The number of vibrations performed by a string in a second of time, is evidently the reciprocal value which we have found for the time of one vibration ; so that if N represent the number of vibrations, we shall have this formula : The frequency of vibration which this equation gives, is found to agree very exactly with the result of expe riments performed with strings, whose vibrations are so slow as to admit of being numbered.
The relation between the number of vibrations per formed by different strings, may be expressed by a more simple formula ; for g and the number 2 being both constant quantities, they may in this case be re jected, and we get the following proportional equa Lion ; ,/ . According, then, as we diminish • 1..,/ the length of a string, and the weight of an inch of it, or increase its tension, we increase its frequency of vi bration ; but equal changes in these circumstances do not produce equal effects. Thus, if in different strings, their tension and the weight of an inch remain the same, their frequency of vibration will be inversely as their or n so that if we make the length one third, we triple the number of vibrations: If the length and tension remain the same, n or the number of V et vibrations is inversely proportional to the square roots of the weights of equal lengths of the respective strings ; and if the le ngth and the weight of equal portions be the same, n or the frequency of vibration is as the square roots ufthe tension to which the respective strings are subjected; the effect which each of these circum stances has in increasing the frequency of vibration is exactly proportional to its effect upon the pitch of the string; for it' we diminish the length of a string to one third, it would require the weight of equal lengths of the chord to be diminished to one-ninth, or the force of tension to be increased nine times, to produce an equivalent effect upon its pitch. As there is no other conceivable mode in uhich the action of these circum stances can correspond to the changes they produce in the pitch of a sonorous body, it is impossible to doubt that the frequency of vibration is the cause on which the pitch of sonorous bodies depends.
If, in the beginning of its vibration, a string has any form ABC, wholly si,uated in one plane, and on one side of its axis AC, it follows from theory, and accords with observation, that at the end of a single vibration it will have assumed on the other side of its axis, a form ADC, perfectly similar, but in an inverted position ; so that the portion DC shall be equal and similar to BA, and the portion DA to BC. The chord will consequent
ly, at the end of a double or complete vibration, return to its initial form ABC.
Every musical string is capable of vibrating laterally, in a mode considerably different, from that by which it produces its fundamental sound.
Let a string, AE, have an initial form, AmBnCryDrE, of which equal and similar portions Am13, BnC, DrE, are on different sides ()fits axis, and let these por tions be arranged in such a manner as that, in any two adjacent portions, their extremities, which meet in the point of division between them, shall be similar to each other: Thus, in the portions AmB,and BnC, which meet in the point B, let the extremities mB and nIl be similar ; and in the portions BnC and CyD, which meet in the point C, let ne be similar to Cry, and hB consequently si milar to (D; and let the same law extend to all other portions.
The several points in which the string cuts its axis must remain at rest, and, at the end of a single v ibrdtion, the string will have assumed the form AiBeCeiDaE si milar to its initial form inverted ; and, at the end of a complete or double vibration, it will have returned to its original position. For if we conceive the points, B, C, D, to be fixed in their present position, by means of pins, then it is evident that, as de se equal and similar por tions, AmB, 137/C, COD, DrE, begin their vibration at the same instant of time, and in similar circumstances, the changes which they may have sustained at any mo ment of time during the vibration will be exactly the same. They will consequently remain similar through out the vibration, and at each instant of time solicit the points of division between them with forces which are equal, and in contrary directions; these points of divi sion therefore will remain at rest, though the pins by which they are fixed should be removed. The sounds which a string titi, s when %ibrating in this manner, arc called its harmonics: the points of the 1:tring whit. h remain at rest ace named zyliruhtpn nodrN, or point8 ; and the vibrating portions intercepted between them are denominated tit Hirs or Avis.