HAKMONICS. 105 



CHAPTER IH. 



Earliest Stages of Harmonics. 



AMONG the ancients, the science of Music was an application of 

 Arithmetic, as Optics and Mechanics were of Geometry. The 

 story which is told concerning the origin of their arithmetical music, 

 is the following, as it stands in the Arithmetical Treatise of Nicom- 

 achus. 



Pythagoras, walking one day, meditating on the means of measur- 

 ing musical notes, happened to pass near a blacksmith's shop, and had 

 his attention arrested by hearing the hammers, as they struck the 

 anvil, produce the sounds which had a musical relation to each other. 

 On listening further, he found that the intervals were a Fourth, a 

 Fifth, and an Octave ; and on weighing the hammers, it appeared that 

 the one which gave the Octave was one-half the heaviest, the one 

 which gave the Fifth was tivo-thirds, and the one which gave the 

 Fourth was three-quarters. He returned home, reflected upon this 

 phenomenon, made trials, and finally discovered, that if he stretched 

 musical strings of equal lengths, by weights which have the proportion 

 of one-half, two-thirds, and three-fourths, they produced intervals which 

 were an Octave, a Fifth, and a Fourth. This observation gave an 

 arithmetical measure of the principal Musical Intervals, and made 

 Music an arithmetical subject of speculation. 



This story, if not entirely a philosophical fable, is undoubtedly in- 

 accurate ; for the musical intervals thus spoken of would not be pro- 

 duced by striking with hammers of the weights there stated. But it 

 is true that the notes of strings have a definite relation to the forces 

 which stretch them ; and this truth is still the groundwork of the the- 

 ory of musical concords and discords. 



Nicomachus says that Pythagoras found the weights to be, as I 

 have mentioned, in the proportion of 12, 6, 8, 9; and the intervals, 

 an Octave, corresponding to the proportion 12 to G, or 2 to 1 ; a Fifth, 

 corresponding to the proportion 12 to 8, or 3 to 2 ; and a Fourth, cor- 

 responding to the proportion 12 to 9, or 4 to 3. There is no doubt 

 that this statement of the ancient writer is inexact as to the physical 

 fact, for the rate of vibration of a string, on which its note depends, is, 



