LOUDNESS 385 



ponent, we now have to determine a single function depending only 

 upon the properties of the reference tone and as stated above this 

 function is the relationship between loudness and loudness level. 

 And since the frequency is always 1000 this function is dependent only 

 upon the single variable, the intensity level. 



This formula has no practical value unless we can determine bk and 

 G in terms of quantities which can be obtained by physical measure- 

 ments. It will be shown that experimental measurements of the 

 loudness levels L and Lk upon simple and complex tones of a properly 

 chosen structure have yielded results which have enabled us to find 

 the dependence of b and G upon the frequencies and intensities of the 

 components. When b and G are known, then the more general 

 function G{f, 13) can be obtained from Eq. (9), and the experimental 

 values of Lk corresponding to fk and ^k- 



Determination of the Relation Between Lk, fk and /3/t 



This relation can be obtained from experimental measurements of 

 the loudness levels of pure tones. Such measurements were made by 

 Kingsbury '^ which covered a range in frequency and intensity limited 

 by instrumentalities then available. Using the experimental technique 

 described in Appendix A, we have again obtained the loudness levels 

 of pure tones, this time covering practically the whole audible range. 

 (See Appendix B for a comparison with Kingsbury's results.) 



All of the data on loudness levels both for pure and also complex 

 tones taken in our laboratory which are discussed in this paper have 

 been taken with telephone receivers on the ears. It has been explained 

 previously how telephone receivers may be used to introduce the 

 reference tone into the ears at known loudness levels to obtain the 

 loudness levels of other sounds by a loudness balance. If the receivers 

 are also used for producing the sounds whose loudness levels are being 

 determined, then an additional calibration, which will be explained 

 later, is necessary if it is desired to know the intensity levels of the 

 sounds. 



The experimental data for determining the relation between Lk and 

 fk are given in Table I in terms of voltage levels. (Voltage level 

 = 20 log V, where V is the e.m.f. across the receivers in volts.) The 

 pairs of values in each double column give the voltage levels of the 

 reference tone and the pure tone having the frequency indicated at 

 the top of the column when the two tones coming from the head re- 

 ceivers were judged to be equally loud when using the technique 



* B. A. Kingsbury, "A Direct Comparison of the Loudness of Pure Tones," Phys. 

 Rev. 29, 588 (1927). 



