LAWS OF SONOROUS VIBRATIONS. 833 



its want of harmony with the 5th excludes it from the common chord. The 1st, 4th, 

 and 8th are harmonious, but to make a complete chord we must use the 6th. These dis- 

 cussions might be extended into the progression of chords and modulation ; but this is 

 not essential to our purpose, as we wish only to ascertain the laws of the vibrations of 

 sounds in harmony and the mechanism of discords. 



Discords. A knowledge of the mechanism of simple accords enables us to understand 

 more easily the rationale of discords, and vice versa. As in the case of harmony, the fact 

 that certain combinations of musical tones produce a disagreeable impression was ascer- 

 ained empirically, with no knowledge of the exact cause of the palpable dissonance. 

 Thanks to the labors of modern physicists, however, the mechanism of discords is now 

 pretty well settled. We shall, in our explanation, begin with a combination of tones 

 slightly removed from perfect unison. 



Suppose, for example, that we have two tuning-forks giving precisely the same num- 

 bers of vibrations in a second ; the tones are then in perfect unison. We load one of the 

 forks with a bit of wax, so that its vibrations are slightly reduced, and start them both 

 in vibration at the same instant. Taking the illustration given by Tyndall, we assume 

 that one fork has 256, and the other, 255 vibrations in a second. While these two forks 

 are vibrating, we have one gradually gaining upon the other ; but, at the end of half a 

 second, one will have made 128 vibrations, while the other will have made 127. At 

 this point the two waves are in direct opposition to each other ; they are moving in 

 exactly opposite directions ; and, as a consequence, the sounds neutralize each other, and 

 we have an instant of silence. The perfect sounds, as the two forks continue to vibrate, 

 are thus alternately reenforced and diminished, and we have what are known in music as 

 beats. As the difference in the number of vibrations in a second is one, we have the 

 instants of silence occurring once in a second ; and in this illustration the beats occur 

 once a second. Unison takes place when two sounds can follow each other indefinitely, 

 their waves blending perfectly ; dissonance is marked by successive beats, or pulses. If 

 we now load forks so that one will vibrate 240 times in a second, and the other 234, there 

 will be six times in a second when the interference will be manifest ; or, to make it 

 plainer, in of a second, one fork will make 40 vibrations, while the other is making 

 39. We shall then have 6 beats in a second. From these experiments, the law may 

 be deduced, that the number of beats produced by two tones not in harmony is equal 

 to the difference between the two rates of vibration. An analogous interference of un- 

 dulations is observed in optics, when waves of light are made to interfere and produce 

 darkness. 



It is evident that the number of beats will increase as we sound two discordant tones 

 higher and higher in the scale. According to Helmholtz, the beats can be recognized up 

 to 132 in a second. Beyond that point they become confused, and we have only a sen- 

 sation of dissonance, or roughness. We can illustrate this point very satisfactorily by a 

 simple experiment upon the piano. Let us take two tones, the highest on the scale, 

 separated from each other by a semitone. When we strike these two notes together, 

 we have a disagreeable sensation of dissonance, but no appreciable beats, because, the 

 rate of vibration of each note being high, the difference is great and the beats are too 

 rapid to be appreciated as such. We strike, now, the two notes an octave below ; still 

 we have dissonance, less disagreeable, but no beats. Passing down, an octave at a time, as 

 the numbers of vibrations become smaller, the difference between them is less, and there 

 are fewer beats in a second, until they are readily appreciated as beats and can even be 

 counted. Beats, then, are due to interference of sound-waves, and the number in a second 

 is equal to the difference in the rate of vibrations. When these are too rapid to be appre- 

 ciated as beats, we have simply a sensation of discord. There is no interference of the 

 waves of tones in unison, provided the waves start at the same instant ; the intensity of 

 the sound being increased by reinforcement. The differences between the 1st and 8th, 

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