OPTICAL TUNING.

931 and then back again; but the curves obtained in returning will be inverted.

923. Optical Tuning. By the aid of these principles, tuning-forks can be compared with a standard fork with much greater precision than would be attainable by ear. Fig. 635 represents a convenient

arrangement for this purpose. A lens f is attached to one of the prongs of a standard fork, which vibrates in a horizontal plane; and above it is fixed an eye-piece g, the combination of the two being equivalent to a microscope. The fork to be compared is placed upright beneath, and vibrates in a vertical plane, the end of one prong being in the focus of the microscope. A bright point m, produced by making a little scratch on the end of the prong with a diamond, is observed through the microscope, and is illuminated, if necessary, by converging a beam of light upon it through the lens c. When the forks are set vibrating, the bright point is seen as a luminous ellipse, whose permanence of form is a test of the closeness of the unison. The ellipse will go through a complete cycle of changes in the time required for one fork to gain a complete vibration on the other.

924. Other Modes of producing Lissajous' Figures.-An arrangement devised in 1844 by Professor Blackburn, of Glasgow, then a student at Cambridge, affords a very easy mode of obtaining, by a slow motion, the same series of curves which, in the above arrangements, are obtained by a motion too quick for the eye to follow. A cord ABC (Fig. 636) is fastened at A and C, leaving more or less

TY

Fig. 636.-Blackburn's Pendulum.

slack, according to the curves which it is desired to obtain; and to any intermediate point B of the cord another string is tied, carrying at its lower end a heavy body D to serve as pendulumbob.

If, when the system is in equilibrium, the bob is drawn aside in the plane of ABC and let go, it will execute vibrations in that plane, the point B remaining stationary, so that the length of the pendulum is BD. If, on the other hand, it be drawn aside in a plane perpendicular to the plane A B C, it will vibrate in this perpendicular plane, carrying the whole of the string with it in its motion, so that the length of the pendulum is the distance of the bob from the point E, in which the straight line AC is cut by DB produced. The frequencies of vibration in the two cases will be inversely as the square roots of the pendulum-lengths BD, ED.

If the bob is drawn aside in any other direction, it will not vibrate in one plane, but will perform movements compounded of the two independent modes of vibration just described, and will thus describe curves identical with Lissajous'. If the ratio of ED to BD is nearly equal to unity, as in the left-hand figure, we shall have curves corresponding to approximate unison. If it be approximately 4, as in the right-hand figure, we shall obtain the curves of the octave. Traces of the curves can be obtained by employing for the bob a

CHARACTER OR TIMBRE.

933

vessel containing sand, which runs out through a funnel-shaped opening at the bottom.1

The curves can also be exhibited by fixing a straight elastic rod at one end, and causing the other end to vibrate transversely. This was the earliest known method of obtaining them. If the flexural rigidity of the rod is precisely the same for all transverse directions, the vibrations will be executed in one plane; but if there be any inequality in this respect, there will be two mutually perpendicular directions possessing the same properties as the two principal directions of vibration in Blackburn's pendulum. A small bright metal knob is usually fixed on the vibrating extremity to render its path visible. The instrument constructed for this mode of exhibiting the figures is called a kaleidophone. In its best form (devised by Professor Barrett) the upper and lower halves of the rod (which is vertical) are flat pieces of steel, with their planes at right angles, and a stand is provided for clamping the lower piece at any point of its length that may be desired, so as to obtain any required combination.

925. Character.-Character or timbre, which we have already defined in § 889, must of necessity depend on the form of the vibration of the aerial particles by which sound is transmitted, the word form being used in the metaphorical sense there explained, for in the literal sense the form is always a straight line. When the changes of density are represented by ordinates of a curve, as in Fig. 603, the form of this curve is what is meant by the form of vibration.

The subject of timbre has been very thoroughly investigated in recent years by Helmholtz; and the results at which he has arrived. are now generally accepted as correct.

The first essential of a musical note is, that the aerial movements which constitute it shall be strictly periodic; that is to say, that each vibration shall be exactly like its successor, or at all events, that, if there be any deviation from strict periodicity, it shall be so gradual as not to produce sensible dissimilarity between several consecutive vibrations of the same particle.

There is scarcely any proposition more important in its application

1 Mr. Hubert Airy has obtained very beautiful traces by attaching a glass pen to the bob (see Nature, Aug. 17 and Sept. 7, 1871), and in Tisley's harmonograph the same result is obtained by means of two pendulums, one of which moves the paper and the other the pen.

to modern physical investigations than the following mathematical theorem, which was discovered by Fourier:-Any periodic vibration executed in one line can be definitely resolved into simple vibrations, of which one has the same frequency as the given vibration, and the others have frequencies 2, 3, 4, 5. . . times as great, no fractional multiples being admissible. The theorem may be briefly expressed by saying that every periodic vibration consists of a fundamental simple vibration and its harmonics.

We cannot but associate this mathematical law with the experimental fact, that a trained ear can detect the presence of harmonics in all but the very simplest musical notes. The analysis which Fourier's theorem indicates, appears to be actually performed by the auditory apparatus.

The constitution of a periodic vibration may be said to be known if we know the ratios of the amplitudes of the simple vibrations which compose it; and in like manner the constitution of a sound may be said to be known if we know the relative intensities of the different elementary tones which compose it.

Helmholtz infers from his experiments that the character of a musical note depends upon its constitution as thus defined; and that, while change of intensity in any of the components produces a modification of character, change of phase has no influence upon it whatever. Sir W. Thomson, in a paper "On Beats of Imperfect Harmonies,"1 adduces strong evidence to show that change of phase has, in some cases at least, an influence on character.

The harmonics which are present in a note, usually find their origin in the vibrations of the musical instrument itself. In the case of stringed instruments, for example, along with the vibration of the string as a whole, a number of segmental vibrations are simultaneously going on. Fig. 637 represents curves obtained by the composition of the fundamental mode of vibration with another an octave higher. The broken lines indicate the forms which the string would assume if yielding only its fundamental note. The continuous lines in the first and third figures are forms which a string may assume in its two positions of greatest displacement, when yielding the octave along with the fundamental, the time required for the

1 Proc. R. S. E. 1878.

2

2 The form of a string vibrating so as to give only one tone (whether fundamental or harmonic) is a curve of sines, all its ordinates increasing or diminishing in the same proportion, as the string moves.

string to pass from one of these positions to the other being the same as the time in which each of its two segments moves across and back

again. The second and fourth figures must in like manner be taken together, as representing a pair of extreme positions. The number of harmonics thus yielded by a pianoforte wire is usually some four or five; and a still larger number are yielded by the strings of a violin. The notes emitted from wide organ-pipes with flute mouth-pieces are very deficient in harmonics. This defect is remedied by combining with each of the larger pipes a series of smaller pipes,1 each yielding one of its harmonics. An ordinary listener hears only one note, of the same pitch as the fundamental, but much richer in character than that which the fundamental pipe yields alone. A trained ear can recognize the individual harmonics in this case as in any other.

1 The stops called open diapason and stop diapason (consisting respectively of open and stopped pipes), give the fundamental tone, almost free from harmonics. The stop absurdly called principal gives the second tone, that is the octave above the fundamental. The stops called twelfth and fifteenth give the third and fourth tones, which are a twelfth (octave+fifth), and a fifteenth (double octave) above the fundamental. The fifth, sixth, and eighth tones are combined to form the stop called mixture.

As many of our readers will be unacquainted with the structure of organs, it may be desirable to state that an organ contains a number of complete instruments, each consisting of several octaves of pipes. Each of these complete instruments is called a stop, and is brought into use at the pleasure of the organist by pulling out a slide, by means of a knob, handle, on which the name of the stop is marked. To throw it out of use, he pushes in the slide. A large number of stops are often in use at once.