Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



135 



tique, p. 252), that when the vibrations of two sounds come together very rarely, we perceive 

 the coincidences like beats (comme des battemens, tres-desagreables...) very disagreeable to the 

 ear in a badly-tuned instrument. The more nearly, he goes on to say, the consonance is 

 made perfect, the more insensible the beats become, until at last they are lost in the sensation 

 of a feeble resonance with a grave sound. And he ends by telling us that an instrument is not 

 in tune if any one of its intervals allow beats to be heard. According to Chladni, then, uni- 

 sons ought to give a grave sound when perfectly tuned; to say nothing of his appearing to 

 believe in some system of tuning a whole instrument in which there are no beats. 



All the modern writers with whom I am acquainted content themselves, at the utmost, 

 with describing the phenomenon, and giving some account of the beats of imperfect unisons, 

 except as I proceed to mention. Some time ago, after detecting the explanation from the 

 formulas, and then unravelling the demonstration of the same formulae with very great diffi- 

 culty, I searched far and wide to see if any writer had appreciated and acknowledged the skill 

 with which Dr Smith had concealed his truth at the bottom of a well of learning. The only 

 writers in whom I found a solution of the problem were as follows. William Emerson, a 

 sound and once well-known, but now nearly obsolete, writer, gave a true solution (p. 484) of the 

 problem in his Algebra, published in 1763. His method is very obscure just at the pinch of 

 the demonstration; we see that certain recurrences are established, but are left wholly in the 

 dark as to why those recurrences should explain the beats; it is quite as likely that two of 

 them should go to a beat as one. Mr Woolhouse, in his Essay on Musical Intervals (Lon- 

 don, 1835, 8vo. p. 84), the best modern manual of mathematical harmonics which I know of, 

 has treated the problem in the same manner, arriving at another variety* of the formula. 

 Both these methods want the introduction of Tartini's beat in its connexion with Smith's; 

 and this the following treatment of the subject will supply. 



Let m and n be two numbers prime to one another, m > n, and let the higher note make m 

 vibrations while the lower note makes n. In the diagram I shall suppose m = 5, n = 3, or the 

 interval a major sixth. I shall also suppose each whole wave to be one of condensation, for 

 simplicity. And first, let two zeros of condensation, one in each wave, be synchronous. 

 The following diagram represents the whole of one wave of Tartini's beat, whether it be the 



• Emerson arrives at the formula which I presently mark as 

 ( 1 — j;) Mn -=- X ; Mr Woolhouse arrives at (.1 — x) Nm. Look- 

 ing at all probabilities, as derived from Emerson's life, habits, 

 and access to books, I very much doubt his method being 

 derived from Dr Smith. He was a musician, and an amateur 

 tuner of instruments; and he was mechanic enough to enrich 

 his own virginal with additional semitones. He was nearly 

 fifty before the first edition of Smith appeared, he lived in the 

 county of Durham on a very small fixed income (about i;60 

 a-year), his writings show very little reading, and the library 

 which he sold before his death, the collection of nearly forty 

 years, was valued by himself under £50. If I could only 

 establish a high probability of acquaintance between Emerson 

 and Thomas Wright, now known as the speculator on the milky 

 way, who lived within twelve miles of Emerson, I should con- 

 sider the united chances of Wright having possessed the book 



and having lent it to Emerson as giving a liigher probability to 

 Emerson having seen it than anything I can create from com- 

 parison of the two methods. It is very likely, then, that he 

 had not seen Smith's Harmonics. The amusing biography of 

 Emerson, which is prefixed to his collected works, and which 

 appears to have been written by some one who had ample in- 

 formation, states that he was a very desultory student till after 

 thirty years of age. Having been treated with contempt by his 

 wife's uncle, he determined to gain a name, that he might 

 prove himself the better man of the two. This he has done : 

 if the name of his relative were now worth inserting, it would 

 only be in connexion with the statement, true or false, that, 

 though possessed of two livings and a stall, he made a large 

 income by the practice of surgery. Emerson died in 1782, in 

 his 81st year. 



