20*2 Prof. J. D. Everett on Resultant Tones. 



m : m + 1 ; and I have satisfied myself, both by my own 

 trials and by a study of Koenig's experimental results, that 

 when the difference-tone and the common fundamental are 

 not identical, the common fundamental is usually the pre- 

 dominant, and often the only audible resultant tone. (See 

 Appendix.) 



8. The common fundamental is, however, not the only 

 resultant tone that can be thus accounted for. Similar 

 reasoning to that employed in reference to A and B suffices 

 to explain the introduction of any or all of the harmonics of 

 the fundamental ; but it is to be expected, from the analogy 

 of ordinary experience in harmonic analysis, that the succes- 

 sive constituents will usually be smaller and smaller as we 

 advance in the series. The octave is likely to be the largest 

 of them ; and Koenig found, in several experiments with 

 primaries in the ratio of 3 : 5, that both the fundamental 1 

 and its octave 2 were distinctly heard as resultant tones. 



9. The following investigation bears on the relation be- 

 tween beats and resultant tones. The expression 



a cos mO + b cos nO 

 can be reduced to the form 



where A and e are given by 



A 2 = a 2 + b 2 + 2ab cos {n - m) 0, 



, a — b. n—ni* 

 tan e = r tan — ~ — 0, 



a + o 2 ' 



and the beating together of two tones not differing much in 

 pitch is explained by the fact, definitely expressed in these 

 formulas, that the whole effect may be regarded as a succession 

 of waves with gradually varying amplitude. The frequency 

 of the beats is the frequency of the maxima of A 2 , and is the 

 difference of m and n. 



We have ascribed resultant tones to alterations made in 

 the wave-form by the action of the ear, such alterations being 

 in general largest at those points at which the excursions of 

 the drumskin are largest. These excursions are measured by 

 ±A, and the above investigation shows that their maxima 

 have a frequency corresponding to the difference-tone. This 

 is true whether m and n are commensurable or incommen- 

 surable. If they are commensurable, their greatest common 



