Januabt 29, 1909] 



SCIENCE 



163 



pieties and variations; speech employs 

 these very qualitative varieties to distin- 

 guish the letters and syllables. 



When we inquire as to the cause of tone 

 quality, since pitch depends upon fre- 

 quency and loudness upon amplitude, we 

 conclude that quality must depend upon 

 the only remaining property of a periodic 

 vibration, namely, the peculiar kind or 

 form of the motion; or, if we represent 

 the vibration by a curve or wave line, the 

 quality is dependent upon the peculiarities 

 represented by the shape of the wave. 

 There is possible an endless variety of 

 motion for the production of sound, and 

 quality is, therefore, almost infinitely com- 

 plicated in its causes, as compared with 

 the other two properties of sounds. 



There can not be a simpler mode of vi- 

 bration than that known as simple har- 

 monic motion, which is represented by the 

 wave line called the sine curve; such mo- 

 tion is often referred to as pendular 

 motion. Tuning forks properly con- 

 structed and mounted on resonance boxes 

 are shown by analysis to produce vibra- 

 tions in the air which are single simple 

 harmonic motions; the resulting tones are 

 called simple tones, and their sensation is 

 markedly simple and pure. If several 

 simple tones of different pitches, as from 

 several tuning forks, are simultaneously 

 sounded, they simultaneously excite differ- 

 ent systems of waves, which exist as varia- 

 tions in density of the air; the resulting 

 displacements, velocities, and changes in 

 density of the air are each equal to the 

 algebraic sum of the corresponding dis- 

 placements, velocities and changes in den- 

 sity which each system of waves would 

 have separately produced had it acted in- 

 dependently. There must, therefore, be 

 peculiarities in the motion of a single 

 particle of air which differ for a single 

 tone and for a combination of tones; and 

 in fact the kind of motion during any 



one period is entirely arbitrary, and may 

 indeed be infinitely various. 



The method by which the ear proceeds 

 in its analysis of tone quality was first 

 definitely stated by Ohm, in Ohm's law of 

 acoustics. Helmholtz states this law in 

 the following forms: 



All musical tones, however complex or peculiar 

 in quality, are periodic; the human ear perceives 

 pendular vibrations alone as simple tones, and it 

 resolves all other periodic motions of the air into 

 a series of pendular vibrations, hearing the series 

 of simple tones which correspond to these simple 

 vibrations. 



Another rendering of this law is : 



Every motion of the air which corresponds to a 

 composite mass of musical tones is capable of 

 being analyzed into a sum of simple pendular 

 vibrations, and to each such simple vibration 

 corresponds a simple tone, sensible to the ear, and 

 having a pitch determined by the periodic time 

 of the corresponding motion of the air. 



The separate component tones are called 

 partial tones, or simply partials; that par- 

 tial having the lowest frequency is the 

 fundamental, while the others are over- 

 tones. However, it sometimes happens 

 that a partial not the lowest in frequency 

 is so predominant as to give the main char- 

 acter to the whole sound, and it may be 

 mistaken for the fundamental. If the 

 overtones have frequencies which are 

 exact multiples of the frequency of the 

 fundamental, they are often called har- 

 monics; otherwise they may be designated 

 as inharmonic partials. 



In stating his law, in 1843, Ohm says: 

 Fourier spread light in our darkness when he 

 brought out (in 1822) his work "La Th€orie 

 Analytique de la Chaleur," and so enabled the- 

 oretical mechanics to solve the most difficult prob- 

 lems of physics with unparalleled ease. 



Fourier had shown in a purely mathe- 

 matical way, and with no idea of acoustical 

 application, that any given regular peri- 

 odic function can always be expanded in 

 a trigonometric series of sines and cosines, 

 and for each case in one single way only. 



