424 Mr. R. H. M. Bosanquet on the Beats 



in a system in which the forces are proportional to the displace- 

 ments. 



15. In the present case, if we find a term present whose 

 whole period is that of the Smith's beat, it must therefore 

 arise by transformation, i. e. through the presence of terms of 

 higher orders than the first. We shall use generally the ex- 

 pression " transformation " to signify the effect on a system 

 of terms higher than the first in the expression for the forces. 

 This is substantially Helmholtz's explanation of the difference- 

 tone, which is identical with the lowest beat-note of Konig. 



16. We shall show that all Konig's beat-notes can be 

 accounted for in a similar manner, by the assumption that 

 terms of higher orders become important in the mechanism of 

 the ear when the displacements are considerable. 



17. We can illustrate further the difference produced in th& 

 curves by the admission of the difference-tone or beat-note 

 as part of the mass of sound. The characteristic difference is, 

 that the medial line is itself bent into a curve, whose whole 

 period is that of the Smith's beat. I have not been able to 

 draw any long curve to show this; but the appearance of the 

 curve at the top of illustration A (p. 425) is very like it in a 

 general way. This illustration represents the square term of 

 the force developed by a fifth (2 : 3) in a transforming system. 

 B is the figure of the total disturbed force in a similar case; but 

 the thro wing-up of the medial line is not so prominent as it 

 would be in a longer curve. I have, however, no machine 

 that will draw the combination; and the construction of a 

 long curve of this kind is not worth the labour it would 

 entail. 



18. We sum up this part in the following conclusions: — - 

 As two notes of equal amplitudes separate from unison, they 



are at first received by the ear in the manner of resultant dis- 

 placements, consisting of the beats of a note whose frequency 

 is midway between that of the primaries. 



When the interval reaches about two commas, the ear 

 begins to resolve the resultant displacements, and the primary 

 notes step in beside the beats. 



When the interval reaches a minor third in the ordinary 

 parts of the scale, neither the beats nor the intermediate pitch 

 of the resultant note are any longer audible, at least as matter 

 of ordinary perception ; but the resultant displacement which 

 reaches the ear is decomposed, and produces the sensation of 

 the two primary notes, perfectly distinct from each other : 

 that is to say, Ohm's law has set in, and is true, for ordinary 

 perceptions and in the ordinary regions of the scale, for the 

 minor third and all greater intervals. 



