of the Musical Scale. 199 



lumns, being respectively equal, only the former are vulgar 



fractions (composed of the musical primes 1,2, 3, and 3, or 



their multiples) expressed in their lowest terms, and the 



latter in decimals, as better adapted to the divisions of the 



string upon a monochord. 



I beg now to return to the hi-equal thirds No. 6, ia 



_ , , __ VIII - III , . , • . , 



1 able II, or — , aa an example tor explammg the 



numbers in the columns of the tables ; and first, in order to 

 express the same in vulgar fractions, we have VI1I--III=-|- 

 x-f-=-|, (or 6th) of which the square root is to be extracted, 

 answering to bisecting or equally dividing the interval ; but 

 as neither the numerator or denominatcr are square num- 

 bers, multiply them both by 2 or by |- (which does not alter 

 the value of the fraction), and we get -f§^, of which the square 



root is , as in the table. If now we take the loora- 



4 

 rithm of 10, or 1*0000000, and halve it (answering to ex- 

 tracting the square root), we have '5000000 for the loga- 

 rithm of the numerator, from which deduct the logarithm 

 of 4, equal 0-6020600, and we have '8979400 for the loga- 



v lo 



rithm of the ratio , as in the table ; while in column 6, 



4 

 •7905694 is the number answering to the last logarithm, as 



. . . . V'lo 



It IS also the decimal value of the fraction in column 4. 



4 



Let us now consider that the logarithm of the true major 

 third, or 4-, is '903090O ; from which deducting '8979400, 

 the logarithm found above, we have '0051500 or 51500 for 

 the difference in cohunn 7, and which is marked as a sharp 

 temperament, or relating to an interval greater than perfect; 

 because the logarithm of the li-equal third in question is less 

 than the logarithm of the III, and the logarithms of inter- 

 vals (see column 7 or 5) decrease as the intervals themselves 

 increase. 



The smallest musical interval which the antients consi- 

 dered, was the difference between the tone major (T) or -J, 

 and the tone minor (/) or -,9^, that is, -,»„ x *•, equal to » J , 



which 



