On the Beats of Consonances of the Form h : 1. 421 



sympathy by considerations of a somewhat indirect character. 

 I wish to point out the important bearing, on the theory, of 

 the direct determination of the interval which separates the 

 region in which two notes are heard only as resultant dis- 

 placements, from that in which they begin to be heard sepa- 

 rately, in accordance with Ohm's law. 



5. The experiments I have made on this point have been 

 mostly conducted by means of my enharmonic organ, in which 

 I have a collection of notes separated for the most part by 

 single commas. 



The results, so far as I have gone, are:— 1. The critical in- 

 terval, at which two notes begin to be heard beside their beats, 

 or resultant displacements, is about two commas, throughout 

 that medium portion of the scale which is used in practical 

 music. 2. This critical interval appears not to be exactly the 

 same for all ears. In my own case two notes two commas 

 apart are not heard distinctly beside the beats. In the case 

 of Mr. Parratt, who has kindly examined the point with me, 

 two notes two commas apart are distinctly heard beside the 

 beats. In both cases the beats alone are heard with an in- 

 terval of one comma, and the two notes are quite clear beside 

 the beats with an interval of three commas. 



I propose to undertake further experiments, with the view T 

 of determining this initial interval more accurately. So far 

 as the above results go, they are quite consistent with Helm- 

 holtz's assumption as to the degree of sympathy of the ear. 



6. Independently of any theory, the fact that at a certain 

 point the ear begins to separate out two independent pendulum- 

 vibrations from the resultant displacements, is one that must 

 be recognized. It is easy to show that it is inconsistent with 

 that representation of facts which assumes that beats arise only 

 from the resultant forms exhibited by the superposition of the 

 two vibrations on one receptive mechanism. This is shown 

 as follows. 



7. If we combine two vibrations of equal amplitude, which 

 we may take =1, cospt and cos (qt—e), on the same recep- 

 tive mechanism, the effect is to produce a resultant displace- 

 ment represented by 



2 cos 0i±|liZL 6 . cos fc|)£±f. 

 2 2 



This would be heard, by a hypothetical ear receiving the whole 

 disturbance on the same sensorium, as a note whose frequency 

 is the arithmetic mean between the frequencies of the two 

 primaries, and having oscillations of intensity whose fre- 

 quency is defined by a pendulum-vibration of frequency equal 



