It is hardly necessary to remark, that the whole seine; performs its vibration in the satin• time with any of its loops ; and that the time occupied in perferming a com plete vibration, will be Panel by considering any of the loops as a separate string fixed at both extremities, and applying to it the formula given above for the fundamen tal vibration of strings. This upplication may he made in the following manner: Let L be the length ()I' the whole string, b the number of bellies or loops, then L di% ided b, 01 each loop, which we muy sub stitute in the formulae instead of L; by substitutinn we transform these formula:. into the following: b h T= G1, n=, g t 2w L V Iv These equations are very general in their application, and evidently include those given for a string vibrating in its fundamental mode ; For in that case h becomes and totally disappears from the expressions. In com paring the number of vibrations performed in a given time, by a string emitting its different sounds, the quan tities, L, g, t, and ev, becoming constant, may be re jected, and we get this proportional equation, or the number of vibrations performed by a given string in a second of time, is proportional to the number of loops in which the string vibrates : the harmonic sounds of a chord should consequently be more acute than its fundamental sound.
The complicated form which a chord assumes when giving its harmonics, would lead us to suppose, that such sounds could hardly CA er be excited ; but this is by no means the case, for harmonic sounds may be easily produced by drawing a bow across the string AE, and lightly touching the point of division D. In some mu sical instruments these are the only sounds employed; and in all such cases we can, by putting bits of paper on the string, prove that the points of division are at rest while the intervening portions are in motion.
A string has, in the foregoing remarks, been consi dered as producing, at a given time, but one simple. sound. This, however, is seldom the case, as sonorous bodies, at the same time that they produce their funda mental, produce also one or more of its harmonics. For such an accompaniment, it appears at first sight difficult to assign a proper reason, as that vibration, which theory attributes to the chord, scents fitted to produce only one sound.
Philosophers have accordingly. in general, been dis posed to attribute the production of these harmonies, to something external to the vibrating string; some sup posing them produced in the transmission of the funda mental to the car; sonic conceiving that they arise nom the peculiar structure of that organ ; others, as La Cl range, referring them to sympathetic vibrations in the different bodies adjacent to the string. These different opinions. however, are untenable ; for some bodies, such as a drink ing glass, when excited by rubbing a moistened finger along its edge, or an jEolian lyre, when acted on be the wind, give exactly the same notes with a vibrating string, but nnaccompanied with harmonics. Dr Tho mas Young has also found, that el en in the same chord it is not universally true, that the fundamental sound must always be accompanied by all the harmonics of which the chord is susceptible ; lur that by inflecting the chord exactly at any point in winch thr 1:110111 may be till ided into a mummer of equal parts, and then sill feting it to vibrate, we lose the effect of the correspond ing harmonic- The just inkrence from this is, that the of the iiihuamental in conjunction with its harmonics, depends neither upon any thing in the trans mission of the sound, nor upon the peculiar structure of the ear, nor upon any agitation of the surrounding bo dies, hut rather upon We Manlier in which the suing vi brates. One of the simplest modes in which we can
conceive harmonics, produced by the peculiar Manner in which the string vibrates, is by supposing (agreeably to the theory Daniel Bernoulli, which we have already mentioned) that while time whole of the string, ABCDE, vibrates on each side of its axis AE, producing its fun damental sound, it serves as a moveable axis to partial vibrations of its aliquot parts AB, BC, CD, DE, in the same plane with it, producing the corresponding harmo nic. That a string can, in some measure, assume such a mode of vibration, is shown by professor Robison ; who, in some experiments with the covered string of a vio lincello, sounding by the friction of an ivory wheel, found, that if he put something soft, such as a lock of cotton, in the way of the wide vibrations of' the chord, at. one-third and two-thirds of its length, so as to disturb them when they became very wide, the string instantly put on an appearance something similar to Fig. 3, per forming at once the full vibration competent to its whole length, and the three subordinate vibrations, correspond ing to one-third of its length, and sounding the hinds mental and the 12th with equal strength. In this man ner all the different accompaniments were produced at pleasure." These experiments show, that harmonics may be pro duced in the manner supposed by Bernoulli; but to de termine whether this be the usual mode of their produc tion, by a string vibrating freely, requires that we trace the actual motion of the string. Dr Thomas Young is, we believe, the only philosopher who has attempted this with any success. He observed, by a microscopic in spection of any luminous point on the surface of a chord, for instance the reflection of a candle in the coil of a fine wire wound round it, that the vibration of a chord de viates from the plane of its first direction, and becomes a rotation or revolution, which may be considered as composed of various vibrations in different planes; and that besides these vibrations of the whole chord, it is also frequently agitated by subordinate vibrations, which constitute harmonic notes oldifferent kinds. It is to be much regretted, that the other avocations of that inge nious gentleman have prevented his prosecuting these observations, so as to refer each separate harmonic to the particular subordinate vibrations on which it de pends, and trace the peculiar mode of division which the chord sustains, in assuming each particular vibra tion. Yet we conceive that we do not transgress the due bounds of philosophic caution, in considering these observations, combined with the experiments of profes sor Robison, as sufficient proof that the harmonics which accompany the fundamental sound of a string, arc occa sioned by the fundamental vibration of the string being combined w ith subordinate vibrations of its aliquot parts, either in the same or in different planes. There arc several sources from which these partial vibrations may supposed to originate, such as inequalities in the thickness of the chord, or in the density or flexibility of its different parts, and also the particular mode of exci tation which has been employed in producing the fun damental sound. All these causes are probably com bined in the production of the effect; but we know little respecting the particularoperation of each of them, and still less concerning the general result of their combined action.
Surfaces in a state of tension, such as the top of a drum, the tambourine, &c. may perhaps be considered as included under this division of the sources of musical sounds. Their mode of vibration, however, is little un derstood, and, we conceive, not very interesting.