1830.] 071 the Physical Foundation of Music. 217 



shall get beats, not between the primary tones, but between the beat- 

 tones. Suppose we add enough wax to reduce the vibration of solg 

 from 3072 to 3070. Then the positive remainder is 1022, and the 

 negative remainder is 1026; the former being ut^ flattened two 

 vibrations, the latter the same note sharpened 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 slightly mistuned, will, 

 like the fifth, yield secondary beats. Or, to put it in 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 from those in the 

 ratios 3 : 4, 3 : 5, 4 : 7, and so forth. 



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

 example of a secondary beat-tone. This is afforded by one of the 

 previous experiments, in which were sounded iUq, and the 11th har- 

 monic of ut^. In this experiment, as in that which followed with the 

 13th harmonic, two (primary) beat-tones were produced, of 768 and 

 1280 vibrations respectively. These are related to one another by 

 the intervals 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 uti. 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 previously. 



In von Helmholtz's " Tonemptindungen," 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 fifth, uti to sol^, 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 perceptible up 

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

 using forks of higher pitches but of narrower interval, one produces 

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

 Doubtless this arises from the greater true intensity of the sounds 

 of higher pitch. With the object of pursuing this matter still more 

 closely. Dr. Koenig constructed a series of 12 forks of extremely high 

 pitch, all within the range of half a tone, the lowest giving s^e and 

 the highest ut^. The frequencies, and the beats and beat-tones given 

 by seven of them, are recorded in Table III. 



The first of these intervals is a diatonic semitone ; the second of 

 them is a quarter-tone ; the third is an eighth of a tone ; neverthe- 

 less, a sensitive ear will readily detect a difference of pitch between 

 the two separate sounds. The last of the intervals is about half a 

 comma. 



