LOUDNESS 411 



condition mentioned above. It is 0.10 instead of zero for AL = — 25, 

 the most probable value of T. For A/ = 100 and Q = 0.88 we will 

 obtain the smallest value of hk without applying the AL factor, namely, 

 0.31. Then when using this factor as given above, all values of bk will 

 be unity for values of AL greater than 12 db. 



Several more complicated functions of AL were tried but none of 

 them gave results showing a better agreement with the experimental 

 values than the function chosen above. 



The formula for calculation of hk then becomes 



hk = [(250 +/, -/.)/1000]10(^'=-^"'>/^()(^. + 30 log/, - 95) (21) 

 where 



fk is the frequency of the component expressed in cycles per second, 



fm is the frequency of the masking component expressed in cycles per 

 second. 



La- is the loudness level of the ^th component when sounding alone, 



Lm is the loudness level of the masking tone, 



(3 is a function depending upon the intensity level /S, and the fre- 

 quency /a of each component and is given in Table VI as a function 

 of X = ^A- + 30 log /a - 95, 



T is the masking and is given by the curve of Fig. 12. 



It is important to remember that hk can never he greater than unity so 

 that all calculated values greater than this must he replaced with values 

 equal to unity. Also all components within the limiting frequency 

 hands must he grouped together as indicated ahove. It is very helpful to 

 remember that any component for which the loudness level is 12 db 

 below the ^th component, that is, the one for which h is being calcu- 

 lated, need not be considered as possibly being the masking com- 

 ponent. If all the components preceding the ^th are in this class then 

 hk is unity. 



Recapitulation 



With these limitations the formula for calculating the loudness level 

 L of a steady complex tone having n components is 



G(L) ^ZhkG{Lk), (10) 



where hk is given by Eq. (21). If the values of /a and jSk are measured 

 directly then corresponding values of La can be found from Fig. 5. 



