B E A 



370 



B E A 



Benr. Example. 



If the conchord be the minor sixth of Earl Stan- 



w 

 hope's monochord system : here | = is the con- 

 chord, and [Phil. Mag. xxvii. 195.) /= a flat tem- 

 perament of 51500 in seven place logs. Also N=240, 

 the vibrations per 1" : and from the first of the lower 

 theorems, we have 

 WljjOOXgx gg = ^7760000 



868600+51500 8737500 



beats in 1". 



4/A Method. 



Let the conchord whose perfect ratio is expressed 

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

 est term,) be tempered so that its acute and grave 

 sounds make M and N complete vibrations in one 

 second of time, respectively ; and let b be the num- 

 ber of beats occasioned by this temperament in one 

 second of time. 



Then, if the temperament be sharp, b=nM niN. 



Or, if the temperament be flat, &=iN nM. 



Example. 



If the conchord be the minor sixth of Earl Stan- 

 hope's monochord system, here -jr = is the con- 

 chord, and (PA/7. Mag. xxx. p. 5.) M= 379.47 and 

 N=240 are the vibrations respectively ; and from 

 the second of the above theorems we have, 



8x2405x379.4.7=1920 1897.35=22.65 the 

 beats in 1". 



5th Method. 



Let the conchord, whose perfect ratio is expressed 

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

 est terms) be tempered by r Schismas (S in the Table, 

 Plate XXX.), neglecting the smaller intervals most 

 minute (m) and lesser fraction (fl), should they oc- 

 cur, ana if great accuracy is sought, substituting 

 their value in decimals of S : also let M and N be 

 the number of complete vibrations in one second of 

 time made 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. 



2t* x m x N 

 Then, if the temperament be sharp, J= -tmq 



1r x X M 



2X10-5X8X240 



1772+r 



Example. 



If the conchord be the minor sixth of Earl Stanhope's 



5 n 



monochord system, here = is the conchord ; and 



' 8 m 



(Phil. Mag. xxviii. 141.) r= a flat temperament of 

 10.5 schismas; and N=240, the vibrations of the bass 

 per second : and from the first of the lower theorems 

 above we have, 



403' '0 Beat, 



: t^tt-. = 22.6199, the beats Beaton. 



1772 + 10.5 

 in 1". 



Note. .0078631 x 2=m, and 127.1905 X m=2 ; 

 also .149()61 X S=/, and 6.5297 X/=2. The near 

 coincidence of the above six results would have been 

 still more complete, but that the first, third, and 

 fifth methods are founded on approximating theo- 

 rems, and the vibrations M, used in the fifth method, 

 are not given to places enough of decimals to insure 

 a result equally accurate with the other calculations. 



By two, at least, of the above methods, the beats 

 produced by every conchord, throughout several 

 tempered systems, have been calculated, and will be 

 given in Tables, under the names of those systems, 

 or that of their respective authors, as Hawkes, 

 Smith, Staxhope, Young, &c. ; reserving an ac- 

 count of such systems as may come to our know- 

 ledge, but under no well-known name, for the arti- 

 cle Tempered Systems of Music, wherein we shall 

 endeavour to draw some comparisons between the 

 different systems of temperament, whose correct re- 

 sults will thus be exhibited, in a form perfectly 

 adapted for comparing their respective merits : and, 

 we propose, to aid these comparisons, by some new 

 and general investigations, on the relations subsisting 

 between the temperaments of the different conchords, 

 in every donzeave, or tempered system of twelve in- 

 tervals, only within the octave. ($) 



BEATON, or Beton, David, archbishop of 

 St Andrews, primate of Scotland, and cardinal of 

 the Romish church, the son of John Beaton of Bal- 

 four, in Fife, was born in 1494. He was educated 

 at the university of St Andrews, and gave early in- 

 dications of strong mental powers. Being destined 

 for the church, he was sent, by his uncle, the arch- 

 bishop of St Andrews, to complete his education at 

 the university of Paris ; and, as soon as he attained 

 the usual age, he was admitted to holy orders. In 

 France he was early received into the favour and ser- 

 vice of the Duke of Albany, regent of Scotland, 

 during the minority of James V. ; and, by him, in 

 1519, he was appointed Resident at that court. 

 About the same year, his uncle bestowed upon him 

 the rectory of Campsie ; and, in 1523, vested him 

 with the abbacy of Aberbrothock. He returned to 

 Scotland in 1525, and received the privy-seal in 

 1528. In 1533, we find him, in conjunction with 

 Sir Thomas Erskine, returning to France, to con- 

 firm the treaty entered into between the two nations, 

 and to demand in marriage for his sovereign the 

 daughter of the French king. During his residence 

 at that court, he insinuated himself into the good 

 graces of Francis; was favoured with a knowledge 

 of the whole political system of that great monarch ; 

 and, by his influence in bringing over his sovereign to 

 adopt the same political views, he laid the founda- 

 tion of his own future greatness. The nuptials ot 

 the Scottish king and the young princess, were cele- 

 brated at Paris, on the 1st of January 1537; and 

 Beaton returned with them to Scotland in the en- 

 suing May ; but the queen having survived her mar- 

 riage only two months, he was again sent to Paris, 



If very great exactness is required, 1772.243 mny be used instead of 177?. 



