CONSONANCE AND DISSONANCE 615 



being the ground tone of the lower note, and the identity diminishes as we 

 pass from the cctave through the thirds to the sixths. The overtones which 

 are identical are shown by black type : 





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 dis- 

 sonance 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 fre- 

 quencies corresponding to 1, |, f , i.e. 4, 5, 6, the major chord from g is g, b, 

 d, i.e. three notes with vibration frequencies corresponding to f , l-f-, f , 

 i.e. 4, 5, 6. The major chord from the fourth, /, is /, a, c, with the vibration 

 frequencies -J, f -|, -^-, i.e. 4, 5, 6. The C major scale is therefore as follows : 



CDEFGABC 

 1 9 5 4 3 5 15 2 



3 2 



Different instruments are tuned to one normal note, i.e.' to A with 440 

 vibrations per second (this note varies somewhat in different countries). 

 Taking this as the normal, the vibration frequencies of the various notes 

 used in music are given in the following Table : 



