that is, the Remainder ( to wit, over and above Tisoo Tones. ) 
But^ in common difcourfe (when we do not pretend to fpeak 
nicely, nor intend to be fo underftood ) it is ulual to call it an 
Hemttone or Half- Note ( as being very near it ) and, the other, 
two Whole- Notes. And this is what Ptolemy calls jDiatonum 
3}itonum, ( of the 2)iatomck kind with Two full Tones. ) 
Againlt this, it is objedled ( as not the moft convenient Divi- 
fion, ) that the Numbers of 8i to are too great fbr that of 
9.7)itone or Greater Third \ Which is not //^r/^ to the Ear ; 
but is rather Sweeter than that of a fingle Tone^ who's propor- 
tion is p to B. And in that of 2 5- 6 to 243, the Number^ are yet 
much greater. Whereas there are many proportions (as |, f, J,^,) 
in fmaller numbers than that of p to 8 ; of which, in this divi- 
fion, there is no notice taken. 
To re<9;ify this, there is another Divjfion thought more con- 
venient ; which is Ttolemfs 'Diatonum Intenjum ( of the 2>/^- 
tonichY^vsM^^ more Intenfeox Acute than that other.. ) Which, 
inffead of Two Full tones for Fa folia ; aflignes ( what we now 
call) a Greater and a Lefier Tone ; (which, by the more nice Mu- 
licians of this and the iafl Age, feems to be more embraced ; ) 
Affigning to Fa Jol^ that of p to 8 ( which they call the Greater 
Tone :) and to Sol la^ that of 10 to p, (which they call the Lejfei' 
Tone : ) And therefore to Fa la ( the 2>itone or Greater Third ) 
that of 5r to 4. ( For ^|x| = ^|r=|. ) And confequently , to 
Mifa{ which is remaining of the Fourth ) that of 16 to ly. For 
i ) 3 ( if- That is ; if out of that of 4. to 3, we take that off to 
4, there remains that of 1(5 to If. 
M Many other waies there are ( with which I fhall not trouble 
you at prefent ) of dividing the T^^r^^ ot T>m-teJfarony or the 
proportion of 4 to 3, into three parts, anfwering to what ( in a 
ioofer way of Expreflion) we call an Half-note^ and two Pinhole' 
notes. But this of || x | x '|=: |, is that which is now received as 
the moft proper. To which therefore I fhall apply my difcourfe. 
Where if is (what we call)thQ Hemitone or Half -note Mifa\ \ 
that of the Greater-Tone^m Fa fol;md '| the Leffer-Tone^m Sol la. 
Onely with this addition ; That each of thole Tones, is (upon 
occafion ) by Flats and Sharps ( as we now fpeak ) divided into 
two Hemitones or Half notes : Which anfxvers to what by the 
Greeks was called Mutatio cjuoad Modos (the change of Mood ; ) 
and what is now done by removing Mi to another Key. Namely 
1? X \l ; and = If X 
Thus, 
