HARMONICS. 105 



CHAPTER III. 

 EARLIEST STAGES OF HARMONICS. 



A MONG the ancients, the science of Music was an application 01 

 * Arithmetic, as Optics and Mechanics were of Geometry. The 

 ftory 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 two-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 

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



