LAWS OF SONOROUS VIBRATIONS. 747 



when the interference will be manifest ; or in other words in of a second, 

 one fork will make 40 vibrations, while the other is making 39. This will 

 give 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 undulations is observed in optics, when waves of light are made to inter- 

 fere and produce darkness. 



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

 notes are produced 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 there is only a general sensation of dissonance. Beats, 

 then, are due to interference of sound-waves. There is no interference of the 

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

 intensity of the sound being increased by re-enforcement. The differences 

 between the 1st and 8th, the 1st and 5th, the 1st and 3d, and other harmo- 

 nious combinations, is so great that there are no beats and no discord, the 

 more rapid waves re-enforcing the harmonics of the primary sound. It is 

 important to remember in this connection, that resultant tones are equal to 

 the difference in the rates of vibration of two harmonious tones. Taking a 

 note of 240 vibrations, and its 5th, with 360 vibrations, these two have a 

 difference of 120, which is the lower octave of the 1st and is an harmonious 

 tone. 



It is evident that the laws just stated are applicable to overtones, resultant 

 tones and additional tones, which, like the primary notes, have their beats 

 and dissonances. 



Tones by Influence. After what has been stated in regard to the laws of 

 musical vibrations, it will be easy to comprehend the production of sounds 

 by influence. If a key of the piano be lightly touched, so as to raise the 

 damper but not to sound the string, and then a note be sung in unison, the 

 string will return the sound, by the influence of the sound-waves of the voice. 

 The sound thus produced by the string will have its fundamental tone and 

 overtones ; but the series of overtones will be complete, for none of the nodes 

 are abolished, as in striking or plucking a string at any particular point. If 

 instead of a note in unison, any of the octaves be sounded, the string will re- 

 turn the exact note sung ; and the same is true of the 3d, 5th etc. If. a 

 chord in harmony with the undamped string be struck, this chord will be 

 exactly returned by influence. In other words, a string may be made to 

 sound by influence, its fundamental tone, its harmonics and harmonious com- 

 binations. To carry the observation still farther, the string will return, not 

 only a note of its exact pitch and its harmonics, but notes of the peculiar 

 quality of the primary note. This is a very important point in its applica- 

 tions to the physiology of hearing and can be readily illustrated. Taking 

 identical notes in succession, produced by the voice, trumpet, violin, clarinet 

 or any other musical instrument, it can easily be noted that the quality of the 

 note, as well as the pitch, is rendered by a resounding string ; and the same 

 is true of combinations of notes. These laws of tones by influence have been 



