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On Musical Temperameni. 189 
perfect, all the Vths on the same letter, in whatever octave 
they are situated, must have the same temperament. 
The reasoning is precisely the same for the IIIds and Sds, 
considering the former as forming 4 distinct series of an oc- 
tave each, beginning with C, C4, D and Eb ; and the latter 
as forming $ distinct series of an actave each, beginning with 
€,C# and D. If the former be made all equal, each will be 
sharpened 343 ; if the latter be made equal, each will be flat- 
tened 392. In every system which renders none of the for- 
mer flat, and none of the latter sharp, the sum of their tempera- 
ments will be 12 x 343, and 12x 392, respectively. 
Cor. The demonstration holds equally true, whatever be 
the magnitude of a, 6, c, &c.: only if they be such that the 
difference —a+b—b+c, &c. of any two successive ones be 
greater than the temperament of the corresponding concord in 
the system of equal semitones, the temperament of that chord 
Must be reckoned negative, and the sum, in the enunciation 
of the proposition, must be considered as the excess of those 
temperaments which have the same sign with those of the same 
concords in the system of equal semitones, above those which 
have the contrary sign. Hence it is universally true that the 
excess of the flat above the sharp temperaments of the Vths 
is equal to 1249; that the excess of the sharp above the flat 
temperaments of the [Ids is equal to 12343; and that the 
excess of the flat above the sharp temperaments of the 3ds is 
12x392. Hence, likewise, we have a very easy method of 
Proving whether the temperaments of any given system have 
been correctly calculated. it is only to add those which have 
the same sign ; and if the differences of the sums be equal to 
the products just stated, the work is right. 
: Proposrrion IX. 
If all the concords of the same name, in a scale of twelve 
intervals to the octave, were of equally frequent occur- 
rence, the best system of temperament would be that of 
equal semitones. 
Itis evidently best, so far as the concords of the same name 
are concerned, that if of equal frequency, they should be 
