the Temperament of Mufical Infrumcnis. 245 
taking its double, or its half, or the double of the double, 
&c. : for inftance, in fig. 1. if cZ be given equal to 90 inches, 
we may find its octave below by taking twice 90, viz. 180, or 
the o&ave of this odtave, which is 360, viz. equal to twice 
180, or to four times 90; and, on the other fide, we may 
find the oftave above of the given note by taking its half, 
which is 45, &c. 
It is now necefiarv to fnew why within the oftave there are 
admitted only thirteen different notes, viz. eight principal 
ones, and five others, called lharps and flats. 
The line XY, fig. 2. reprefents a mufical firing, the length 
of which is fuppofed to be divided into 13286025 equal parts *, 
On one fide of this line there are the divifions of feven fuccef- 
five octaves, viz. the half of XZ, a quarter of it, &c. ; 
and on the other fide are the divifions of a feries of fifths, 
viz. the fifth of the whole firing, the fifth of this fifth, 
and fo on, which are found by taking two-thirds of the whole 
firing, then two-thirds of thofe two thirds, andfoon. 
Here we take notice only of the ofiaves and fifths, becaufe 
they are the principal and the befi concords ; fo that a tempera- 
ment being required, it is neceffary firfi to take care, that thele 
concords be not rendered infufferable to the ear, the refi admit- 
ting of a greater latitude in the temperament or deviation from 
the perfect fiate. Befides, it will appear in the fequel, that all 
the other notes are derived from the feries of fuccefiive fifths. 
In whatever key a piece of mufic is performed, its filth is 
the mofi predominant of its concords; and as the notes ot 
* This number has been chofen, becaufe the line may in that cafe be divided Into 
the neceffary number of fuccefiive o£taves and fifths, without any common frac- 
tions, which renders the operation more eafy aad more perfpicuous ; otherwiic 
there mieht have been afTumed any other number. 
L 1 2, 
mufic 
