On Musical Temperameni. i9f 
name. But itis demonstrated in the Lemma, that the sum 
of the temperaments of each parcel ef concords, in the system 
of equal semitones, is the least possible. Hence no changes in 
the Vths can diminish the average temperaments of the IIIds 
and 3ds. : 
Cor. Hence we derive an important practical conclusion: 
That whatever irregularities are introduced into the scale, must 
be such as are demanded by the different frequency of occur- 
rence of the several concords. If we 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 different concords. And those who maintain that the 
frequency of different intervals 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 tobe 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. standing 
against those adjacent degrees which have but one sound in this 
“cale. They will then stand as in the following table : 
