246 MEMOIRSoffhe 
S-fa-b-mi, which is called the natural Icale) the lengths for the 
extremes la^ la, an octave, are as 2 to r, or 1 2 to 5 • thofe for 
la, la, (in la,fa,fol, la) or mi, la (in mi, ja.fol, la) a fourth, as 
4 to 5, or 12 to 9, or 8 to 5; thofe for la, mi (in la,fa,Jol, la^ 
mi) or la, la, (in la, m,fa,fol, la) a fifth, as 3 to 2, or 12 to 8, 
or 9 to 5 ; thole for la, mi, the iHazeutic tone (or diffe-rence of 
a fourth and fifth) as 9 to 8 j thus we have for thefe four notes, 
la, la, mi, la, their proportional lengths in the numbers, 12, % 8, 
6 : Then if we proceed in like manner to divide a fifth, or 'J)ia- 
pente, la,fa,fol, la, mi, or la, mi, fa,fol, la, or the ratio of 5 
to 2, into near equals, taking double numbers in the fame ratio, 
6 to 4, and interpofing the middle number 5 5 of thefe three 
numbers,^, 5, 4, that'of 6 to 5, is the ratio of a leffer third, 
called a Tri-hemitGue, or tone and a half, as la, fa, (in la, mi, 
fa-^) and that of 5 to 4 is the ratio of the greater third, com. 
, fa, la, their propo 
tional lengths in numbers, as 5, 5, 4: In like manner, if we di- 
vide a 'Ditotie, or greater third, as fa la, (in fa, fol, la) whofe 
ratio is as 5 to 4, or 10 to 8, into two near equals, by means of a 
middle number 9, then we have in thefe three numbers, 10, 9, 8, 
that of 10 to 9, for what is called the leflertone^ and that of p 
to 8, for what is called the greater tone : But whether y^,/(?/, 
fliall be made the leffer as 10 to 9 • and/(?/, la, the greater, as 9 
to 8; or this the lefler, as 10 to 9, and that the greater, as 9 to 
8 j or fometimes this, lometimes that, as there" is occafion, to 
avoid what they call a Schifm, is Ibmewhat indifterentj for 
either way the compound will be as 5 to 4, and the difference, 
which is called a Comma, as 81 to 805 that is, f x ^-^ x | — -J^ 
:= J^ and Y)\ CB: Lalfly, if from that of the Jri-hemi-toKe, 
or lefler third, la, mi, fa, whofe ratio is as <^ to 5, we take that 
of the tone, la, mi, which is the difference of a fourth and fifth, 
as 9 to 8, there remains for the Hemi-tone, mi, fa, or la, fa, that 
of !<; to 15 . that is, f) f (if :=, 1-1: Qr, the ^Tri-hemi-tone, or 
lefler third, whofe ratio is 6 to 5, may be divided into 9 near 
equals, by taking triple numbers in the iame ratio 18, 15, and 
intfrpofing the two intermediates 17, i6- which will therefore 
be^as 18 to 17, and as 17 to 16, and as 16 to 15; that is, j4 
? 1 6 ^ It- ■=. If izz f : Where alio the greater tone, whofe ratio 
IS as 9 to 8, or 18 to 16, is divided into its two near equals, 
commonly called Hemi-tones, that of 1 8 to j t, and that of 1 7 to 
i6t 
