

on the Physical Work of von Helmholtz. 485 



note would be in the proportion 8 : 7, which are sufficiently near to 

 make the beats very prominent and disturbing. 



In cases where the notes are pure, that is, are not accompanied by 

 upper partials, the explanation of dissonance is based upon another 

 phenomenon. 



When two notes are sounded simultaneously a third tone is often 

 perceived, the frequency of which is equal to the difference of their 

 frequencies. The number of vibrations of this tone is equal to the 

 number of beats, and as there has been controversy as to whether the 

 beats when they become rapid can produce a note, and if so, whether 

 this note is or is not the same thing as the difference tone, it is neces- 

 sary to distinguish between the two. This distinction is to be found 

 in the mode of their production ; but for the moment it is sufficient 

 to remember that they may be distinguishable, and to reserve for 

 them two names, viz. the beat-note, and the first difference tone 

 respectively. 



Helmholtz drew attention to the fact that besides the differ- 

 ence tone there is also produced a note, the frequency of which is 

 equal to the sum of those of the two primaries, and this he called the 

 first summation tone. 



Together with these he believed that there existed summation and 

 difference tones of higher orders, the whole series being included 

 under the name of combination tones. Our sense of dissonance 

 between pure notes was explained as dependent on beats produced by 

 the combination tones. 



Up to the time of Helmholtz it was generally thought that these 

 tones were produced in the ear itself, and had no objective existence 

 in the external air. They are thus often called subjective, but as that 

 adjective is usually reserved for impressions produced in the brain 

 itself, it is better to say that they were regarded as ear-made. Helm- 

 holtz himself gave a theory, which showed that it is probable that a 

 membrane like the drum-skin of the ear, which is forced out of shape 

 by pressure, and that bones, like those in the ear, which can rattle, 

 would, if acted upon by two notes, manufacture by their own proper 

 movements all the varied combinational tones which his theory 

 postulated. He therefore believed that combinational tones were 

 largely ear-made. 



You will observe that his theory of discord is quite unaffected by 

 the question whether the combination tones are or are not sometimes 

 objective. Provided only they are produced at all, it is immaterial 

 whether they are produced in the ear itself. Von Helmholtz admitted 

 that the phenomena we observe are in most cases ear-made tones ; 

 but he also asserted that they were sometimes objective, and could 

 set bodies tuned to vibrate with them in resonant motion. This latter 

 statement has been denied with singular unanimity, sometimes, I 

 think, without due regard to the limitations which Helmholtz himself 

 placed on the conditions under which the objective character of the 

 notes can be realised. 



Vol. XIV. (No. 89.) 2 l 



