250 Mr, Cavallo’s Obfervations cn 
the difference between the perfeCt fifths of the latter, and the 
tempered ones of the former. 
The divifions, thus ascertained, form a feries of notes, in 
which the oftaves only are perfect ; but all the fifths, all the 
thirds, and in fhort all the chords of the fame denomination, 
are equally tempered throughout : fo that whichever of them 
is taken for the key-note, its fifth, lixth, &c. will have al- 
ways the fame proportion to it, and confequently will always 
produce the fame harmony when founded with it. 
It is evident, that, befides this, there can be no other tem- 
perament capable of producing equal harmony ; for when the 
extremes of a geometrical feries and number of mean propor- 
tionals are given, there can be but one fet of thofe means : 
thus, if we are to find two mean proportionals between the 
numbers 2 and 16, thefe are neceffarily 4 and 8; nor is it 
poffible to aflign any others. 
If, on the other hand, we endeavour to find a better tem- 
perament by introducing more than thirteen notes within the 
limits of an oCtave, we fhall find it impracticable, becaufp it 
has been fhewn, in the preceding pages, that after the number 
thirteen, if the fuccefiion of fifths be carried farther on, they 
will recede more from a coincidence with any one of the 
oCtaves. 
This explanation of the nature, origin, and neceffity of the 
temperament has been thought neceflary for the fake of per- 
spicuity ; but the fame end may be obtained by the following 
eafier method. As the thirteen notes of an oClave muff be 
arranged fo, that whichever of them be taken for the firff or 
key-note, the fecond, third, fourth, &c. may bear the fame 
confiant proportion to it; therefore it follows, that they muff 
be in a geometrical proportion one of the other, fo as to form a 
feries 
