n86 



THE EAR. 



pillars of Corti lengthen ; that from base to apex both inner and 

 outer hair cells increase in size ; that from base to apex the hairs 

 increase in length, while the increase is greater in the hairs of the outer 

 cells ; and that the membrana tectoria increases in breadth from base to 

 apex. 1 Further, if we take man, the cat, and the rabbit, all of which 

 presumably hear tones, but probably have very different powers of 

 discrimination of tones, we find remarkable differences in detail, as given 

 in the following table by Ketzius : 2 — 



Upper partial tones and the theory of dissonance. — We have seen 

 that a compound tone is built up of a number of partial tones, the 

 frequency numbers of which are multiples of that of the first or funda- 

 mental tone. It has also been shown that when two tones sufficiently 

 near in pitch are simultaneously sounded, beats are produced. If such 

 beats are few in number, so as to be readily counted, the sensation of 

 waxing and waning is not disagreeable ; but if they are sufficiently 

 numerous it may be impossible to count them, and the sensation is 

 disagreeable. Such a sensation is dissonance. The sensation is most 

 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 vibra- 

 tions required to produce the sensation of a tone, the sensation is never 

 uniform, but is of a rough intermittent character. In other words, the 

 sensation of a tone cannot be produced by the intermittent impulses on 

 the drum-head, due to beats. If now we sound an interval on an instru- 

 ment giving forth compound tones, such, for example, as an octave, each 

 tone will have its corresponding partials ; and as these come closer and 

 closer together 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 

 on low tones, and contract for intervals on high ones. 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. As a rule, the partials up to the seventh are beyond beating 

 distance, but above this they soon come close together. In the neigh- 

 bourhood of the tenth, the interval may be about a tone, of the sixteenth a 

 semitone, and still higher they come so close together as to cause disson- 

 ance. This fact explains the harsh but brilliant quality of some of the 

 tones of a trumpet. Suppose that one of the tones of the interval is slightly 

 out of tune, then all the partials will be correspondingly affected, and the 



1 M'Kendrick, op. tit., p. 782. 



2 Retzius, " Das Gehororgan der Wirbelthiere, " Bd. ii. S. 356. 



