Theory of the Beats of Mistuned Consonances. 273 



formed their idea of Smith's work. From our present point 

 of view we admire his painstaking appreciation of Smith, 

 while we think he overestimates the value of Smith's analysis 

 of beats. He does not seem to see that it amounted only to a 

 numerical or geometrical analysis, and left the reader a long 

 way off even the idea of the existence of a deeper-seated expla- 

 nation. The obstacle opposed to Smith's explanation by the 

 resolution in the ear of the resultant or complex forms into 

 their elements according to Ohm's law, escaped De Morgan 

 entirely. 



De Morgan is behind Young in thinking that Tartini (com- 

 bination) tones only exist in the case of consonances ; this 

 vitiates fundamentally all his notions of the phenomena, and 

 causes him to give defective rules for them. Further, he 

 condemns Young for the statement that other resultant tones 

 (as we now call them) arise out of the beats of consonances 

 besides the Tartini tone (difference-tone of Helmholtz, first 

 combination-tone). In fact Young here clearly describes the 

 phenomena subsequently described by Konig as beat-notes, 

 and was right in the main as to the facts. 



I think we may fairly summarize the position taken up by 

 Helmholtz on this question as amounting to the admission of 

 the principle — all beats consist of variations of intensity of 

 musical notes. I am not quite sure whether Helmholtz was 

 the first to enunciate principles which amount to this, as the 

 history of these doctrines is voluminous and rather obscure. 

 And I do not think that even Helmholtz enunciates this as a 

 new dogma in so many words, though it is assumed throughout 

 as the basis of his treatment. Some of the older writers, par- 

 ticularly Smith and Young, so guard their exposition that it 

 is quite capable of adaptation to this principle. On the other 

 hand, certain modern writers, professing or appearing to fol- 

 low in the footsteps of Smith and Young, so modify their 

 exposition as to make it inconsistent with the above principle, 

 and to require the admission that beats consist of some form 

 of perception other than that of the variation of musical notes. 



In treating of the position of Helmholtz, it will be desirable 

 to discuss somewhat minutely a treatise on Beats, consisting 

 of two papers by W. Pole, F.R.S., Mus. Doc, in 'Nature,' 

 Jan. 13 and 20, 1876. I regard these papers as the most 

 important manifesto of the school last alluded to, before the 

 paper of Konig to be subsequently considered. The clearness 

 and knowledge of the subject make them a good text for dis- 

 cussion ; yet I consider the exposition they contain to be 

 erroneous. 



Pole describes three kinds of beat — the unison-beat, the 



