Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



133 



which Smith shows how to make of his own beat; namely, the deduction of the number of 

 vibrations in a note. It is also true that Sauveur applies the term hattemens to both, and 

 quite correctly; for both are hattemens, though arising from different sorts of cycles. But it 

 is not true that Sauveur confounds the phenomena by imagining them to be the same, by put- 

 ting one in the place of the other, or by giving to either the reason of the other. His object is 

 (Mem. Acad. So. 1701, Paris, 1719, p. 359) to find the son fixe, as he calls it, which makes 100 

 vibrations in a second. He directs us to take organ-pipes, at least two feet long, and to tune 

 diatonic intervals so perfect that not the smallest battement shall be perceived. Here he 

 speaks of what I call Smith''s beats, of which he clearly knows the negative use, namely, the 

 acquisition of perfect concords by avoiding them. Having thus procured a perfect major and 

 minor third to one note, he sounds them together, the interval being 25 : 24 in ratio of vibra- 

 tions, and thus procures a battement (but this is Tartini's beat) at each 25th vibration of the 

 upper note. By taking nearer* consonances, though certainly not harmonic ones, he procures 

 beats which can be easily counted. Dr Smith {Harmonics, 2nd Ed. p. 96) complains that he 

 cannot count Sauveur's beats: but, though he used low notes, he took the prominent concords 

 or discords of the scale, which are not near enough. 



Dr Young pronounced Smith's work " a large and obscure volume, which for every pur- 

 pose except the use of an irapracticablef instrument leaves the whole subject precisely where 

 it found it." If Dr Young had said that the work was largely obscure, he would have 

 been correct: had the volume been larger, it had probably been less difficult; it is a small 

 volume for the quantity of subject-matter. It leaves the subject where it found the subject 

 only in the minds of those who do not master it; in which number we must place Young 

 (Peacock, Life of Young, pp. 128, 129; Works, Vol. i. pp. 83, 84, yS, 134 — 139; Robison, 

 Mech. Phil., Brewster's edition, Vol. iv. pp. 408, 411, 412). One sentence from Young will 

 make it clear that he confounded Tartini's beat with Smith's, though Smith had distinctly 

 stated (p. 97) that "a judicious ear can often hear, at the same time, both the flutterings and 

 the beats of a tempered consonance, sufficiently distinct from each other." But Young says 

 (i. 84), " The greater the difference in the pitch of two sounds the more rapid the beats, till 

 at last, like the distinct puffs of air in the experiments already related, they communicate the 

 idea of a continued sound; and this is the fundamental harmonic described by Tartini." 



• He inserts between the two, 24 and 25, the pipe 24J, and 

 making the three sound together, gets a three-pipe beat of 48, 

 49, 50 vibrations. He then inserts 48.J and 49J, and gets a five- 

 pipe beat of 96, 97, 98, 99, 100 vibrations. These are the beats 

 which he proposes to count ; so that, though he sets out with 

 Tartini's beat, his experiment is as far removed as can be, even 

 from the mere use of this, and has nothing to do with Smith's 

 theory. Strange that Young, who actually refers to Sauveur, 

 should call Smith's theory nothing but an extension of this 

 multipipe clatter : strange also that Robison should imply the 

 same thing. It is said that Sauveur's musical ear was very 

 bad. That he sounded these pipes together is clear ; for of the 

 three first mentioned he says, that the beat of the first and 

 third is faintly audible through the beat of the three. When 



his five pipes sounded together, each of the consecutive inter- 

 vals was something less than the fifth part of a mean semitone. 

 Any one whose ear was thus guillotined might well have ex- 

 claimed. Oh ! Mtisique ! que de crimes on commei en ton 

 nom ! 



f This entirely relates to the second edition. No doubt 

 some readers of Dr Young have searched their copies of Smith's 

 first edition for this instrument, without finding it. It is the 

 account of an enharmonic harpsichord, which is described in 

 the work, and with improvements in a postscript to the second 

 edition, with a separate title page, in 1762, three years and a half 

 after the publication of the work. The enharmonic piano-forte 

 would not be impracticable, if people cared enough about the 

 accession to pay for it. 



