BIS AURAL LOCATION OF COMPLEX SOUNDS 39 



combination may be anixcd at in which the algebraic sum of all the 

 components is zero for all instants before and after the period occupied 

 by the sound and equal to the instantaneous value of the sound 

 wave for instants within that period. This combination is known to 

 mathematicians as the Fourier Integral corresponding to the wave, 

 and the formula for the phase and amplitude of each component 

 sinusoid is known. It is an extension of the well known Fourier 

 series expansion used for resolving sustained periodic disturbances. 



The physical interpretation of this integral may be facilitated by 

 reviewing the steps in its evolution from the Fourier series. It is 

 well known that if the sound in question were repeated at regular 

 intervals the resulting periodic wave could be resolved by Fourier 

 analysis into a series of sinusoidal components, the frequencies of all 

 of which are integral multiples of the frequency of repetition of the 

 sound. Successive components therefore differ in frequency by an 

 amount equal to this frequency of repetition. Now it is not essential 

 that the repetitions of the sound follow each other immediately. 

 Instead, they may be separated by intervals of silence. The effect 

 of such silent intervals is to reduce the frequency of repetition and 

 therefore also the fundamental frequency. As a result the com- 

 ponent frequencies are brought closer together and the number within 

 any particular frequency range is increased. 



Suppose now that the interval between repetitions is indefinitely 

 increased. As this is done the efTect of any one occurrence of the 

 sound becomes more and more independent of the others, and in the 

 limit when the sounds next preceding and next following the one 

 under consideration are infinitely far removed, we have the case of a 

 discrete sound. As this limiting case is approached the fundamental 

 frequency becomes smaller and smaller and the component frequen- 

 cies, which are multiples of it, are separated by infinitesimal frequency 

 differences. While the amplitude of each component also decreases, 

 the number of components increases at such a rate that the aggregate 

 energy of all the components within a given frequency range remains 

 finite. In this way, the distribution of the sound energy over various 

 frequencies — that is, the "energy spectrum" — can be obtained. 



It is evident, then, that when an aperiodic complex sound is resolved 

 mathematically there results an infinity of component tones, each 

 having a characteristic intensity and phase. If an observer were 

 capable of an equally complete resolution he would have at his dis- 

 posal an infinity of sets of data from which an infinity of images 

 could be formed. In the absence of distortion these should all co- 

 incide. 



