484 Professor A. W, Rucker [March 8, 



The most characteristic work of Helmholtz was, as I have already 

 hinted, that in which his knowledge of physics and his knowledge of 

 anatomy were both directed to a common end. He dealt in turns with 

 the external physical phenomena, with the mechanism of the organs 

 which the phenomena affect, with the relations between the mechanical 

 effect on the organ and the sensations which it excites, and, lastly, with 

 the connection between the sensations in those simple cases which 

 can alone be investigated in the laboratory, and the complex laws of 

 aesthetics and art. 



The two books in which these problems were chiefly treated were 

 the ' Physiological Optics,' and the ' Sensations of Sound.' It is im- 

 possible to do more than lay before you a sample which may afford 

 some idea of the intricacy of the problems with which he dealt, and 

 of the pitfalls amongst which he walked so warily. For this purpose 

 I have chosen one branch of his work on ' Sound.' 



I have deliberately selected that particular portion which has 

 been most questioned, that on which the verdict of most of those who 

 have sat in judgment on his views has been against him. 



In discussing this question I must give a general description of 

 the principle phenomena ; but if I were to attempt an exhaustive 

 catalogue of all the facts disputed and undisputed, and of all the 

 theories which have been based upon or upset by them, not only 

 would time fail me, but those who have not given special attention to 

 the subject would, I fear, become hopelessly confused amid the chaos 

 of opposing statements and views. Another reason which urges me 

 to be brief, is that a few years ago Prof. Silvanus Thompson explained 

 the whole subject to the members of the Royal Institution, having 

 kindly consented to act as the mouthpiece of the celebrated in- 

 strument maker, Konig, who has played so large a part in these 

 controversies. 



Among the chief achievements of Helmholtz was an explanation 

 of the physical difference between pairs of notes which we recognise 

 as concords and discords respectively. When two neighbouring 

 notes are sounded, alternate swellings and fallings off of the intensity 

 are heard which are called beats. These produce an unpleasant 

 effect, which depends partly on their number, partly on the relative 

 pitches of the beating notes. When two notes beat badly, they 

 form an intolerable discord. When they become separated by a 

 wider interval, the beats are so rapid that they cease to be un- 

 pleasant. 



The sense of dissonance produced by many of these wider intervals, 

 such as the seventh (4 : 7), requires further explanation. In general, 

 the fundamental musical note is only the first and loudest of a series 

 of so-called partials, whose vibration frequencies are 2, 3, 4, &c. times 

 that of the fundamental, and the consonance and dissonance of two 

 notes is shown to depend on the presence or absence of beats between 

 important members of these series. Thus in the case of the seventh 

 the frequencies of the octave of the lower note and that of the upper 



