( 2/1 j 
of an OBave^ 2 to i ; then that of a Ftftb^ j to 2 ; and then 
that of a /^^r/^, 4 to 3. 
And thus, that of a Fourth Fifth^ do together make an 
Eighth \ For|x|z=r| = | = 2. That is, /t^^r thirds of /z&rd'^ 
halves^ is the fame as /^//^r halves^ that is T^tJ^?. Or ( in other 
words to the fame fenfe ) the proportion of 4 to 3, compounded 
with that of 3 to 2, is the fame with that of 4 to 2, or 2 to t. 
And, confequently, the T>'tffeYence of thofe Two, wiiich is that 
of a Tone or Full-Note^ is that of 9 to 8. For | ) |( | ; that is, 
three /S/xfej divided by four thirds^ is nine eights ; or, if out of 
the proportion of 3 to 2, we take that of 4 to 3 ; the Refult is 
that of pto8. 
Now, according to this Computation, it is manifefi:, That an 
OEiave is fomewhat le(s than Six Full-notes. For ( as was firit 
deraonftrated by Eudide^ and fince by others ) the Proportion 
of p to 8, being iix times compounded, is fomewhat more than 
" that of 2 to I. For |x|x|x|x|x|zz:||i||i, is more than 
524288 — 2 
262T44 — 1* 
This being the Cafe ; they allowed ( indifputably ) to that of 
the jDia-zeu^ick Tone ( La miy ) the fuli proportion of p to 8 ; 
as a thing not to be altered ; being the Di&rence of T>ia-pente 
and Dia-teJSaron^ or the Fifth and Fourth. 
All the Difficulty, was. How the remaining Fourth {Mi fa 
Jol la ) fliould de divided into three parts, fo as to anfwer (pretty 
near ) the Ariftoxenians TvooTones and an Half\ and might, al- 
together, make up the proportion of 4 to g ; which is that of a 
Fourth - or T>ia'teJkron, 
Many attempts were made to this purpofe : And , according 
to thofe, they gave Names to the different Genera or Kinds of 
Mufick, ( the T)iatonichy Chromatick^y and Enarmonick Kinds, ) 
with the feveral Specie s^ or leffer Diflindtions under thofe Ge- 
nerals. All which to enumerate, would be too large, and not ne- 
ceilary to our bufinefs. 
The firfl was that of Euclide ( which did moft generally ob- 
tain for many ages : ) Which allows to Fa Jol^ and to Sol la^ the 
full proportion of p to 8 ; And therefore to Fa folk ( which 
we call xh&greater 77;/r</5) that of 8 1 to 64. (For|x|z=|i ) And, 
confequently, to that of ^//^ ( which is the Remainder to a 
Fourth ) that of 256 to 243 . For I5 ) | ( Hf ; that is, if out of the 
proportion of 4 to 3,we take that of 81 to <^4, the Refult is that 
of 25'<5 to Z45. To this they gave the name of Limma {h^i^fM ) 
P p 2 that 
