1890.] on the Physical Foundation of Music. 213 



more rapid and become undistinguishable ; but these are succeeded 

 by another set which gradually emerge from the intolerable vacarme, 

 and, though rapid at first and indistinct, grow slower and stronger as 

 the pitch is raised, until, when it reaches 80I2, the frequency of which 

 is exactly three times that of uti, all beats again vanish. This rauge 

 between the octave and the twelfth tone may be called the second 

 " period," to distinguish it from the period from unison to the first 

 octave, which was our first period. Similarly, the range from the 

 twelfth tone to the second octave is the third period, and from thence 

 to the major third above is the fourth period, and so forth. In each 

 period, up to the sixth or seventh of such periods, a set of inferior 

 and a set of superior beats may be observed ; and in every case the 

 frequency of the beats corresponds, as I have said, to one or other 

 of the two remainders of the frequencies of the two tones. Ko beat 

 has ever been observed corresponding to the sum of the frequencies, 

 even when usinec the slowest forks. None has ever been observed 

 corresponding to the diiference of the frequencies, save in the first 

 period ; where, of course, the positive remainder is simply the 

 difference of the two numbers. 



That you may hear for yourselves the beats belonging to one of 

 the higher periods, I will take a pair of forks which will give us 

 some of the superior beats in the fourth period. One of the 

 forks is the great ut^, 64 as previously used. The other is mi^ 

 = 320 ; their ratio being 1 : 5. Sounded together they give a 

 pure consonance, but if the smaller one is loaded with small 

 pellets of wax to lower its pitch slightly, and I then bow it, at once 

 you hear beats. It was in studying the beats of these higher periods 

 that Dr. Koenig made the observation that whereas the beats of an 

 imperfect unison are heard as alternate silences and sounds, the beats 

 of the imperfect consonances of higher periods — twelfth tone, double 

 octave, &c. — consist mainly in variations in the loudness of the lower 

 of the two primary tones ; an observation which was independently 

 made by Mr. Bosanquet, of Oxford. 



Passing from the beats themselves, I approach the question, what 

 becomes of the beats when they occur too rapidly to produce on the 

 ear a discontinuous sensation ? On this matter there have been 

 several conflicting opinions : some^ holding with Lagrange and 

 Dr. Thomas Young, that they blend into a separate tone ; others, with 

 von Helmholtz, maintaining that the combinational tones cannot be 

 so explained, and arise from a different cause. Let it be observed 

 that, even if beat-tones exist, it is quite possible for beats and beat- 

 tones to be simultaneously heard. A similar co-existence of a 

 continuous and discontinuous sensation is afforded by the familiar 

 experiment of producing a tone by pressing a card against the 

 periphery of a rapidly rotating toothed wheel. There is a certain 

 speed at which the individual impulses begin to blend into a 

 continuous low tone, while yet there are distinguishable the discon- 

 tinuous impulses ; the degree of distinctness of the two co-existing 



