110 



theory of sound. When once that theory has been ac- 

 cepted, it needs nothing more than inspection of diagrams 

 to see that the waves produced by the sound-vibrations of 

 two simple musical tones interfere less and reinforce each 

 other more exactly in proportion to the simplicity of the 

 ratio between their respective number of vibrations in a 

 given time ; and the investigation, partly physical and 

 partly physiological, of the relations of the more complex 

 musical tones and their effect upon the ear and the mind 

 leads to the same general result, that the only case in 

 which two tones can be simultaneously heard by the ear 

 without mutual disturbance is when the vibration-numbers 

 of those tones bear to each other certain very simple 

 ratios. By the vibration number of a tone is meant the 

 number of vibrations in a unit of time necessary to pro- 

 duce that tone. 



The simplest ratios are of course those expressed by 

 the smallest numbers. The simplest ratios from 1 to 2 

 are, therefore, in their order, 



1, 2, f, |, , |, , f, -I, f, , f, !-, f , etc. 



These ratios do not include all possible, nor even all 

 actual relations ; but they embrace all the so-called con- 

 sonant intervals and several of the so-called dissonant 

 intervals. An attempt has been made to show that these 

 intervals are pleasing exactly in proportion to the small- 

 ness of the numbers expressing the ratios. This theory 

 has not been maintained, probably because of the working 

 of some physiological principles together with the physi- 

 cal. 



The relation 2 : 1 is the ratio between the vibration- 

 number of any tone and that of its octave ; and the 

 problem of temperament is practically confined to the 

 limits of an octave, because the key-board of any octave 



