Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



137 



questions of temperament, shows that the theory agrees with other theories, and with practice. 

 Smith's beat* is a kind of disturbed orbit, of which Tartini's beat is the instantaneous orbit. 



The phenomenon itself is different to different ears. To some it consists in alternations of 

 louder and softer: and undoubtedly there are changes from condensation reinforcing conden- 

 sation, and rarefaction rarefaction, to condensation balanced by rarefaction, and rarefaction by 

 condensation. To others it consists in alternate perception of the two sounds of the conso- 

 nance; and this also is intelligible, as the stronger parts of the two waves alternate. For 

 myself, though I can perceive both the effects above mentioned when I look out for them, the 

 phenomenon which forces itself on my ear is an alternation of vowel-soundsf, as in u-a u-a 

 u-a, &c. pronounced in the Italian way. 



The time of a beat depends upon a circumstance which I suppose, by the manner in 

 which many writers have confined themselves to the case of imperfect unisons, has not been 

 clearly apprehended. The diagrams are only detached portions of a succession unlimited in 

 both directions. If the times of vibration be 3a and 5a, (so that a represents the greatest 

 common measure of the times of vibration, which is repeated 15 times in Tartini's beat,) and 

 if one of the shorter waves begin at zero with one of the longer ones, the first, third, and 

 fifth of the shorter waves are advanced 0, a, 2a, upon the several longer waves. If the first 

 of the shorter waves be advanced ,v (<a) upon the longer one with which it began, then the 

 advances just spoken of become x, a + w, 2a + ai. Accordingly, looking at the full succes- 

 sions, while w progresses from to a the consonance passes through all its phases. Every 

 possible variety of Tartini's cycle is exhibited during the motion of the beginning of one wave, 

 not through the whole of the other, but through that portion of the other which is performed 

 in the time which is the greatest common measure of the times of the two waves. In disen- 

 tangling Dr Smith's explanation, I thought it looked much as if he had first counted on the 

 whole of the other wave, had found that his results would not agree with experiment, had 

 detected tiie true submultiple of the other wave by comparison of his theory with experiment, 

 and had then corrected his former theory. This may be fancy: but, should any one ever read 

 Dr Smith's book again, I should recommend his attention to this point. And should he find 



" 1 have looked in many places to see If 1 could discover the 

 two beats in any other connexion than that of confusion between 

 tile two. I once thought I had succeeded; for there is a paper 

 by Lord Stanhope (Tilloch's Phil. Mag. Vol. xxviii. 1807, 

 June — September, p. 150) which is sometimes cited in a manner 

 which would make one suppose he had the distinction. But on 

 looking at this paper, I find that his distinction between a beat 

 and a beating is this, that the first is merely the vibration of tlie 

 not«, the second is Smith*s beat. There is a rather obscure 

 paragraph at the end, in which he speaks of the beating of a 

 beating occasioned by two badly turned fifths DA, DA, in 

 difTerent octaves, sounding together. 



f And so it struck Emerson, who says (I. c.) — " Its noise 

 is such as this, waw, aw, aw, aw, or ya, ya, ya, ya, ya. Our 

 business is to find out in how many vibrations this perturbation 

 happens, or how many yaws in a second of time." In the 

 organ-pipe, in which the efl'ect is much coarser than in the 

 string, the alternation of vowels is not very self- asserting. 

 Since the text was written, I have tried tlie comparison of pipes 



Vol. X. Part I. 



again, which I had never heard, with any reference to beats, for 

 many years. Mr Davison (of the firm of Gray and Davison) 

 instructed one of his tuners to prepare an octave of equal 

 temperament, by ear in the usual way. On trying one of the 

 fifths, the first thing which struck me was that the beating 

 seemed to be about double what it ought to be. Without say- 

 ing anything, I asked tlie tuner to count the beats in a minute 

 in his own way ; his counting gave the half of mine, and 

 agreed with the theory, nearly. So little did the alternation of 

 vowels present itself, that is, so like was each half beat to the 

 other, that I did not even remember the phenomenon. It was 

 only on a subsequent day, after more practice, that I caught 

 the two vowels. The equal temperament, tuned by a practised 

 tuner, brings the beats near enough to the theory to prove that 

 the complete cycle of Tartini's beat, and not any multiple or 

 sub-multiple of it, is the cause of the phenomenon called 

 Smith's beat, namely, the sound of pulsation which is heard in 

 two halves, with different vowels in the two. 



18 



