, where m is the mass in grams and d the diameter in centimetres. For example the mass per unit length (1cm.) of a steel string 0.1 centimetres thick (mwg. 17½) will be about 0.0652 grams if we take 7.8 as the value of ϱ. We can thus eliminate m, a quantity we cannot practically measure, from our equation and substitute d and l, which we can measure with the micrometer and the ruler. We shall do this after dealing with T, the tension. Tension—In our basic formula T, the tension, is expressed in dynes, the dyne being the unit of force in the CGS system of units we have decided to use. One dyne is the force required to accelerate a mass of 1 gram by one centimetre per second per second and is equivalent to ca. 980.62 gram force, this number, “g45” being the acceleration due to gravity at latitude 45°. We are accustomed to express tension in pounds force or kilograms force, so we need to convert this also to familiar units, and we calculate that one pound force equals ca. 444802 dynes (see above) Having converted the quantities in the original formula to useful units for our purpose, we are now in a position to produce useful and memorable formulæ that can be used to perform the calculations using a pocket calculator or used in spreadsheets. Practical Formulæ1. Calculating tension, given length, diameter and frequencyTo calculate the tension of a given string we use the formula on the left. The constant K will be the same for all the plain wire strings and vary somewhat for the covered strings. It is presumed that if you are using a pocket calculator you will look up the frequency in a table or calculate it beforehand; a formula for calculating the frequency is given below. In this equation n is the number of the note, f_{A49} is the frequency of A-49 (say 440 cps) and f_{n} is the frequency of note n. =fA49 * 2^(1/12)^(n-49) In a spreadsheet with cells named f, l, d and k the following would be entered in the cell that calculates the tension: =(f*l*d)^2/k And on a pocket calculator the following steps will give the result: f x l x d x = ÷ k = The value of K is determined by dividing our conversion factor (pounds force to dynes, 444801.713718088) by πϱ. Since the value of π never changes, K is thus equal to 141584.78286796 divided by ϱ. In the table below are given values for K for the range of values of ϱ that may be encountered. Formulæ will be given later to enable the relative density of a given covered string to be calculated. Values of K for a range of ϱ
2. Calculating diameter, given length, frequency and tensionGiven a known speaking length and frequency, this formula is used to calculate the diameter of the wire needed for a given tension. As an example, take the top note of the piano, C88, having a frequency (at A49=440cps.) of 4186.01 cps. and a length of 51 mm, and suppose that the target tension is 150 lb. Taking 7.8 as the value of ϱ, we have 18152 as our constant K. We multiply this by the tension (150), take the square root of the product and divide this by the frequency and by the length in centimetres (5.1) =sqrt(18152 * 160) / 4186.01 / 5.1 which gives 0.0798 cm. The nearest wire to our target is therefore mwg. 13½ with a diameter of 0.0800 cm. The final tension, using our tension formula above, will be =(4186.01 * 5.1 * 0.08)^2 / 18152 giving us an actual tension of 160.69 lbs. As another example, the speaking length of a single-covered bichord string on note E-20 is 90 cm. and we want a pair of strings at a tension of about 190 lbs. =sqrt(20000 * 190) / 82.41 / 90.0 and get the result 2.63 mm. Supposing a Nº 18 core, we need to make up the diameter using a copper cover of 0.85mm (taking into account a factor for the reduction, due to stretching and ovalization, in the effective diameter of the copper wire). In practice a value of 20,000 for K in the calculations for covered strings is quite adequate as an average figure and one very easy to remember. Although the formulæ have been deduced using very precise data, there is no point in using too much precision in the final calculations, since there is not an infinite number of wire sizes. In the above calculation one could use K=19655 instead of 20000, but the target diameter would still be 2.61 mm, 0.02 mm less than the 2.63 mm. The same copper covering will be required, and two string-makers making the same string would be most unlikely to achieve the same diameter. 3, 4. Calculating length or frequencyThe remaining two formulæ in this set, those to deduce the length and the frequency respectively, are given here without comment, since by now they should be self-explanatory. Some Observations on Covered StringsIt is possible to calculate a precise theoretical value for ϱ and hence K from the diameter of the core and the cover(s), but this is rather a futile exercise. It has been shown above how little difference in the result is made by a significant change in the value of K, and that 20000 is a good working value. In reality the relative density of covered strings ranges from about 6.9 for the thickest double-covered string to about 7.4 for the thinnest single-covered. If you choose a figure between 6.9 and 7.4 from the table equivalent to the position of the string in the bass scale, you will achieve a precision exceeding the call of duty and good craftsmanship. Bear in mind that any theoretically calculated figure for the relative density of covered strings will be inexact for several reasons : a) unless account is taken of the lengths of bare wire at the bridges, an inaccuracy will creep in; b) unless both the steel and the copper are guaranteed to be drawn to precisely the advertised diameter another inaccuracy will occur; c) When the copper is wound onto the steel it loses its roundness and adopts more the shape of an egg with a flat on one side and d) Unless you have precisely weighed samples of both materials under laboratory conditions to determine their relative density, you cannot be sure the values you are using are exactly right. John Delacour © October 2006 |