338 DR. KCENIG's RKSEARCIIES UN 



periodic function, liowever coini)lex, could be analyzed out and ex- 

 jn-essed as the sum of a certain seiies of periodic functions liavinj^ 

 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 a fundamental tone, the octave of that 

 tone, the tweltth, the ilouble octave, etc. Von Ilebnholtz, 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 Kameau, Smith, and Young, who had all observed 

 the coexistence, in the tones of musical instruments, of higher partial 

 tones. Tliese 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 part ial tones, 

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

 respond to vibrations in ^Mris. It is to be regretted that Professor 

 Tyndall, in his lectures on sound, rendered von llebnholtz's Ohcrpar- 

 tialtone 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. Kd'uig in speaking of these as sounds 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 jjresently. 



Returning to the subject of beats, the question arises. What 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. KoMiig is that they blend to make a tone 

 of their own. Earlier acousticians have propounded, in accordance* 

 with this view, that the f/rarc harmonic of Tartini (a sound which cor> 

 responds to a fretpiency of vibration that is the difierence between 

 those of the two tones producingit) isdueto thiscause. Von llelmholtz; 

 Las taken a different view, denying th;it the beats can blend to forni 

 a sound, giving reasons i)resently to be examined. Von Helmholtz; 

 considered that he had discovered a new s[)ecies of combinational 

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

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

 fore him by Sorge) corres[)onded to their difference. Accordingly, he 

 includes under the term of combinational tones tlie differential tone of 

 Tartini and the summational tone which he considered himself to have 

 discovered. To tin' existence of such combinational tones he ascribed 

 a very imi)ortant part in determining the character, harmonious or 

 olheiwise, of cords j and to them also he attributes the ability of the 



