154 BELL SYSTEM TECHNICAL JOURNAL 



ation. In the sensation area chart of Fig. 1, the intensity line cor- 

 responding to one dyne was used as the zero level, that is, pi was 

 chosen equal to 1 so that 



AL=20 log£ 



The choice of the base of logarithms for the pitch scale is dictated 



by the fact mentioned before, that the ear perceives octaves as being 



very similar sensations. Consequently the base 2 is the most logical 



choice for expressing pitch changes. If the logarithm of the frequency 



to the base 2 were used, perceptible changes in pitch would correspond 



to inconveniently small values of the logarithm. It is better to use 



ioo / _ 

 the logarithm to the base V2 which is 100 times as large. On 



this scale the smallest perceptible difference in pitch is approxi- 

 mately unity — somewhat more for frequencies greater than 100 cycles 

 or somewhat less for lower frequencies, according to Knudsen's data. 

 The scale on the charts is chosen so that the change in pitch is given by 



AP = 100 1og 2 N 



where N is the frequency of vibration. 



It is now evident why such pitch and loudness scales were used 

 in Fig. 1. With these scales, the number of units in any area gives 

 approximately the number of tones that can be ordinarily appreciated 

 in that area. For example, there are approximately 2,000 distin- 

 guishable tones in each square, there being more near the centre and 

 fewer near the boundary lines than this number. 



Experiments have shown that pure tones of different frequencies 

 which are an equal number of units above the threshold value sound 

 equally loud. This statement may require modification when very 

 loud tones are compared, but the data indicated that throughout the 

 most practical range this was true. Consequently, the absolute 

 loudness of any tone can be taken as the number of units above the 

 threshold value. 



Loudness of Complex Sounds 



In the measurement of the loudness of complex tones, the situation 

 is not so simple. It has been found that if two complex tones are 

 judged equally loud at one intensity level and then each is magnified 

 equal amounts in intensity, they then may or may not sound equally 

 loud. The curves shown in Figs. 4, 5 and 6 will illustrate this. The 

 first (Fig. 4) shows the comparisons at different intensity levels of 

 two sounds whose pressure spectra are shown in the two figures at 



