338 DR. koenig's rksearches on 



periodic function, however complex, could be analyzed out and ex- 

 pressed as the sum of a certain series of periodic functions having 

 frequencies related to that of the fundamental or first number of the 

 series, as the simple numbers 2, 3. 4, 5, etc. Thirty years later, G. S. 

 Ohm suggested that the human ear actually performs such an analysis, 

 by virture of its mechanical structures, upon every complex sound of a 

 periodic character, resolvingit into afundamental tone, the octave of that 

 tone, the twelfth, the double octave, etc. Von Helmholtz, arming him- 

 self with a series of tuned resonators, sought to pick up and recognize 

 as members of a Fourier series the higher harmonics of the tones of 

 various instruments. In his researches he goes over the ground pre- 

 viously traversed by Rameau, Smith, and Young, who had all observed 

 the co-existence, in the tones of musical instruments, of higher partial 

 tones. These higher tones correspond to higher modes of vibration in 

 which the vibratile organ — string, reel, or air column — subdivides into 

 two, three, four, or more parts. Such parts naturally possess greater 

 frequency of vibration, and their higher tones, when they co-exist along 

 with the lower or fundamental tone, are denominated upper imrtial tones, 

 thereby signifying that they are higher in the scale and that they cor- 

 respond to vibrations i/i2^«ris. It is to be regretted that Professor 

 Tyndall, in his lectures on sound, rendered von Helmholtz's Oberpar- 

 iialtone by the term overtones, omitting the most significant half of the 

 word. To avoid all confusion in the use of such a term I shall rather 

 follow Dr. Kwnig in speaking of these as son7ids of subdivision. And 

 I must protest emphatically against calling these sounds harmonics, 

 for the simple reason that in many cases they are very inharmonious. 

 It is a matter to which I shall recur ])resently. 



Keturning to the subject of beats, the question arises, Wliat becomes 

 of the beats when they occur so rapidly that they cease to produce a 

 discontinuous sensation upon the ear? The view which I have to put 

 before you i:i the name of Dr. Kcenig is that they blend to make a tone 

 of their own. Earlier acousticians have propounded, in accordance 

 with this view, that the grave harmonic of Tartini (a sound whicli cor- 

 responds to a frequency of vibration that is the difference between 

 those of the two tones producing it) is due to this cause. Von Helmholtz 

 has taken a different view, denying tliat the beats can blend to form 

 a sound, giving reasons presently to be examined. Von Helmholtz 

 considered that he had discovered a new species of combinational 

 tone, namely, one corresponding in frequency to tlie sum of the fre- 

 quencies of the two tones, whereas that discovered by Tartini (and be- 

 fore him by Sorge) corresponded to their difference. Accordingly, he 

 includes under the term of combinational tones the differential tone of 

 Tartini and the summational tone which he considered himself to have 

 discovered. To the existence of such combinational tones he ascribed 

 a very important part in determining the character, harmonious or 

 otherwise, of cords; and to them also he attributes the ability of the 



