ON HARMONICS. 391 



beat in ei'ery second, it is obvious that the second sound must consist of 

 101 vibrations in a second. Again, if we have two portions of a similar 

 chord equally stretched, or two simple pipes, of whicli the lengths are in 

 the proportion of 15 to 16, they will produce a beat in 15 vibrations of the 

 longer; and if we count the numbei' of beats in 15 seconds, we shall find 

 the number of vibrations in a single setond. The easiest way of procuring 

 two such strings or pipes, in practice, is to tune them by a third, so that 

 they may be respectively 4- and -J of its length; the vibrations of the third 

 pipe in a second will also be equal to the number of beats of the first two in 

 12 seconds. (Plate XXV. Fig. 353.) 



When the beats of two sounds are too frequent to be heard as distinct 

 augmentations of their force, they have the same effect as any other impulses 

 which recur in regular succession, and produce a musical note, which has 

 been denominated a grave harmonic. Thus, two sounds in the proportion of 

 4 to 5, produce, when they are both very low or grave, an audible succes- 

 sion of beats; but when they are higher or more acute, a grave harmonic, 

 which may be separately distinguished as a third sound by an attentive ear. 

 Those combinations of sounds, which produce beats distinctly audible, have 

 in general a harsh and coarse effect, and are called discords; but those of 

 which the vibrations are so related, as to have a common period after a few 

 alternations, and which may be observed to produce a third sound, constitute 

 concords, which are in themselves the more perfect as the common periods 

 are shorter. (Plate XXV. Fig. 353.) 



The natural association of the secondary sounds, which generally ac- 

 company almost all musical notes, serves in some measure as a foundation 

 for the science of harmonics, the same sounds, as are tlms frequently con- 

 nected in nature, being found to be agreeable when united by art. But it 

 appears to depend still more immediately on a love of order, and a predilec 

 tion for a regular recurrence of sensations, primitively implanted in the 

 human mind. Hence, when two sounds are heard together, those propor- 

 tions are the most satisfactory to the ear, which exhibit a recurrence of a 

 more or less perfect coincidence at the shortest intervals, expressed by the 

 smallest numbers of the separate vibrations: for though Ave cannot im- 

 mediately estimate the magnitude of the vibrations, yet the general effect of 



