THE PHYSICAL BASIS OF MUSICAL HARMONY. 345 



II. 



So far we Lave been dealiug with primary beats aucl beat-tones; but 

 there are also secondary beats and secondary beat-tones, which are pro- 

 duced by the interference of primary beat-tones. An example of a sec- 

 ondary beat is afforded by the following experiment. Recurring to the 

 preceding table of experiments, it may be observed that when the two 

 shrill notes, n /g, sole, ft'iviug the interval of the fifth, are sounded to- 

 gether, the inferior and superior beat-tones are both present and of the 

 same pitch. If, now, one of the two forks is lightly loaded with pel- 

 lets of wax to put it out of adjustment, we shall get beats, not between 

 the i^rimary tones, but between the beat-tones. Suppose we add enough 

 wax to reduce the vibration of sola from 3,072 to 3,070. Then the posi- 

 tive remainder is 1,022 and the negative remainder is 1,02G, the former 

 being uf-, flattened two vibrations, the latter the same note sharjieued 

 to an equal amount. As a result there will be heard four beats per 

 second — secondary beats. Similarly, the intervals 2 : 5, 2 : 7, if slightlj^ 

 mistuned, will, like the fifth, yield secondary beats. Or, to put it iu 

 another way, there may be secondary beats from the (mistuned) beat- 

 tones that are related (as in our experiment) in the ratio 1 : 1 or in the 

 ratios 3 : 4, 3 : 5, etc., and even by those of 1 : 2, 4 : 5, 4 : 7, etc. 



I have given you an example of secondary beats; now for an exam- 

 ple of a secondary beat tone. This is afforded by one of the previous 

 experiments, in which were sounded ute and the 11th harmonic of uts. 

 In this experiment, as iu that which followed with the 13th harmonic, 

 two (primary) beat-tones w^ere produced, of 768 and 1,280 vibrations re- 

 spectively. These are related to one another by the interval 3:5. If 

 we treat these as tones that can themselves interfere, they will give us 

 for their positive remainder the number 256, which is the frequency of 

 w<4. As a matter of fact, if you listen carefully you may, now that your 

 attention has been drawn to it, hear that note, in addition to the two 

 primary tones and the two beat-tones to which you listened prcviouslj^. 



In von Helmholtz's Tonempfindungen he expresses the opinion 

 that the distinctness with which beats are heard depends upon the 

 narrowness of the interval between the primary tones, saying that 

 they must be nearer together than a minor third. But, as we have 

 seen, using bass sounds of a sufficient degree of intensity and purity, 

 as is the case with those of the massive forks, beats can be heard with 

 every interval from the mistuned unison up to the mistuned octave. 

 Even the interval of the fiftli, utx to soli, gave strongly marked beats 

 of 32 per second. When this number is attained or exceeded, the ear 

 usually begins to receive also the effect of a very low continuous tone, 

 the beats and the beat-tone being simultaneously i)erceptible up to 

 about 60 or 70 beats, or as a roughness up to 128 per second. If, using 

 forks of higlier pitches, but of narrower interval, one i)roduccs the 

 same number of beats, the beat-tone is usually more distinct. Doubt- 



