HARMONY. 189 



duced by such combinations was explained mathematically. 

 We do not propose to enter into a full discussion of the laws 

 of harmony, but a knowledge of certain of these laws is es- 

 sential to the comprehension of the physiology of audition. 

 These are very simple, now that we have analyzed the tone 

 of a single vibrating body. 



It is a law in music, that the more simple the ratio be- 

 tween, the number of vibrations in two sounds, the more per- 

 fect is the harmony. The simplest relation, of course, is 

 1:1, when the two sounds are said to be in unison. The 

 next in order is 1 : 2. If we sound C and its 8th, we have, 

 for example, 24 vibrations of one to 48 of the other. These 

 sounds can produce no discord, because the waves never inter- 

 fere with each other, and the two sounds can be prolonged 

 indefinitely, always maintaining the same relations. The 

 combined impression is therefore continuous. The next in 

 order is the 1st and 5th, their relations being 2:3. In 

 other words, with the 1st and 5th, for two waves of the 

 1st, we have three waves of the 5th. The two sounds may 

 thus progress indefinitely, for the waves coincide for every 

 second wave of the 1st and every third wave of the 5th. 

 The next in order, if we sound at the- same time the 1st, 

 5th, and 8th, is the 3d. The 3d of has the 8th of C for 

 its 5th, and the 5th of C for its minor 3d. The 1st, 3d, 5th, 

 and 8th form the common major chord ; and the waves of 

 each tone blend with each other at such short intervals of 

 time that the ear experiences a continuous impression ? and no 

 discord is appreciated. This explanation of the common ma- 

 jor chord illustrates the law that, the smaller the ratio of vi- 

 bration between different tones, the more perfect is their har- 

 mony. Sounded with the 1st, the 4th is more harmonious 

 than the 3d ; but its want of harmony with the 5th excludes 

 it from the common chord. The 1st, 4th, and 8th are har- 

 monious, but to make a complete chord we must use the 6th. 

 These discussions might be extended into the progression of 

 chords and modulation ; but this is not essential to our pur- 



