We have already mentioned, that the physical pro perties by which bodies vibrate arc not always the smile ; sonic vibrating in consequence of their cohesion ; others from a strong repulsion, which the particles exert on each other, such as the different kinds of air ; and a third class, such as the metals, being capable of vibrating, by either of these forces, separately, or by the combined action of both.
It is convenient to arrange the sonorous bodies, which produce musical sounds, rather according to the powers by which they actually vibrate at the time when under our consideration, dividing them into the three following classes,—those which vibrate by cohesion alone,—those which vibrate by repulsion,—acid those which vibrate by the combined action ol both. According to this mode of din ision, the same body may successively appear under each of these different classes. But as its mode of vibra tion and of sounding follows different laws, according to the division in which for the time it appears, we may consider it in each as a different sounding body. The first class includes all bodies vibrating by tension, such as musical strings, when vibrating laterally ; the second class includes wind instruments, and the longitudinal vibrations of rods, strings, &c. ; and the third class com prehends the lateral vibrations of elastic rods, bells, plates, rings, cylinders, Ste.
A musical string is of an uniform thickness, and stretched between two points, by a force much greater than its weight. The stretching force which is applied, is generally conceived as measured by the weight, w hick would occasion an equal tension. In the usual mode of exciting a musical string, it vibrates on each side of its quiescent position, the extremities being the only points of the string which remain at rest. The sound which the string gives in this mode of vibration is called its funda mental sound.
The pitch of the fundamental sound of musical strings is found by experience to depend on three circum stances ; the length of the string, the weight of a given portion of it, and the force of tension to which it is sub jected. The tone becomes more acute as we increase their tension, or diminish their length, and the weight of a given portion. Thus the diminution of the length of strings, and of the weight of equal portions, produce the same effect upon their pitch as if we had increased the force of tension. if strings, therefore, differ from each other in arty two of these circumstances, we can, by a pro per adjustment of the third, produce from them sounds whose pitch will be the same, or which shall differ in any degree we choose. On this fact depend, for the most
part, the various modes of producing the several musical sounds in stringed instruments.—These circumstances also affect the time occupied by the vibration of an uni form string.
Let AF13 vibrate between the points G and K, we call its motion in one direction, from G to K, a single vibra tion ; and its motion in reuniting from K to G, another single vibration ; and these two motions, which it per forms between the time when it leaves G and returns to the same point, arc, when taken together, called a dou ble vibration.
It has been demonstrated, that the time of a double vibration, expressed in parts of a second of time, will be found by the following operation : Multiply the number of inches described by a falling bony in a seem d of time, that is 193 nearly, by the weight which is equal to the foi cc of tension ; and, by this pronuct, divide the weight of two inches of the string, exLract the smiure root of the quotient, and multiply. the root thus mound by the length of the string in ihches ; the yy ,i1 be the time of a double ibration expressed in parts el a second of time.
The same thing may be expressed more by an algebraic lOrmula. Let L represent the length of the string in inches ; the weight of an inch of the string ; r, a wr ight equivalent to the force of tension ; the iiumber of inches through which a body falls in a second of time, by the action of gravity ; and T, the time of a double vibration expressed in seconds. Then As the distance of the string from its quiescent posi tion does not i1)1111 an element of the algebraic expres sion, which is thus found for the time of a vibration, it follows, that this time is independent of the distance, and that a string performs eat', of its vibrations in equal times, whether in these vibrations its excursions on each side of the axis be great or small. So long, then, as the string continuc s vibrating in the manner which pro duces its fundamental sound, its vibrations will be iso ehronous. Upon this isochroni.m depends the unifor mity- of its tone ; for, it we employ a string of unequal thickness, and whose vibrations are consequently per formed in different times, the sound which we procure is confused and variable ; and any other mode by which we destroy the isochronism produces a similar effect. The same law has been found to extend to the other cases of musical sounds being produced by vibration ; and therefore we may conclude, that isochronism, in the vibrations of sonorous bodies, is essential to their pro ducing musical sounds.