164 



SCIENCE 



[N. S. Vol. XXIX. No. 735 



Eacli sine or cosine term in the series may 

 be considered as representing a single 

 vibration; then in Fourier's series, the suc- 

 cessive terms have frequencies which are 

 exact multiples of the first, but the ampli- 

 tudes and phase differences are arbitrary 

 and can always be found in every given 

 ease, by peculiar methods of calculation 

 which Fourier has shown. 



So far as Fourier's theorem is concerned 

 this method of analyzing sound vibrations 

 might be merely a mathematical form, not 

 necessarily having any corresponding 

 actual meaning in the sounds themselves. 

 Moreover, in actual musical sounds, many 

 of the important partials are not exact 

 multiples of the fundamental in frequency, 

 that is, they are inharmonic, and with these 

 Fourier's theorem has nothing whatever to 

 do, although Ohm's law still applies to 

 them. 



Helmholtz fully demonstrates that the 

 ear unaided can thus analyze tones ; he also 

 developed several methods for assisting the 

 ear, chiefly by the use of resonators. His 

 monumental work, " Tonenempfindungen, " 

 was referred to by our chairman last year 

 as 



produced by a masterful knowledge of physiology, 

 physios and mathematics, and a scholar's knowl- 

 edge of the literature of music, remarkable for its 

 breadth, completeness and wealth of detail. 



A large part of this work is concerned 

 with the demonstrations of Ohm 's law that 

 the quality of a musical sound is dependent 

 upon the particular combination of partial 

 tones which make up the sound under ex- 

 amination. He held that 



Differences in musical quality of tone depend 

 solely on the presence and strength of partial 

 tones, and in no respect on the difference in phase 

 under which these partial tones enter into com- 

 position. 



A few historical references may be inter- 

 esting as showing how clearly these ideas 

 were perceived by the earliest investigators. 



Descartes (1618) says: 



No musical soimd can be heard which does not 

 appear to the ear to be accompanied by the octave 

 above it. 



Mersenne (1636) says of Aristotle: 

 He seems to have been ignorant of the fact that 

 every string produces five or more different sounds 

 at the same time, the strongest of which is called 

 the natural tone of the string, and alone is accus- 

 tomed to be taken notice of, for the others are so 

 feeble that they are perceptible only to delicate 

 ears. 



Perrault (1680) says: 



Every noise, although apparently simple, is in 

 effect a system and an assemblage of an infinity 

 of partial noises that compose a total, in which 

 no confusion is remarked on account of the affinity 

 that all these partial noises have together. 



Sauver (1702) remarks that 

 The organ only imitates by the combination of 

 its stops the natural harmony of sonorous bodies. 



La Grange and Bernoulli (1760) both 

 state clearly the cause of quality: 



The same single sonorous ray may be moved at 

 the same time by many species of vibrations which 

 do not interfere with each other in any manner; 

 in the place of a node with regard to one species, 

 a segment may be formed with regard to another. 



Monge (1800) says that 



Quality is due to the order and number of the 

 vibrations of the aliquot parts of a string, and if 

 the vibrations of these aliquot parts could be sup- 

 pressed, all strings, of whatever material, would 

 yield tones of the same quality. 



Young (1800), describing his experi- 

 ments for rendering visible the vibrations 

 of a string by means of a ray of light, says : 



According to the various modes of applying the 

 bow an immense variety of orbits are produced; 

 more than enough to account for all the differences 

 of tone by different performers. 



Biot (1817) says: 



All sonorous bodies yield simultaneously an 

 infinite number of sounds of gradually decreasing 

 intensity, but the law for the series of harmonics 

 •is different for bodies of different forms; it is this 

 difference which produces the particular character 

 of sound called timbre. And may not the quality 

 of each particular substance, wood or metal, for 



