1890.] on the Physical Foundation of Music, 225 



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

 von Helmholtz does in his great work, that all timbres possess a 

 purely periodic character ; with the necessary corollary that all 

 timbres consist 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 Helmholtz constructs 

 a series of resonators, accurately tuned to correspond to the terms of 

 a Fourier-series (the first being tuned to some fundamental tone, the 

 second to one of a frequency exactly twice as great, the third to a 

 frequency exactly three times, and so forth), and applies such resona- 

 tors to analyse the timbres of various musical and vocal sounds, he 

 is trying to make Nature fit to an ideal system which Nature does 

 not herself follow. He is trying to make his resonators pick up 

 things which in many cases do not exist — upper partial tones which 

 are exact harmonics. If they are not exact harmonics, even though 

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

 them imperfectly, and he is thereby led into an erroneous appreciation 

 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 repro- 

 duce the supposed efifects. The failure is inherent in the instrument ; 

 for it cannot reproduce those natural timbres which do not fall 

 within the circumscribed limits of its imposed mathematical principle. 

 Nature does not sort men out into rigidly defined sets, one set exactly 

 four feet high, another set exactly five feet high, another exactly six 

 feet high. Neither does she, in the vibrations of strings, reeds, and 

 air-columns impose rigid mathematical relations between the funda- 

 mental notes and the sounds of subdivision, though in many cases 

 such mathematical relations are approximately attained. Harmony 

 depends, beyond contest, on the approximate fulfilment of exact 

 mathematical relations, and it is the grand achievement of von Helm- 

 holtz to have shown us why this is so. But the question of timbre 

 involves the more subtle question of the minuter details of vibration 

 by virtue of which the sound of a notfe in one instrument differs from 

 that of the same note in an instrument of another kind, and depends 

 therefore on the mechanism of the small vibrating parts. In these 

 matters of delicate detail the natural departures from mathematical 

 relations assert themselves. He who neglects these departures, or 

 tries to square them to his preconceived theory, misses one of their 

 most important characteristics, and can only render an imperfect 

 account of them. 



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

 particularly in those of such instruments as the harp and the piano- 

 forte, in which the impulse, once given, is not sustained, the relations 



Vol. XIII. (No. 84.) q 



