136 



Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



pulse of a grave harmonic, or only one of Smith's flutters: namely, five waves of the upper 

 note, three of the lower, and the resultant wave. 



The abscissa represents 15 equal portions of time, of which the component waves take suc- 

 cessive threes and fives; the ordinates represent the condensations at the end of the times 

 represented by the abscissas. The thick line, whose ordinate is always the sum of the other 

 two, represents the wave of Tartini's beat, which is repeated in the next fifteen portions of 

 time. 



The united effect of the two waves is one particular phase of a major sixth: a pulse of 

 the grave harmonic in which gradations of loudness and faintness are distributed in a certain 

 manner through 15 portions of time, to be strictly repeated in the next 15 portions, and so on. 

 An unlimited number of other phases exist, one for every mode in which the zero of conden- 

 sation of the shorter wave can be laid down in the longer wave, so as to produce a law of 

 loudness and faintness which is not found in any other mode. Thus the following is the dia- 

 gram in which the maximum condensation of the shorter wave synchronises with the zero of 

 condensation of the longer wave. 



We have now Tartini's beat under a different type, in which the loudness and faintness 

 are distributed in another way: the consonance of a major sixth, as before, with a different 

 kind of pulse for the grave harmonic, if there be one. Whether the ear would acknowledge 

 any difference between two major sixths of these different types, cannot be settled; for it is not 

 in our power to start the pulses as we please. But the ear does acknowledge the gradual 

 progression through all the types, by recognizing what I have called Smith''s beat. If the 

 consonance be a very little mistuned, Tartini's cycle is not sensibly altered in character, but 

 its recommencement undergoes a very small change. If the higher note be tuned a little too 

 sharp, for example, so that the shorter wave is a very little less than three-fifths of the longer 

 wave, Tartini's cycle, or something excessively like it, begins a little sooner the second time 

 than it should do; and the zero of condensation of the shorter wave is thrown back a little. 

 This effect is doubled at the next commencement, trebled at the next one, and so on: accord- 

 ingly, in a consonance slightly mistuned, the approximate compound pulse goes through all the 

 phases which variations in the mode of setting off can give to the true one. This is the 

 most marked geometrical effect upon the pulses; and Smith's beat is the most marked acousti- 

 cal effect upon the ear. The connexion of the two is then of the highest probability : and this 

 becomes certainty so soon as, and not until, the study of the beats, and their application to 



