( «3 ) 
Then, if we proceed in like^ manner, eo divide a Fifth (or 
Dia^ftnte) la fa fil la mi^ or la mi fa fol la, or the Proportion 
of ; to 2, into near-equals^ f taking double Numbers in the 
lame Proportion, 6 4; and interpofing the middle Number 
f ;) of thefe Three Numbers, 6^ j 45 that of 6 toy, is the 
Proportion of a leffer Third (called a Tri hemitone^ or Tone^ 
and half,) as la fa (\n la mi fa.) And that of f to 4, is the 
Proportion of the Greater Third ( commonly called a Ditone^ 
or Two Tones Jas/<j la (in fa jol la) which Two put together, 
makeaFifch,as 3 to 2 ;That is fxj=|=f And their difference 
is as 2 y to 24 ; That is f ) J (|J. So have we for thefe 3 Notes 
la fa la, their proportionate Lengths in Numbers, as 6 j- 4, 
In like manner, if we divide a Dttone (or Greater Thirdj 
as /i la (in fa fol la) whofe Proportion is as po 4 (or 10 to 8) 
intoTjvo near equals {by help of a middle Number 9 0 then 
have we (in thele Three Numbers 1098) that of 10 to % 
lor (what they calU the LeJJer Tone: And that of 9. to 8, for 
(what they call), the Greater Tone. 
But, whether fa fol fhall be madethe Leff^ (as 10 to 9) and 
fol ia ihQGreater, as 9 to 8 or. This the LeJfer(pLS 10 to ^ 
and That the Greater (as p to 8) or fometimeThis ^ndfome- 
time That, as there is occafion, (to avoid what they call a 
Schi/m; ) isfomewhat indifferent : For, either way, the Com- 
pound will be, as 5 to 4 5 and the Difference ( which they 
call a Comma) as 8 1 to So. That is, | x y = y x | = V = I • 
And',^)f(|i. 
Lafily, If from that of thQ Tri^hemitone (or Leffer Third) la 
mi fa; whole Proportion is as 6 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 Hcmi-tone mi fa (or la fa) 
ihat of 1 6 to I y. That is %) ? (||=:If 
Or, the Tri^hemitone (or Lefifer Third) whofe Proportion 
is as 6 to 5* ; may be divided into Three Near equals^ (by xsl'- 
king Triple Numbers, in the fame Proportion, 18 15 ; and 
interpofing the Two Intermediates, 17 16; ) which will 
therefore be as 18 to 17, and^s 17 to 16, and as 16 to if j 
That is, |ix^Jx!?=i?=f. 
Where alio the Greater Tone, who's Proportion is as 9 to 8, 
or 18 to 16, is divided into its Two Near-equals (commonly 
calkd Hemi-tonesy) that of x8 to 17, and that of 17 to 16: 
Thatis, ||x|J=:^J=|. And^ 
