VII. On the Beats of Imperfect Consonances. By Augustus De Morgan, F.R.A.S. 

 of Trinity College, Professor of Mathematics in University College, London. 



[Read Nov. 9, 1857.] 



The subject of this paper was treated in full, for the first and only time, by Dr Robert 

 Smith, in the two editions of his Harmonics (Cambridge, 1749, 8vo. ; London*, 1759, 8vo.). 

 The results are the same in both editions, but the improvements of the second edition add 

 considerably to the learned obscurity in which the subject is involved. Dr Smith presents, so 

 far as I know, the strongest union of the scholar, mathematician, physical philosopher, and 

 practical musician, who ever treated of mathematical harmonics: and his book is not only the 

 most obscure and repulsive in its own subject, but it would be difficult to match it in any sub- 

 ject. The consequence has been that the point in which Robert Smith made an important 

 addition to acoustics has been little more than a resultf in the hands of some of the organ- 

 tuners. Dr Young certainly did not understand Smith's theory. He was also a remarkable 

 union of the scholar, mathematician (a character in which he deserves to stand much higher 

 than he is usually placed), and physical philosopher: and was a successful student in music; 

 but he wanted a musical ear (Peacock's Life, pp. 59, 79, 81). I have my doubts whether 

 Robison had read more of Smith's theory than its results. For myself, I made out what 

 ought to have been the theory from the formulce, and then was successful in mastering Smith's 

 explanations. 



Before proceeding to the subject, I make some remarks upon the method of dividing the 

 octave. Should this paper fall into the hands of any mathematician unused to musical mea- 

 surement, he must be informed that proximity and longinquity are measured by ratio, not by 

 difference. Thus notes of p and q vibrations per second are at the same interval as notes of 

 hp and kq vibrations per second, be k what it may. Consequently, an interval remains con- 

 stant, not with p - q, but with logp - log g. The octave of any note, which has with that 

 note a sort of identity of effect which no words can describe, makes two vibrations while 

 the note makes one vibration. Any note makes p vibrations while its upper octave makes 

 Zp vibrations : hence log 2p - log^, or log 2, is the measure of every interval of an octave. 



• It is worthy of note that at this period the book bears the 

 name of the place where it is printed, not of the place where 

 the publisher sells it. Both these editions are printed for 

 Cambridge publishers (the Merrills). 



■f- So long as unequal temperament was in use, and even 



now when it is adopted, the beats were and sometimes are used 

 in tuning: but when equal temperament is required (and this 

 system has gained ground rapidly) the tuners have nothing to 

 do with beats, except to get perfect octaves by destroying them. 

 I speak of the organ, and of this country. 



Vol. X. Pakt I. 17 



