HARMONY IN MUSIC, 89 



partials are not exactly alike, and hence will beat and roughen 

 the tone. 



It is very easy to hear the beats of such imperfect fifths, 

 because, as our pianos and organs are now tuned, all the fifths 

 are impure, although the beats are very slow. By properly 

 directed attention, or still better with the help of a properly 

 tuned resonator, it is easy to hear that it is the particular upper 

 partials here spoken of that are beating together. The beats are 

 necessarily weaker than those of the fundamental tones, because 

 the beating upper partials are themselves weaker. Although we 

 are not usually clearly conscious of these beating upper partials, 

 the ear feels their effect as a want of uniformity or a roughness in 

 the mass of tone, whereas a perfectly pure fifth, the pitches being 

 precisely in the ratio of 2 to 3, continues to sound with perfect 

 smoothness, without any alterations, reinforcements, diminutions, 

 or roughnesses of tone. As has already been mentioned, the siren 

 proves in the simplest manner that the most perfect consonance 

 of the fifth precisely corresponds to this ratio between the pitches. 

 We have now learned the reason of the roughness experienced 

 when any deviations from that ratio has been produced. 



In the same way two tones which have their pitches ex- 

 actly in the ratios of 3 to 4, or 4 to 5, and consequently form 

 a perfect fourth or a perfect major third, sound much better 

 when sounded together, than two others of which the pitches 

 slightly deviate from this exact ratio. In this manner, then, 

 any given tone being assumed as fundamental, there is a pre- 

 cisely determinate number of other degrees of tone which can 

 be sounded at the same time with it, without producing any 

 want of uniformity or any roughness of tone, or which will 

 at least produce less roughness than any slightly greater or 

 smaller intervals of tone under the same circumstances. 



This is the reason why modern music, which is essentially 

 based on the harmonious consonance of tones, has been compelled 

 to limit its scale to certain determinate degrees. But even in 

 ancient music, which allowed only one part to be sung at a time, 

 and hence had no harmony in the modern sense of the word, 

 it can be shown that the upper partial tones contained in all 



