THE PHYSICAL BASIS OF MUSICAL HARMONY. 



341 



strongly at 16 per second. If it is raised once more to 6ii=120 (the 

 seventh of the ordinary scale), the beats are still stronger and slower 

 at 8 per second. Finally, when we bring the pitch up to the octave 

 nt2=128, we find that all beats have disappeared : there is a perfectly 

 smooth consonance. The facts so observed are tabulated for you as 

 follows : 



Table I. — rrimary beats. 



Suppose now, keeping the lower fork unaltered, we raise the pitch 

 of the higher note (taking a uew fork that starts at the octave) from 

 iit2 to soli by gradual steps, we shall find that there begins a new set of 

 primary beats, an inferior set, which are. at first slow, then get more 

 rapid and become undistinguishable, but succeeded by another rapid 

 and indistinct, which grow stronger and slower, until as the pitch rises 

 to S0I2, the frequency of which is exactly three times that of uti, all 

 beats again vanish. This range 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 jjeriod up to the sixth or seventh of such 

 l^eriods, 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 fre<iueucies of the 

 two tones. No beat has ever been observed corresi)onding to the sum 

 of the frequencies, even when using the slowest forks. None has ever 

 been observed corresponding to the difference 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^ Dr. Kcenig 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 wfi=64, as previously used, the other is wi/;i=32(), 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 



