70 DYNAMICAL THEOKY OF SOUND 



and so varies inversely as the length and as the square root of 

 the line-density, and directly as the square root of the tension. 

 These statements, which were formulated as the result of 

 experiment long before the mathematical theory had been 

 developed, are known as Mersenne's laws*. The determination 

 of absolute pitch by the formula (5) does not admit of very 

 great accuracy owing to the difficulty in measuring the tension, 

 which is apt (owing to friction) to be slightly different on the 

 two sides of a bridge. 



The principles that the frequency diminishes with increase 

 of length and with increase of line-density have a familiar 

 illustration in the pianoforte, where longer and intrinsically 

 heavier strings are used for the graver notes. If the relation 

 of pitch were adjusted by length alone the strings corresponding 

 to the lower notes would have to be at least 100 times as long 

 as those belonging to the highest. In order to secure a suffi- 

 ciently low pitch within practical limits of length, and with 

 a sufficient degree of tension, the string is loaded with a coil of 

 wire wrapped closely round it. This has the effect of increasing 

 the inertia without seriously impairing the flexibility, which is 

 an essential point. The influence of tension, again, is illustrated 

 in the process of tuning, which consists in tightening up the 

 wires when these have stretched, or the pegs have yielded, so 

 that the instrument has fallen in pitch, or become " flat." 



In the next normal mode after the fundamental the middle 

 point x \l is at rest (Fig. 29). And in the 5th mode, whose 

 frequency is by (3) s times that of the fundamental, there are 

 s 1 internal points of rest, or " nodes," in addition to the 

 ends. Midway between these we have the points of maximum 

 amplitude, or " loops." Each segment into which the string is 

 divided by the nodes vibrates as in the fundamental mode of a 

 string of 1/sth the length. 



As already stated ( 2) the sequence of simple vibrations 

 with frequencies proportional to the natural numbers 1,2,3,-..., 

 which we here meet with, has important properties, musically 



* M. Mersenne (15881648), a Franciscan friar, was a schoolfellow and 

 lifelong friend of Descartes, and maintained an extensive correspondence with 

 him and other men of science of the day. 



