LOUDNESS 409 



value of bk = 0.26. These three values must belong to the same 

 condition Af = 10. It is evident then that the formulae for b given by 

 Eq. (17) will lead to very erroneous results for such components. 



In order to cover such cases it was necessary to group together all 

 components within a certain frequency band and treat them as a single 

 component. Since there was no definite criterion for determining 

 accurately what these limiting bands should be, several were tried and 

 ones selected which gave the best agreement between computed and 

 observed results. The following band widths were finally chosen : 



For frequencies below 2000 cycles, the band width is 100 cycles; for 

 frequencies between 2000 and 4000 cycles, the band width is 200 

 cycles; for frequencies between 4000 and 8000 cycles, the band width 

 is 400 cycles; and for frequencies between 8000 and 16,000 cycles, the 

 band width is 800 cycles. If there are k components within one of 

 these limiting bands, the intensity / taken for the equivalent single 

 frequency component is given by 



I = Zlk = Z 10^*/^". (18) 



A frequency must be assigned to the combination. It seems reasonable 

 to assign a w^eighted value of / given by the equation 



/ = Zf>ch/i = Z A-io^^/^VZ 10^/1°. (19) 



Only a small error will be introduced if the mid-frequency of such 

 bands be taken as the frequency of an equivalent component except for 

 the band of lowest frequency. Below 125 cycles it is important that 

 the frequency and intensity of each component be known, since in 

 this region the loudness level Lk changes very rapidly with both changes 

 in intensity and frequency. However, if the intensity for this band 

 is lower than that for other bands, it will contribute little to the total 

 loudness so that only a small error will be introduced by a wrong choice 

 of frequency for the band. 



This then gives a method of calculating bk when the adjacent com- 

 ponents are equal in loudness. When they are not equal let us define 

 the difference AL by 



AL = Lk- Lm. (20) 



Also let this difference be T when Lm is adjusted so that the masking 

 component just masks the component k. Then the function for 

 calculating b must satisfy the following conditions : 



bk = [(250 + A/)/1000](3 when AL = 0, 

 bk = when AL = — T. 



