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THEORY OF HELMIIOLTZ.

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ing with resultant tone. And besides this, we have the embers of each class interfering with the members of ery other class. The imagination retires baffled from y attempt to realise the physical condition of the atmohere through which those sounds are passing. And, as

shall learn in our next lecture, the aim of music, rough the centuries during which it has ministered to e pleasure of man, has been to arrange matters empially, so that the ear shall not suffer from the disrdance produced by this multitudinous interference. he musicians engaged in this work knew nothing of the ysical facts and principles involved in their efforts; ey knew no more about it than the inventors of gunwder knew about the law of atomic proportions. They ied and tried till they obtained a satisfactory result, and ow, when the scientific mind is brought to bear upon the bject, order is seen rising through the confusion, and the sults of pure empiricism are found to be in harmony ith 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

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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.

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 TO BEATS INTERFERENCE OF PRIMARY TONES AND OF OVERTONES GRAPHIC REPRESENTATION OF CONSONANCE AND DISSONANCE-MUSICAL CHORDSTHE DIATONIC SCALE-OPTICAL ILLUSTRATION OF MUSICAL INTERVALSLISSAJOUS' 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 vibrations 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

COMBINATION OF MUSICAL SOUNDS.

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present that the combination of sounds now heard is that of a fundamental note and its octave; nor need I tell them that, next to the perfect unison, these two notes, the ratio of whose vibrations is as 1 : 2, blend most harmoniously together.

I now throw another pair of forks into contemporaneous vibration. The combination of the two sounds is very pleasing to the ear, but the consonance is hardly so perfect as in the last instance. There is a barely perceptible roughness here, which is absent when a note and its octave are sounded. The roughness, however, is too insignificant to render the combination of the two notes anything but agreeable. I look to the numbers stamped upon these two forks, and find that one of them executes 256 and the other 384 vibrations in a second. These two numbers are to each other in the ratio of 2:3; one of the forks, therefore, sends two waves and the other three waves to the ear in the same interval of time. The musicians present know that the two notes now sounding are separated from each other by the musical interval called a fifth. Next to the octave this is the most pleasing combination.

I once more change the forks, and sound two others simultaneously. The combination is still agreeable, but not so agreeable as the last. The roughness there incipient is here a little more pronounced. One of these forks executes 384 vibrations, the other 512 vibrations, in a second; the two numbers standing to each other in the ratio of 34. This interval the musicians present recognise as a fourth. Thus, then, with perfect unison the ratio of the vibrations is as 1: 1; with a note and its octave it is 1:2; with a note and its fifth it is 2:3; and with a note and its fourth it is 3: 4. We here notice the gradual development of the remarkable law that the combination of two notes is the more pleasing to the ear,

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