Text: The first law: When a string fixed at each end like the piano string, is struck at one end, it vibrates in a complex form, most strongly in its full length but also perceptibly in segments of that length such as 1/2, 1/3, 1/4, and 1/5. The second law is equally important. It may be stated as follows: Length and Frequency: The frequency of a string - that is to say, the number of vibrations per unit of time (c.p.s.) which it can perform, is proportional inversely to its length. Thus, since an octave above a given tone has twice as many cycles per second as the original tone, it follows that to obtain a sound an octave above a given sound we must have a string one-half as long. Weight, Tension and Thickness: Similar laws exist with regard to the influence upon string vibrations, of weight, tension and thickness. Without undertaking to prove these completely, we may state them briefly as follows: Weight: The frequency of a string's vibration is inversely proportional to the square root of its weight. In other words, if the weight be divided by 4 (the square of 2) the frequency will be multiplied by 2. To produce a tone one octave above the original tone the weight of the string must be only 1/4 its original weight. Tension: The frequency of string vibrations is directly proportional to the square root of their tension. In other words, to get twice as many cycles per second you must multiply the tension by 2^2=4. To get 4 times as many cycles per second you must multiply the tension by 4^2=16. So if a string be stretched with a weight of 10 lbs. and it is desired to make it sound an octave higher, this can be done by making the stretching weight (4 X 10 =) 40 lbs. Thickness: The frequency of string vibrations is inversely proportional to the thickness of the string. If a string of a given length and weight produces a sound of a given frequency, a string of the same length and of twice the thickness will give a sound one octave lower; that is, of half the frequency. Mechanical Variable Factors: All these laws, be it remembered, are based upon the assumption of mathematical strings, in which weight and stiffness remain constant throughout all changes in length. In the case of the actual piano string, in which the weight and tension do vary with the length, some compensation must therefore be made. Thus, to illustrate, it is found that whereas the acoustical law of frequency requires a doubling of the string length at each octave downwards, or halving at each octave upwards, the practical string, where weight and tension vary with length, is better served by a proportion of 1:1.875 or 1:1.9375, instead of 1:2. This difference must be kept in mind by designers of "scales" for pianos. [NOTE: We have also to consider volume, density and pressure as influenses on frequency. DP 11/18/03] [Piano Tuning and Allied Arts, White, 1938]

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