HAEMONY m MUSIC. 101 



quence pass as a consonance. Even when the fund imental 

 tones have such widely-different pitches that they cannot 

 produce audible beats, the upper partial tones may beat 

 and make the tone rough. Thus, if two tones form a 

 fifth (that is, one makes two vibrations in the same time 

 as the other makes three), there is one upper partial in 

 both tones which makes six vibrations in the same time. 

 Now, if the ratio of the pitches of the fundamental tones 

 is exactly as 2 to 3, the two upper partial tones of six 

 vibrations are precisely alike, and do not destroy the 

 harmony of the fundamental tones. But if this ratio is 

 only approximatively as 2 to 3, then these two upper 

 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 smooth- 

 ness, 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 devia- 

 tion from that ratio has been produced. 



