352 DR. kcenig's researches on 



vibration is precisely four times as rapid as the fuiidainental vibration 

 of the strinjif as a whole. This would be true if the string were abso- 

 lutely uniform, homogeneous, and devoid of rigidity. Strings never 

 are so; and even if uniform and homogeneous, seeing that the rigidity 

 of a string has the eftect of making a short piece stitfer in proportion 

 than a long piece, can not emit true harmonics as the sounds of subdi- 

 vision. In horns and open organ pipes the width of the column (which 

 is usually neglected in simple calculations) affects the freiiucncy of the 

 nodal modes of vibration. Wertheini found the partial tones of pipes 

 higher than the supposed harmonics. 



These things being so, it is manifestly insufficient to assume, as von 

 Helmholtz does in his great work, that all timbres i)ossess a purely 

 periodic character ; with the necessary corollary that all timbres con- 

 sist merely in the presence, with greater or less intensity, of one or more 

 members of a series of higher tones corresponding to the terms of a 

 Fourier series of harmonics. When, therefore, following ideas based 

 on this assumption, von Ilelmholtz constructs a series of resonators, 

 accurately tuned to corresi)ond to the terms of a Fourier series (the 

 first l)eing tuned to some fundamental tone, the second to one of a fre- 

 quency exactly twice as great, the third to a frequency exactly three 

 times, and so forth), and applies such resonators to analyze the tim- 

 bres of various musical and vocal sounds, he is trying to make his reson- 

 ators pick up things which in many cases do not exist — upper partial 

 tones which are exact harmonics. If they are not exact harmonics, 

 even thougli they exist, his tuned resonator does not hear them, or only 

 hears them imperfectly, and he is thereby lead into an erroneous appre- 

 ciation of the sound under examination. 



Further, when in pursuance of this dominant idea he constructs a 

 system of electro-magnetic tuning-forks, accurately tuned to give forth 

 the true mathematical harmonics of a fixed series, thinking therewith 

 to reproduce artificially the timbres not only of the various musical 

 instruments but even of the vowel sounds, he fails to reproduce the 

 supposed effects. The failure is inherent in the instnimcnt ; for it can 

 not rei)roduce those natural timbres which do not fall within the cir- 

 cumscribed limits of its imposed mathematical principle. 



Nothing is more certain than that in the tones of instruments, partic- 

 ularly in those of such instruments as the harp and the pianoforte, in 

 which the impulse, once given, is not sustained, the relations between 

 the component partial tones are continually changing, both in relative 

 intensity and in phase. Tlie wavelets, as they follow one another, are 

 ever changing their forms; in other words, the motions are not truly 

 periodic — their main forms may recur, but with modifications ever 

 changing. 



To estimate the part played in such phenomena by mere differences 

 of phase — to evaluate, in fact, the influence of i>liase of the constitu- 

 cuts upon the integral effect of a compound sound — Dr. Kojnig had 



