On Musical Temperament. 177 
% he for example; a be greater than a’, and let x be any va- 
- riable distance to which C is moved, so as to diminish the 
temperament m, of the chord whose frequency is expressed 
ya. Then the temperament of a will become =m~z, an 
that of a'=m'+2. Hence, as the dissonance head in each, in 
a given time, is in the compound ratio of its frequency of oc« 
currence and its temperament, their aggregate dissonance will 
se 
be as am~x+a'm'+a3 a quantity which, as a is supposed 
greater than a’, evidently becomes a minimum when z=m, 
or the chord, whose frequency is a, is made perfect. But 
in this way we render the harmony of the chords very un- 
equal, which is ceteris paribus, a disadvantage. As these 
considerations are heterogeneous, it must be a matter of 
judgment, rather than of mathematical certainty, what pre- 
tise weight is to be given toeach. We will give so much 
Weight to the latter consideration, as to make the temperament 
each concord inversely as its frequency. We have then 
per 3 
ae so iy : : ; which gives f= 
@:@23:m—* mito od 
But there are six concords to be accommodated, instead of 
\ two; and it is evident that all the pairs cannot have their 
_ temperament inversely as their frequency, since the num- 
bers a, b, &. and m, n, &c. have no constant ratio to each 
other. This, however, will be the case, at a medium, if z 
be made such, that the sem of the products of the numbers 
expressing the frequency of those chords whose temperaments 
are increased by 2, into their respective temperaments, shall 
be equal to the sum of the corresponding products belonging 
to those chords whose temperaments are diminished by 2 
Applying this principle to the system of temperament in Prop. 
Ul. Which flattens all the concords, it is plain that raising any 
given degree by a, will increase the temperaments of the con- 
cords above that degree, and diminish those of the con- 
below it. Hence it ought to be raised’ till : 
m)at(n—2)b+(p—n)e=(m'+x)a+(n'+a)b + (P+x)e’: 
from which x is found = 2m —@'m + bn=bnit Poe 
ah ata’ +b+b'+e+e 
alg either of the temperaments be sharp, the sign of that 
ob LNo, 9. 23 
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