.6324554--.625X 210 X8=•6324554 the beats in 1".
Corollary. If in this method, the bass-note. be considered as unity, then S= , and our theorems be ns .
come, For sharp temperaments, b=C—m) x N For fat temperaments, N ( 632455 the above example will stand thus ; viz. )— 210— 094306 x 240=22.6331, the beats in 1".
3d Method.
Let the conchord whose perfect ratio is expressed by , (a being the least term of the ratio in its lowest terms,) be tempered by l 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 he the number of beats occasioned by this temperament in one second.
Then, if the tempe-1 x x N 27 x n X M rament be sharp, .1 8686000—r °r 8686000+ i* Or, if the tempera-1 b— x xN2/XnXM ment be flat, or + e 8686000—r Example.
If the conchord be the minor sixth of Earl Stan hope's monochord system : here is the con chord, and (Phil. Mag. xxvii. 195.) l= a flat tem perament of 51500 in seven place logs. Also N=210, the vibrations per 1" : and from the first of the lower theorems, we have 2 X 51500 x 8 x 240197760000 = 22.6335, the 868600+51500 = 8737.500 beats in 1".
4th Method.
Let the conchord whose perfect ratio is expressed by (n being the least term of the ratio in its low ea 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—mN. Or, if the temperament be fiat, b=nzN—nM. Example.
If the conchord be the minor sixth of Earl Stan.
n hope's monochord system, here 5 — = is the con 8 nichord, and (Phil. Mag. xxx. p. 5.) M=379.47 and N=240 are the vibrations respectively ; and from the second of the above th'eorems we have, 8 x 240--5 x 379.47=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 nr est terms) be tempered by r Schismas (E in the Table, Plate XXX.), neglecting the smaller intervals most
minute (tn) and lesser,fraction (f), should they oc cur, and if great accuracy is sought, substituting their value in decimals of M : 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.
Then, if the temperament be sharp, b= 2r xmxN 1772—r 2rXnXM •r 1772+r *• Example.
If the conchord be the minor sixth of Earl Stanhope': monochord system, here 5 —= the conchord; ane 8 ni xxviii. 141.) r= a flat temperament of 10.5 schismas; and N=240, the vibrations of the bas; per second : and from the first. of the lower theorerri; above we have, 2 X 10.5 X 8 X 240 40320 —2— = 2.619,, the beats 1772 +10.5 — 1782.5 in 1".
Note. .0078631 X E=nt, and 127.1905 x m=m ; also .149661 x 2=f, and 6.5297 Xf=z. 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 FlAwKr.s, SMITH, STANTIOPE, 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 SvsTEms 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 donxeave, or tempered system of twelve in tervals, only within the octave. (g)