BEAT 



369 



Beat. the advancement of this sublime and beautiful science, 

 to exhibit theo"ms for calculating the number of 

 beats made in a given time, divested as much as pos- 

 sible of the difficulties likely to deter the practical 

 tuner and musician from attempting to understand 

 and apply them to use, illustrated by an example in 

 each case. For the satisfaction of such as are unable 

 or unwilling to go into the nice and difficult theory 

 on which these theorems are founded, nothing is so 

 likely to inspire confidence in their truth, as well as 

 in the right application of the rules they furnish to 

 particular cases they may undertake to calculate, as 

 the having several such theorems, involving different 

 data, yet by means of which the same results are to 

 be obtained. 



Of the five methods given below, for calculating 

 the beats of any tempered conchord from different 

 data, the two first only have hitherto been published, 

 as far as we are acquainted ; the first is the original 

 method of Dr Smith, Harmonics, prop, xi.; and the 

 second is that of Mr Emerson, Algebra, prob. ccii. 



1st Method of calculating the Beats of an Imperfect 

 Conchord. 



Let the conchord, whose perfect ratio is expressed 

 by > ( " being the least term of the ratio in its lowest 



terms) be tempered by the fraction % of a major com- 



P 

 via, (a being the least term of this fraction ;) also let 

 M and N be the number of complete vibrations in 

 one second of time, made or excited by the acute and 

 the grave notes of the above tempered conchord re- 

 spectively : and let b be the number of beats occasion- 

 ed by thi8 temperament in one second. 



Then, if the tempera 

 rr.ent be sharp, or the 

 chor&greaterthzn per- 

 fect, 



Or, if the tempera 

 ment he fat, or the 

 chord less than per 

 feet, 





2<7 X m x N 2q x X M 



z or 



161p q 16lp + q 



_2y X w X N Qq x n X M 

 161p+q r 16\pq 



Example. 



Iftheconchordproposed,bctheminorsixth(C4A)of 

 Earl Stanhope's monochord system : here I = is the 



. VI 



ratio of the perfect conchord, and (Phil. Mas. xxvii. 

 195.) tt=- is the part of a comma nearly, (not ?, 



as erroneously printed), by which the same is flatten- 

 ed : also N=240, the number of complete vibrations 

 cl C, the bass note ml": and from the first of the 

 lowest of the theorems above, 



we have 2 *21x8x240 80640 

 161x22+21" 3563 

 beats in 1". 



VOL. III. PAHTir. 



= 22.6326, the 



2c? Method. 



Let the conchord, whose perfect ratio is expressed 



by , (n being the least term of the ratio in its lowest 



terms) be tempered so that its string, which, for sound- 

 ing the treble note of the perfect conchord was S in 

 length, is altered to be * length : also let N be the 

 number of complete vibrations in one second of time 

 made by the bass-note of the conchord ; and let b be 

 the number of beats occasioned by this temperament 

 in one second of time. 



Then, if the temperament be sharp, b= x N X m. 



s g 



Or, if the temperament be fat, I x N X * 



Example. 



If the conchord proposed, be the minor sixth of 



Earl Stanhope's monochord system : here != is 



8 m 



the ratio of the perfect conchord, and (Phil. Mao: 

 xxvii. 196, and xxx. p. 1.) S=.625, and $=.6324554, 

 are the lengths of string for sounding this perfect and 

 tempered conchord with the bass-note =1, respective- 

 ly : also N=240, the vibrations of C the bass-note in 

 1", and from the second theorem above, we have ' 



.6324554625 n4n B 14.314368 



X 240 X 3= ^,,.., =22.6330, 



6324554 

 the beats in 1" 



6324554" 



( 



Corollary. If in this method, the bass-note be 

 considered as unity, thenS= , and our theorems be- 

 come, 



Torsharptemperaments, b=(- w) x N 

 For fat temperaments, b=fm ) xN 



And the above example will stand thus ; viz. 



8 ^32455l) =:24<) =- 094306 X 240=22.6334, the 

 beats in 1". 



3d Method. 

 Let the conchord whose perfect ratio is expressed 

 by ~ (n being the least term of the ratio in its 



lowest terms,) be tempered by I logarithms, (of seven 

 places, wherein 1.0000000 expresses the key, and 

 .6989700 the octave : ) also let M and N be the num- 

 ber of complete vibrations in one second of time, 

 made or excited by the acute and grave notes of the 

 above tempered conchord respectively ; and let b be 

 the number of beats occasioned by this temperament 

 in one second. 

 Then, if the tempe-1 2/x0<N 2/x X M 



rament be sharp, J - 8686000 C r 8686000 + f 

 Or, if the tempera- 7 2/xixN _ 2xxM 



mentbe>f, J "- 8 ggg 



Bcftt. 



' 8686000 +f 

 3 a , 



8686000 C 



