140 



Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



Let the notes of the imperfect consonance be P, Q, and let P' be the octave above P. 

 If the interval PQ be tuned too flat, then QP^ is two sharp, and vice versa. All remaining 

 as above for PQ, in passing from PQ to QP^ we must change N, M into M, ZN. If m be 

 an odd number, we must change n, m, into m, 2w; but if m be even, n, m, must change 

 into im, n; since the fundamental ratio must be in its lowest terms. And we must also 

 change the sign of q, neglecting the negative sign of the value of /3, when it occurs. Conse- 

 quently, jS' being the number of beats of QP^ in a second, we have 



(m odd) /3 = 



(m even) /3 = 



2q 



l6lp + q 



Nm, /3'= 



2? 



I6lp - q 



M.Zn = 2/3; 



9.q 



\6\p + q 



Nm, /3' = 



^q 



l6lp - q 



M.n = fi. 



That is, when the fundamental number (in the ratio m : n) of the mistuned note is odd, the 

 interval complemental to the octave beats twice as fast as the lower interval first given. But 

 when this fundamental number is even, the interval and its octave complement have the 

 same rate of beating. This is one of Smith's* experimental verifications, and is a very easy 

 one. He is of opinion that an octave might probably be tuned with more perfection by the 

 isochronous beats of a minor and major concord composing it, than by the judgment of the 

 most critical ear. 



What precedes is a particular case of the following theorem: — Let N, M, L, be three ascend- 

 ing notes represented by their numbers of vibrations per second. Let N make n vibrations 

 while M makes m: let M make m vibrations while L makes I: the fractions m : n and I : m 

 being in their lowest terms. Let the imperfect consonances NM, ML, NL, beat severally /3, 

 j8', B, times per second: )3 being positive when the higher note is flat, and negative when it is 



• There must needs be some way of explaining the excessive 

 difficulty of this one work of Dr Smith's. His Optics, if not 

 a model of perspicuity, is by no means notable for obscurity; 

 on the contrary, I find it abounding insufficiently good descrip- 

 tions of machinery, a point in which an obscure writer is 

 generally most perplexed and perplexing. I take the cause 

 of Dr Smiih's failure of clearness in the Harmonics to be that 

 he was a practical musician, well versed in the practical writers. 

 I suppose others have agreed with myself in noting tliat the 

 worst explainers are those who have to describe the purely con- 

 ventional, without having had it distinguished from the natural 

 or the essential in their education. First come the writers 

 on games of chance, who all, or with the rarest exception, 

 proceed to explain whist or hazard by commencing at the point 

 at which they imagine a priori knowledge of the arrangements 

 ceases. Next come the musicians, with whom a five-line stave, 

 &c. are In the nature of things. Now Dr Smith had got into 

 the way of interchanging the practical and theoretical, the 

 accidental and the essential, &c. The manner in which he 

 treats the theorem on which this note is written is perhaps 

 the easiest instance to produce. He gets into the theorem in a 

 way which leads him to the table of ratios of vibrations, and 

 he arrives at this result, that wlien the minor consonance is 

 above the major, the higher consonance beats twice as quick as 



the lower, but when the minor consonance is below the major, 

 the beats are the same. And not until he has pointed this out, 

 does he proceed to note that the greater term of the ratio of 

 a minor consonance is even, &c. And his final theorem is 

 stated in terms of major and minor consonance, it being merely 

 accidental, so far as our knowledge is concerned, that the nume- 

 rators of minor consonances happen to be even, in the cases in 

 which they are useful. The usual minor intervals are the 



tone (-g); the third \t) ; the sixth lA; the seventh 

 (— -). The usual major intervals are the tone (5); the third 

 (- j ; the fourth (-1, in which there is a failure; the fifth 



(- j ; the sixth (- 1 ; the seventh (-0-). In the minor and 



lib 16\ ^ , . . 

 major semitone I — , jr I the rule is inverted ; and also in the 



minor fifth ( 55 )• Keeping, however, to common intervals 



used in tuning, and calling ihe fourth a minor to ihe fifth, it is 

 a pretty practical rule that the duplication of beats takes place 

 when the minor interval is above the major. 



