Qn Musical Temperament. 191 



name. But it is demonstrated in the Lemma, that the sum 

 ©f the temperaments of each parcel of concords, in the system 

 of equal semitones, is the least possible. Hence no changes 

 in the Vths can diminish the average temperaments of the 

 Illds and 3ds. 



Cor. Hence we derive an important practical conclusion : 

 that whatever irregularities are in'roduced into the scale, must 

 be such as are demanded by the different frequency of occur- 

 rence of the several concords. If wc make any alterations in 

 the scale of equal semitones, this must be our sole criterion. 

 A given system of temperament is eligible, in proportion to the 

 accuracy with which it is deduced from the different frequency 

 of the diflferent concords. And those who maintain that the 

 frequency of different interveils does not sensibly vary, or that 

 it is of such a nature as not to be susceptible of calculation, 

 must, to be consistent, adhere to the scale of equal semitones. 



Proposition X. 



To determine the best distribution of the temperaments of 

 the concords in the Douzeave Scale. 



As the scale of equal semitones has been demonstrated to be 

 the best, on supposition that all the concords of the same name 

 occurred equally often, it ought to be made the standard from 

 which all the variations, required by their unequal frequency, 

 are to be reckoned. To find a set of numbers expressing the 

 relative frequency of the several concords in the common 

 scale, we have only to unite the numbers in Table IV, stand- 

 ing against those adjacent degrees which have but one sound in 

 ^his scale. They will then stand a? in the foll«wing table : 



