HELMHOLTZ IN BONN 



disagreeable when the ear is affected by about 33 

 beats per second ; if they are more numerous, the 

 sensation is rough and unpleasant. Further, even 

 when the frequency of beats is much greater than 

 the number of vibrations required to produce the 

 sensation of a tone, the sensation is never uniform, 

 but is of a rough, intermittent character. 



If now we sound an interval on an instrument 

 giving forth compound tones, such, for example, 

 as an octave, each note will have its corresponding 

 partials ; and as these come closer and closer to- 

 gether the higher they are in the series, it is 

 clear that they may come within beating distance, 

 and thus give a certain harshness to the sound. 

 The beating distance may, for tones of medium pitch, 

 be roughly fixed at a minor third ; this interval, 

 of course, will expand for intervals in low, and 

 contract for intervals in high ranges of the 

 scale. Thus, the same interval in the lower 

 part of the scale may give slow beats that are 

 not disagreeable, while in the higher part it 

 may cause harsh and unpleasant dissonance. Two 

 given notes will produce this l beating ' harshness 

 when the difference of their vibration number is 

 about 70 or 80. Thus a minor third will sound 

 pleasant in the higher ranges of the scale. There 

 will be a slight roughness at medium pitches ; while 

 in the lower ranges there will be harshness and 

 possibly perceptible beating. On the other hand, if 

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