CONCORD. 407 



of their vibrations, 62-}-f and 66, give 3 T ^) ; in this case 

 the sound is not less discordant than in the case of V and 

 c", but the discord is due to the rapid beats of the over- 

 tones which accompany the deeper notes. The discord 

 produced by c' and V is as bad as that of l> and c", 

 although the two notes, which are distant by nearly an 

 octave from one another, do not themselves give rise to 

 beats ; in this case rapid beats are produced by the com- 

 bination of one fundamental note, viz., 6', with the over- 

 tone c" contained in c' '. A similar jcause of discord may 

 be traced in other combinations ; thus/' and V sound dis- 

 agreeable, because the overtone d" (with 3 x 352 = 1056 

 vibrations) is contained in/ 7 , and d" together with b" 

 (2x495 = 990 vibrations ) r which is contained as octave 

 in b', produce 1056 9'90' 6'6 : vibrations per second. 



If the relation of those notes is investigated which, 

 when sounded together, produce a pleasing effect, or con- 

 cord (consonance), it will be found that neither their 

 fundamental tones nor their overtones give rise to rapid 

 beats. A given note together with its octave produces 

 no beats ; together with, the fourth or fifth, only weak 

 beats caused by overtones which are pretty high 

 and therefore not very perceptible; somewhat stronger 

 beats originate in the remaining concords (the third 

 and sixth, minor and major) : hence the sound is. in these 

 cases not quite so agreeable as in the first-mentioned 

 concords. 



The existence or absence of strong and rapid beats 

 is thus the sole cause of the dissonance or consonance of 

 notes which are sounded together. 



This may be strikingly proved if the student possesses a delicate 

 and well-trained ear. An interval which produces a most un- 



