which produce a large amount of disturbance give birth to secondary waves, which appeal to the ear as resultant tones. Having proved this, Helmholtz inferred further that there are also resultant tones formed by the sum of the primaries, as well as by their difference. He thus discovered his summation tones before he had heard them; and bringing his result to the test of experiment, he found that these summation tones have a real physical existence. They are not to be explained by Young's theory, but they receive a complete explanation by that of Helmholtz. Another consequence of this departure from the law of superposition is, that a single sounding body, which disturbs the air beyond the limits of the law of the superposition of vibrations, also produces secondary waves, which correspond to the harmonic tones of the vibrating body. For example, the rate of vibration of the first overtone of a tuning-fork, as stated in our fourth lecture, is 6 times the rate of the fundamental tone. But Helmholtz distinctly proves that a tuning-fork, not excited by a bow, but vigorously struck against a pad, emits the octave of its fundamental note, this octave being due to the secondary waves set up when the limits of the law of superposition have been exceeded. These considerations make it probably evident to you that a coalescence of musical sounds is a far more complicated dynamical condition than you have hitherto supposed it to be. In the music of an orchestra, not only have we the fundamental tones of every pipe and of every string, but we have the overtones of each, sometimes audible as far as the sixteenth in the series. We have also resultant tones; both difference tones and summation tones; all trembling through the same air, all knocking at the self-same tympanic membrane. We have fundamental tone interfering with fundamental tone; we have overtone interfering with overtone; we have resultant tone inter THEORY OF HELMHOLTZ. 283 fering with resultant tone. And besides this, we have the members of each class interfering with the members of every other class. The imagination retires baffled from any attempt to realise the physical condition of the atmosphere through which those sounds are passing. And, as we shall learn in our next lecture, the aim of music, through the centuries during which it has ministered to the pleasure of man, has been to arrange matters empirically, so that the ear shall not suffer from the discordance produced by this multitudinous interference. The musicians engaged in this work knew nothing of the physical facts and principles involved in their efforts; they knew no more about it than the inventors of gunpowder knew about the law of atomic proportions. They tried and tried till they obtained a satisfactory result, and now, when the scientific mind is brought to bear upon the subject, order is seen rising through the confusion, and the results of pure empiricism are found to be in harmony with natural law. SUMMARY OF LECTURE VII. When several systems of waves proceeding from distinct centres of disturbance pass through water or air, the motion of every particle is the algebraic sum of the several motions impressed upon it. In the case of water, when the crests of one system of waves coincide with the crests of another system: higher waves will be the result of the coalescence of the two systems. But when the crests of one system coincide with the sinuses, or furrows, of the other system, the two systems, in whole or in part, destroy each other. This mutual destruction of two systems of waves is called interference. If in two The same remarks apply to sonorous waves. systems of sonorous waves condensation coincides with condensation, and rarefaction with rarefaction, the sound produced by such coincidence is louder than that produced by either system taken singly. But if the condensations of the one system coincide with the rarefactions of the other, a destruction, total or partial, of both systems is the consequence. Thus, when two organ-pipes of the same pitch are placed near each other on the same wind-chest and thrown into vibration, they so influence each other, that as the air enters the embouchure of the one it quits that of the other. At the moment, therefore, the one pipe produces a condensation the other produces a rarefaction. sounds of two such pipes mutually destroy each other. When two musical sounds of nearly the same pitch are sounded together the flow of the sound is disturbed by beats. The These beats are due to the alternate coincidence and interference of the two systems of sonorous waves. If the two sounds be of the same intensity their coincidence produces a sound of four times the intensity of either; while their interference produces absolute silence. The effect, then, of two such sounds, in combination, is a series of shocks, which we have called 'beats,' separated from each other by a series of 'pauses.' The rate at which the beats succeed each other is equal to the difference between the two rates of vibration. When a bell or disc sounds, the vibrations on opposite sides of the same nodal line partially neutralise each other; when a tuning-fork sounds the vibrations of its two prongs in part neutralise each other. By cutting off a portion of the vibrations in these cases the sound may be intensified. When a luminous beam, reflected on to a screen from two tuning-forks producing beats, is acted upon by the vibrations of both, the intermittence of the sound is announced by the alternate lengthening and shortening of the band of light upon the screen. The law of the superposition of vibrations above enunciated is strictly true only when the amplitudes are exceedingly small. When the disturbance of the air by a sounding body is so violent that the law no longer holds good, secondary waves are formed which correspond to the harmonic tones of the sounding body. When two tones are rendered so intense as to exceed the limits of the law of superposition, their secondary waves combine to produce resultant tones. Resultant tones are of two kinds; the one class corresponding to rates of vibration equal to the difference of the rates of the two primaries; the other class corresponding to rates of vibration equal to the sum of the two primaries. The former are called difference tones, the latter summation tones. LECTURE VIII. ΤΟ COMBINATION OF MUSICAL SOUNDS-THE SMALLER THE TWO NUMBERS WHICH EXPRESS THE RATIO OF THEIR RATES OF VIBRATION, THE MORE PERFECT IS THE HARMONY OF TWO SOUNDS-NOTIONS OF THE PYTHAGOREANS REGARDING MUSICAL CONSONANCE-EULER'S THEORY OF CONSONANCE-PHYSICAL ANALYSIS OF THE QUESTION-THEORY OF HELMHOLTZ-DISSONANCE DUE BEATS-INTERFERENCE OF PRIMARY TONES AND OF OVERTONES GRAPHIC REPRESENTATION OF CONSONANCE AND DISSONANCE-MUSICAL CHORDSTHE DIATONIC SCALE-OPTICAL ILLUSTRATION OF MUSICAL INTERVALS— LISSAJOUS' FIGURES-SYMPATHETIC VIBRATIONS-MECHANISM OF HEARING --SCHULTZE'S BRISTLES-THE OTOLITES--CORTI'S FIBRES-CONCLUSION. THE HE subject of this day's lecture has two sides, the one physical, the other æsthetical. We have this day to study the question of musical consonance-to examine musical sounds in definite combination with each other; and to unfold the reason why some combinations are pleasant and others unpleasant to the ear. Here are two tuning-forks mounted on their resonant cases. I draw a fiddle-bow across them in succession: they are now sounding together, and their united notes reach your ears as the note of a single fork. Each of these forks executes 256 vibrations in a second. Two musical sounds flow thus together in a perfectly blended stream, and produce this perfect unison when the ratio of their vibra tions is as 1 : 1. Here are two other forks, which I cause to sound by the passage of the bow. These two notes also blend sweetly and harmoniously together. By means of our syren I have already determined the rates of vibration of the forks, and found that this large one executes 256 vibrations a second, while the small one executes 512. For every single wave, therefore, sent to the ear by the one fork two waves are sent by the other. I need not tell the musicians |