LOUDNESS 387 



described in Appendix A. For example, in the second column it will 

 be seen that for the 125-cycle tone when the voltage is + 9.8 db above 

 1 volt then the voltage level for the reference tone must be 4.4 db 

 below 1 volt for equality of loudness. The bottom set of numbers in 

 each column gives the threshold values for this group of observers. 



Each voltage level in Table I is the median of 297 observations 

 representing the combined results of eleven observers. The method of 

 obtaining these is explained in Appendix A also. The standard 

 deviation was computed and it was found to be somewhat larger for 

 tests in which the tone differed most in frequency from the reference 

 tone. The probable error of the combined result as computed in the 

 usual way was between 1 and 2 db. Since deviations of any one 

 observer's results from his own average are less than the deviations of 

 his average from the average of the group, it would be necessary to 

 increase the size of the group if values more representative of the 

 average normal ear were desired. 



The data show^n in Table I can be reduced to the number of decibels 

 above threshold if we accept the values of this crew^ as the reference 

 threshold values. However, we have already adopted a value for the 

 1000-cycle reference zero. As wall be show^n, our crew obtained a 

 threshold for the reference tone which is 3 db above the reference 

 level chosen. 



It is not only more convenient but also more reliable to relate the 

 data to a calibration of the receivers in terms of physical measurements 

 of the sound intensity rather than to the threshold values. Except in 

 experimental w-ork where the intensity of the sound can be definitely 

 controlled, it is obviously impractical to measure directly the threshold 

 level by using a large group of observers having normal hearing. For 

 most purposes it is more convenient to measure the intensity levels 

 /3i, 182, • • • /3jt, etc., directly rather than have them related in any way 

 to the threshold of hearing. 



In order to reduce the data in Table I to those which one would 

 obtain if the observers were listening to a free wave and facing the 

 source, we must obtain a field calibration of the telephone receivers 

 used in the loudness comparisons. The calibration for the reference 

 tone frequency has been explained previously and the equation 



^, = iSx-h \0 log (Wr/W,) (5) 



derived for the relation between the intensity ^r of the reference tone 

 and the electrical power Wr in the receivers. The calibration consisted 

 of finding by means of loudness balances a power Wi in the receivers 

 w^hich produces a tone equal in loudness to that of a free wave having 

 an intensity level /3i. 



