PHYSICAL MEASUREMENTS OF AUDITION 173 



The degree of deafness is measured by the fractional part of the 

 normal area of audition in which the sensation is either lacking or false. 



Appendix A 



The calculations of the number of pure tones perceivable as being 

 different in pitch at a given intensity or being different in loudness at 

 a given pitch involves a line integral. The calculation of the number 

 of pure tones perceivable as being different either in loudness or 

 pitch involves a surface integral. 



Let the coordinates used in Fig. 1 corresponding to AL and AP 

 be designated a and /?, respectively. Then the relations shown in 

 Figs. 2 and 3 can be expressed by the equations 



— =/(« — a ) and (1) 



f = ^(/3) (2) 



where ao is the value of a along the normal minimum audibility 

 curve shown in Fig. 1. Knudsen's data indicated that the curve 

 shown in Fig. 3 held only for values of a — ao corresponding to the 

 flat part of the curve in Fig. 2. For lower intensities the pitch dis- 

 crimination fell off in about the same way as that shown for the 

 intensity discrimination. To represent this mathematically, <p(/3) 

 can be multiplied by a factor which is unity for the loud tones and 

 which increases similarly to f(a — a ) for the weaker tones. Such a 



factor is 10 f (a — an) since f (a — « ) is approximately — for the louder 



AN . 

 tones. So the corrected formula for -t^ is 



N 



^ = l<V(/3)./(a-a ). (3) 



Let dx be the number of perceivable tones of constant intensity 

 corresponding to a in the pitch region between /3 and j3-f-<f/3 and let 

 dy be the number of perceivable tones of constant pitch corresponding 

 to /3 in the region between a and a-\-da. Then 



A dN {AS 



dx = AN (4) 



dy = IE . (5) 



