248 Mr. Cavallo’s Obfervations on 
eafily find all its oClaves above and below, it follows, that by 
finding; all the octaves of thofe twelve divifions we (hall have 
twelve difiinCt notes within half the firing, tvs. within the firfi 
octave of the whole firing ; to which, if the found of the whole 
fir in o- be added, we (hall have thirteen different founds, which 
(hews why an octave comprehends neither more nor lefs than 
thirteen notes. 
Without dwelling any longer upon the names or number of 
thofe notes, I (hall immediately proceed to find out the tem- 
perament. 
It has been (hewn above, and it is exprefied in fig. 2. that 
the length of the firing for the lafi fifth is (horter than the 
length of the lafi oCtave, and alfo that one of them mufi be 
neceflarily taken for both purpoles ; but here we mufi confult 
nature, examining by the ear which of the two is leaft dif- 
agreeable. This, however, is foon decided ; for imperfeCl oCtaves 
are quite infufferable, whereas a certain degree of imperfe&ion 
in the fifths is tolerable ; therefore we are neceflitated to leave 
the o&aves perfeCV, and to let the feventh o&ave ferve for the 
fifth of F. In this cafe it is evident, that each of the notes 
in the fucceflion of fifths is a perfect fifth to its preceding 
note, excepting the lafi, which would be by much too flat, and 
therefore it is necelTary to divide the imperfection equally 
among them all. 
For this purpofe it mufi: be confidered, that as the twelve 
fucceflive fifths, together with the whole firing or firfi note* 
are each two-thirds of its preceding note ; they form a geome- 
trical feries, the ratio of which is f, its extremes are 132^6025 
and 102400, and the number of terms is 13. But becaufe 
mfiead of 102400, which is the lafi fifth. We mufi take the 
number 103797, 07031 2 5 (vAs. the length of the lafi: oClave) 
for 
