Theory of the Beats of Mistimed Consonances. 275 



on Youncr's theory, Pole fails to notice the real nature of the 

 discovery of Helmholtz, which is that the two notes are sepa- 

 rately received by the ear ; so that the disappearance of the 

 beats is to be regarded as arising, in great measure at least, 

 from the removal of the two disturbances from each other's 

 influence, and not entirely or even principally from their 

 rapidity. I have dealt with this point in my recent paper 

 (Phil. Mag. June 1881, p. 422). 



Pole's second kind of beat is Smith's beat (variation of 

 maximum displacement in resultant forms). It is assumed 

 that the theory of Smith constitutes a complete explanation of 

 the existence of certain beats in mistuned consonances, as to 

 the nature of which nothing further is said. Smith's investi- 

 gation has been sufficiently given for its nature to be fully, 

 realized. It will be seen that Smith does not say any thing 

 as to what the ultimate nature of the beat is : he only examines 

 the superposition, and says, " wherever there is a certain irre- 

 gularity there must he a beat." The classification of beats 

 given by Pole appears therefore to import into Smith's inves- 

 tigation, siib silentio, something not really contained in it — 

 namely, the implication that the beat thus demonstrated is 

 something sui generis, and different in kind from the variation 

 of a musical note. For all beats formed by the variations of 

 musical notes appear to be referred to Pole's third class, 

 viz. the beats of Helmholtz. But before w r e can admit the 

 existence of beats not consisting of variation of musical notes, 

 we must learn what their sounds really consist of, and have 

 these sounds strictly deduced from the theory ; which, to say 

 the least, has not been accomplished so far. 



The third kind of beat described by Pole is the overtone- 

 beat of Helmholtz. It is too well understood for any discus- 

 sion to be needed here. 



The combination-tone beats, which were shown in my recent 

 paper to be the actual principal element in the beats of mis- 

 tuned consonances produced by pure notes, are brought in by 

 Pole at the end, whether regarded as part of the third class or 

 not is not clearly stated. 



The next contribution of importance is the famous paper of 

 Konig on beats and beat-notes, which has been regarded by 

 some authorities as establishing results inconsistent with the 

 views originated by Helmholtz. This paper was to a great 

 extent ihe origin of my work in connexion with this subject. 

 The question which suggests itself continually in reading 

 Konig's paper is, "What did all these beats consist of?** — a 

 question which I have endeavoured to answer in my recent 

 paper, and hope to answer more fully on some future occasion. 



