398 BELL SYSTEM TECHNICAL JOURNAL 



tion these points should be represented by 



G(y) = W{x). 



Any one of these curves which was accurately determined would be 

 sufficient to completely determine the function G. 



For example, consider the curve for two tones. It is evident that 

 it is only necessary to deal with relative values of G so that we can 

 choose one value arbitrarily. The value of G(0) was chosen equal to 

 unity. Therefore, 



GiO) = 1, 



Giyo) = 2G(0) = 2 where 3^0 corresponds to x = 0, 



G{y\) = 2G(xi) = 2G{yn) = 4 where yi corresponds to Xi = yo, 



G{yi) = 2G{xi) = 2G{yi) = 8 where 3^2 corresponds to ^2 = yi, 



G{yk) = 2G(xk) = 2G(yK-i) = 2^+' where yic corresponds to Xk = yk-i- 



In this way a set of values for G can be obtained. A smooth curve 

 connecting all such calculated points will enable one to find any value 

 of G(x) for a given value of .r. In a similar way sets of values can be 

 obtained from the other two experimental curves. Instead of using 

 any one of the curves alone the values of G were chosen to best fit all 

 three sets of data, taking into account the fact that the observed 

 points for the 10-tone data might be low at the higher levels where b 

 would be less than unity. The values for the function which were 

 finally adopted are given in Table III. From these values the three 

 solid curves of Figs. 6 and 7 were calculated by the equations 



G(y) = lOG(x), G{y) = 2G(x), G(y) = ^G(.t). 



The fit of the three sets of data is sufficiently good, we think, to justify 

 the point of view taken in developing the formula. The calculated 

 points for the 10-component tones agree with the observed ones when 

 the proper value of bk is introduced into the formula. In this con- 

 nection it is important to emphasize that in calculating the loudness 

 level of a complex tone under the condition of listening with one ear 

 instead of two, a factor of ^ must be placed in front of the summation 

 of Eq. (10). This will be explained in greater detail later. The values 

 of G for negative values of L were chosen after considering all the data 

 on the threshold values of the complex tones studied. These data will 

 be given with the other loudness data on complex tones. It is in- 

 teresting to note here that the threshold data show that 10 pure tones, 

 which are below the threshold when sounded separateh', will combine 



