AUDITORY SENSATIONS 567 



numbers. If we put down not only the fundamental tones of these 

 notes but also their overtones, we shall see that there is considerable 

 identity as regards the latter. In the case of the octave the two are 

 almost identical, the only difference being the ground tone of the lower 

 note, and the identity diminishes as we pass from the octave through 

 the thirds to the sixths. The overtones which are identical are shown 

 by black type : 



Fundamental tone Overtone 



f 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 

 Octave 1 : 2 . . ( g 4 g g 1Q 



f 2 . 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 

 \ 3 6 9 12 15 18 



{ 3 . 6 . 9 . 12 15 . 18 . 21 . 24 . 27 . 30 

 Fourth 3:4. - { 8 12 16 20 24 28 



f 4 8 . 12 16 . 20 . 24 . 28 . 32 . 36 . 40 

 Major third 4: 5 .| g 1Q lg 2Q 25 3Q 35 4Q 



| 5 . 10 . 15 . 20 . 25 . 30 . 35 . 40 . 45 . 50 

 Minor third 5 : 6 .< Q ^ ^ ^ ^ ^ ^ 



f 3 6 . 9 . 12 . 15 , 18 . 21 . 24 . 27 . 30 

 Major sixth 3 : 5 .[ g J() ^ , (j ^ 3Q 



( 8 . 16 . 24 . 32 . 40 . 48 . 50 . 64 . 72 . 80 

 &eC01ld8:J -\ 9 18 27 36 45 54 63 72 



I 8 . 16 . 24 . 32 . 40 . 48 . 56 . 64 . 72 . 80 

 Seventh 8: 15 .| ^ ^ ^ 6() ?5 9Q 



In the second and seventh, which are discordant, it is only the 

 eighth overtones which are identical, while the fundamental tones will, 

 as a rule, be so close together that beats will be produced of a number 

 calculated to give dissonance. Since the phenomenon of beats depends 

 on the absolute number of vibrations per second they are more easily 

 produced by two notes near together at the lower end of the scale 

 than at the upper end. Thus the dissonance is quite perceptible in a 

 major third at the lower end of the piano, but disappears at the upper 

 part, since here the beats produced are so rapid that they become 

 imperceptible. 



The various notes used in music are obtained by employing the 

 consonant intervals which we have given above. The major chord 

 is composed of the fundamental tone, the major third and the fifth. 

 If we take ' c ' as the fundamental tone, the notes of the chord are 

 c, e, g, with vibration frequencies corresponding to 1, |, |^, i.e. 4, 5, 6. 

 the major chord from g is g, b, d, i.e. three notes with vibration fre- 

 quencies corresponding to |^ J^-, ^, i.e. 4, 5, 6. The major chord from 



