1890.] on the Physical Foundation of Music. 215 



the second fork you hear this clear ut^ sounding out. I am not going 

 to waste your time in a disputation as to whether the sound you hear 

 is objective or subjective. It is enough that you hear it, pure and 

 unmistakable in pitch. It is the grave harmonic ; and the number 

 256, which is its frequency, corresponds to the positive remainder 

 when you divide 2304 by 2048. 



Now let me give you a beat-tone belonging to the superior set : 

 it also will be a grave harmonic, if you so please to call it ; but its 

 frequency will correspond neither to the difference nor to the sum 

 of the frequencies of the two primary tones. I take utg = 2048 

 as previously, and with it sig = 3840. Let us calculate what the 

 superior beats ought to be. 2048 goes into 3840 twice, less 256. 

 Then, 256 being the negative remainder, we ought to hear from these 

 two forks the beat-tone of 256 vibrations, which is ut^, the same 

 note as in our last experiment. I strike the forks, and you hear the 

 result. The beat-tone, which is neither a differential tone nor a 

 summational tone, corresponds to the calculated number of beats. 



If I take utQ = 2048 and soIq = 3072, the two remainders both 

 come out at 1024, which is utr,. Let me sound ut^ itself, separately, 

 on an tit^ fork, that you may know what sound to listen for. Its 

 sound has died away ; and now I strike utg and soIq, when at once 

 you hear M^5 ringing out. That sound which you all heard corre- 

 sponds in frequency to the calculated number of beats. That is 

 enough for my present purpose. 



The next illustration is a little more complex. I select a case in 

 which the beat-tones corresponding to the inferior and the superior 

 beats will both be present. We shall have four tones altogether — 

 two primary tones and two beat-tones. The forks I select are 

 iUq = 2048 as before, and a fork which is tuned to vibrate exactly 

 11 times as rapidly as ut^ — it is the lltli harmonic of that note, but 

 does not correspond precisely to any note of the diatonic scale. It 

 has 2816 vibrations, and is related to utQ as 11 : 8. The two 

 remainders will now be 768 and 1280, which are the respective fre- 

 quencies of sol^^ and mir,. I will first sound those notes on two other 

 forks, that you may know beforehand what to listen for. Now, on 

 striking the two shrill forks in rapid succession, the two beat-tones 

 are heard. 



If I select, instead of the 11th harmonic, the 13th harmonic of ut^, 

 vibrating 3328 times in the second, to be sounded along with ut^, I 

 shall produce the same two beat-tones as in the preceding case ; but 

 mi^ = 1280 is now the inferior one, corresponding to the positive 

 remainder, while sol^ = 768 is the superior tone, corresponding to 

 the negative remainder. It is certainly a striking corroboration of 

 Dr. Koenig's view that the beat-tones actually heard in these last 

 two experiments should come out precisely alike, though on the old 

 view, that the combination tones were simply the summational and 

 differential tones, one would have been led to expect the sounds in 

 the two experiments to be quite different. 



