HARMONY IN MUSIC. 55 



other ratios expressible by small whole numbers. But why I 

 What have the ratios of .small whole numbers to do with con- 

 cord 1 This is an old riddle, propounded by Pythagoras, and 

 hitherto unsolved. Let us see whether the means at the com- 

 mand of modern science will furnish the answer. 



First of all, what is a musical tone? Common experience 

 teaches us that all sounding bodies are in a state of vibration. 

 This vibration can be seen and felt ; and in the case of loud 

 sounds we feel the trembling of the air even without touching 

 the sounding bodies. Physical science has ascertained that any 

 series of impulses which produce a vibration of the air will, if 

 repeated with sufficient rapidity, generate sound. 



This sound becomes a musical tone, when such rapid im- 

 pulses recur with perfect regularity and in precisely equal times. 

 Irregular agitation of the air generates only noise. The pitch 

 of a musical tone depends on the number of impulses which 

 take place in a given time ; the more there are in the same time 

 the higher or sharper is the tone. And, as before remarked,, 

 there is found to be a close relationship between the well-known 

 harmonious musical intervals and the number of the vibrations 

 of the air. If twice as many vibrations are performed in the 

 same time for one tone as for another, the first is the octave 

 above the second. If the numbers of vibrations in the same 

 time are as 2 to 3, the two tones form a fifth ; if they are as 4 

 to 5, the two tones form a major third. 



If you observe that the numbers of the vibrations which 

 generate the tones of the major chord C E G c are in the ratio 

 of the numbers 4:5:6:8, you can deduce from these all 

 other relations of musical tones, by imagining a new major 

 chord, having the same relations of the numbers of vibrations,. 

 to be formed upon each of the above-named tones. The num- 

 bers of vibrations within the limits of audible tones which 

 would be obtained by executing the calculation thus indicated 

 are extraordinarily different. Since the octave above any tone 

 has twice as many vibrations as the tone itself, the second octave 

 above will have four times, the third has eight times as many. 

 Our modern pianofortes have seven octaves. Their highest 



