384 BELL SYSTEM TECHNICAL JOURNAL 



and |8. For the reference tone, / is 1000 and j8 is equal to the loudness 

 level L, so a determination of the relation expressed in Eq. (7) for the 

 reference tone gives the desired relation between loudness and loudness 

 level. 



If now a simple tone is put into combination with other simple tones 

 to form a complex tone, its loudness contribution, that is, its con- 

 tribution toward the total sensation, will in general be somewhat less 

 because of the interference of the other components. For example, if 

 the other components are much louder and in the same frequency 

 region the loudness of the simple tone in such a combination will be 

 zero. Let 1 — 6 be the fractional reduction in loudness because of its 

 being in such a combination. Then bN is the contribution of this 

 component toward the loudness of the complex tone. It will be seen 

 that h by definition always remains between and unity. It depends 

 not only upon the frequency and intensity of the simple tone under 

 discussion but also upon the frequencies and intensities of the other 

 components. It will be shown later that this dependence can be 

 determined from experimental measurements. 



The subscript k will be used when/ and /3 correspond to the frequency 

 and intensity level of the ^th component of the complex tone, and the 

 subscript r used when / is 1000 cycles per second. The "loudness 

 level" L by definition, is the intensity level of the reference tone when 

 it is adjusted so it and the complex tone sound equally loud. Then 



Nr = G(1000, L) = L%,7V, = ZbkGiU /?,). (8) 



k = l fc=l 



Now let the reference tone be adjusted so that it sounds equally loud 

 successively to simple tones corresponding in frequency and intensity 

 to each component of the complex tone. 



Designate the experimental values thus determined as Li, L2, L3, • • • 

 Lk, • • ' Ln. Then from the definition of these values 



iV, = G(1000, L.) = G(/,, /3a-), (9) 



since for a single tone hk is unity. On substituting the values from 

 (9) into (8) there results the fundamental equation for calculating the 

 loudness of a complex tone 



G(1000, L) = if 6,G(1000, Lk). (10) 



k=l 



This transformation looks simple but it is a very important one since 

 instead of having to determine a different function for every com- 



