LOUDNESS 403 



than one of lower pitch. For example, in most cases a tone which is 

 100 cycles higher than the masking tone would be masked when it is 

 reduced 25 db below the level of the masking tone, whereas a tone 100 

 cycles lower in frequency will be masked only when it is reduced from 

 40 to 60 db below the level of the masking tone. It will therefore be 

 assumed that the neighboring component on the side of lower pitch 

 which causes the greatest masking will account for all the reduction 

 in bk-. Designating this component with the subscript m, meaning 

 the masking component, then we have bk expressed as a function of 

 the following variables. 



bu = B(f,,U,S,,S„,), (15) 



where / is the frequency and 5 is the level above threshold. For the 

 case when the level of the ^th component is T db below the level of 

 the masking component, where T is just sufficient for the component to 

 be masked, then the value of b would be equal to zero. Also, it is 

 reasonable to assume that when the masking component is at a level 

 somewhat less than T db below the ^th component, the latter will 

 have a value of bk which is unity. It is thus seen that the fundamental 

 of a series of tones will always have a value of bk equal to unity. 



For the case when the masking component and the ^th component 

 have the same loudness, the function representing bk will be con- 

 siderably simplified, particularly if it were also found to be independent 

 of /a- and only dependent upon the difference between /t and fm- From 

 the theory of hearing one would expect that this would be approxi- 

 mately true for the following reasons: 



The distance in millimeters between the positions of maximum 

 response on the basilar membrane for the two components is more 

 nearly proportional to dilTerences in pitch than to differences in fre- 

 quency. However, the peaks are sharpest in the high frequency 

 regions where the distances on the basilar membrane for a given A/ 

 are smallest. Also, in the low frequency region where the distances 

 for a given A/ are largest, these peaks are broadest. These two 

 factors tend to make the interference between two components having 

 a fixed difference in frequency approximately the same regardless of 

 their position on the frequency scale. However, it would be extra- 

 ordinary if these two factors just balanced. To test this point three 

 complex tones having ten components with a common Af of 50 cycles 

 were tested for loudness. The first had frequencies of 50-100-150- • • 

 500, the second 1400-1450- - - 1900, and the third 3400-3450- - -3900. 

 The results of these tests are shown in Fig. 8. The abscissae give the 

 loudness level of each component and the ordinates the measured loud- 



