LOUDNESS 383 



Formulation of the Empirical Theory for Calculating the 

 Loudness Level of a Steady Complex Tone 



It is well known that the intensity of a complex tone is the sum of 

 the intensities of the individual components. Similarly, in finding a 

 method of calculating the loudness level of a complex tone one would 

 naturally try to find numbers which could be related to each com- 

 ponent in such a way that the sum of such numbers will be related in 

 the same way to the equally loud reference tone. Such efforts have 

 failed because the amount contributed by any component toward the 

 total loudness sensation depends not only upon the properties of this 

 component but also upon the properties of the other components in 

 the combination. The answer to the problem of finding a method of 

 calculating the loudness level lies in determining the nature of the ear 

 and brain as measuring instruments in evaluating the magnitude of an 

 auditory sensation. 



One can readily estimate roughly the magnitude of an auditory 

 sensation; for example, one can tell whether the sound is soft or loud. 

 There have been many theories to account for this change in loudness. 

 One that seems very reasonable to us is that the loudness experienced 

 is dependent upon the total number of nerve impulses per second going 

 to the brain along all the fibers that are excited. Although such an 

 assumption is not necessary for deriving the formula for calculating 

 loudness it aids in making the meaning of the quantities involved more 

 definite. 



Let us consider, then, a complex tone having n components each of 

 which is specified by a value of intensity level /St and of frequency fk. 

 Let N he di number which measures the magnitude of the auditory 

 sensation produced when a typical individual listens to a pure tone. 

 Since by definition the magnitude of an auditory sensation is the loudness, 

 then N is the loudness of this simple tone. Loudness as used here must 

 not be confused with loudness level. The latter is measured by the 

 intensity of the equally loud reference tone and is expressed in decibels 

 while the former will be expressed in units related to loudness levels in 

 a manner to be developed. If we accept the assumption mentioned 

 above, N is proportional to the number of nerve impulses per second 

 reaching the brain along all the excited nerve fibers when the typical 

 observer listens to a simple tone. 



Let the dependency of the loudness A^ upon the frequency/ and the 

 intensity /3 for a simple tone be represented by 



N = G{f /3), (7) 



where G is a function which is determined by any pair of values of / 



