184 



On Musical Temperament. 

 Proposition VIII. 



To comgare the harmoniousness of the foregoing systein 

 with that of several others, which have been most known 

 and approved. 



The aggregate of dissonance, heard in any tempered con- 

 cord, is as its temperament (Prop. I.) when its frequency of 

 occurrence is given, and as its frequency of occurrence, when 

 its temperament is given : hence, universally, it is as the 

 product of both. The whole amount of dissonance heard in 

 all the concords of the same name must consequently be as 

 the sum of the products of the numbers denoting their tem- 

 peraments, each into the number in Table IV. denoting its 

 frequency. These products, for the scale of Huygens which 

 divides the octave into 31 equal parts, of which the tone is 

 5 and the semi-tone 3 ; for the system of mean tones, and for 

 Dr. Smith's system of equal harmony, compared with the 

 scale of the last proposition, (cutting oflf the three right-hand 

 figures) stand as follows : 



TABLE VIII. 



Were we to adhere to Dr. Smith's measure of equal har- 

 mony, the rows of products belonging to the Vths, Illds, and 

 3ds, must be divided, respectively, by i, j\, and -f'j (the 

 reciprocals of half the products of the terms of their perfect 

 ratios,) before they could be properly added to express the 

 whole amount of dissonance heard in all the concords ; but, 

 according to Prop. I. the simple products ought to be added, 

 aud tlie sums at the bottom of the table will express the true 



