400 BELL SYSTEM TECHNICAL JOURNAL 



to give a tone which can be heard. When the components are all in 

 the high pitch range and all equally loud, each component may be 

 from 6 to 8 db below the threshold and the combination will still be 

 audible. When they are all in the low pitch range they may be only 

 2 or 3 db below the threshold. The closeness of packing of the com- 

 ponents also influences the threshold. For example, if the ten 

 components are all within a 100-cycle band each one may be down 

 10 db. It will be shown that the formula proposed above can be made 

 to take care of these variations in the threshold. 



There is still another method which might be used for determining 

 this loudness function G{L), provided one's judgment as to the magni- 

 tude of an auditory sensation can be relied upon. If a person were 

 asked to judge when the loudness of a sound was reduced to one half 

 it might be expected that he would base his judgment on the experience 

 of the decrease in loudness when going from the condition of listening 

 with both ears to that of listening with one ear. Or, if the magnitude 

 of the sensation is the number of nerve discharges reaching the brain 

 per second, then when this has decreased to one half, he might be able 

 to say that the loudness has decreased one half. 



In any case, if it is assumed that an observer can judge when the 

 magnitude of the auditory sensation, that is, the loudness, is reduced to 

 one half, then the value of the loudness function G can be computed 

 from such measurements. 



Several different research workers have made such measurements. 

 The measurements are somewhat in conflict at the present time so 

 that they did not in any way influence the choice of the loudness 

 function. Rather we used the loudness function given in Table III 

 to calculate what such observations should give. A comparison of 

 the calculated and observed results is given below. In Table I\' is 

 shown a comparison of calculated and observed results of data taken 

 by Ham and Parkinson.^ The observed values were taken from 

 Tables la, lb, 2a, 2b, 3a and 3b of their paper. The calculation is 

 very simple. From the number of decibels above threshold 5 the 

 loudness level L is determined from the curves of Fig. 3. The frac- 

 tional reduction is just the fractional reduction in the loudness function 

 for the corresponding values of L. The agreement between obser\ed 

 and calculated results is remarkably good. However, the agreement 

 with the data of Laird, Taylor and Wille is very poor, as is shown by 

 Table V. The calculation was made only for the 1024-cycle tone. 

 The observed data were taken from Table YII of the paper by Laird, 



* L. B. Ham and J. S. Parkinson, " Loudness and Intensity Relations," Jour. Acous. 

 Soc. Am. 3, 511 (1932). 



