27S ME M O IR 5 of the 
^h fay /oh /^> ^lould be divided into three parts, fo as to anfwer 
pretty near the y4riftoxenlans two tones and a half 5 and might al- 
together make up the ratio of 4 to 3, which is that of a fourth 
or 2)ia-teJJaro?i: Many attempts were made to this purpoie ^ and 
according to thofe, they gave names to the different Genera or 
kinds of mufick, the 1)iatonky Chromatic and Enarmonic 
kinds, with the feveral fpecies, or lefTer diftinflions under thole 
generals : The firft was that of Euclid^ which mofl generally 
obtained for many ages, and allows to fa^fol, and to/ol, lay the 
full ratio of 9 to 8 , and therefore to fa, fol^ la^ which we call 
the greater third, that of 8 r to ^4^ for | x J = f^:, and confe- 
quently to that of mi, fa, which is the remainder to a fourth, that 
of 2 5 (J to 245 5 for f J) f (fif, that is, if out of the rario of 4 to 
3 we take that of 81 to 6^, the refult is that of Z)6 to 249 5 to 
this they gave the name of Limma^ that is the remainder, to wit 
over and above two tones 5 but in common difcourle, when we 
do not pretend to fpeak nicely, nor intend to be ib underflood, it 
is ulual to call it an Hemi-tone, or half-note, as being very near 
it, andihe other, two whole notes ^ and this is what 'Ptolemy 
calls 2Jiafonum iDitonum, of the i)iatonic kind with two full 
tones: Againil this it is ob]e£led, as not being the moll conve- 
nient divilion, that the numbers of 81 to 6^ are too great for that 
of a 2)itoney or grearer third, which is not harlli to the ear, but 
is rather fweeter than that of a fingle tone, whole ratio is 9 to 8 j 
and in that of 25^ to 243, the numbers are yet much greater 5 
whereas there are many ratio's, as |, f, -g, f , in fmaller numbers 
than that of 9 to 8, of which in this divilion, there is no notice 
taken: To rectify this, there is another divilion thought more 
convenient, which is ^tolemy\ 2}iatonum intenfumy of the '^lHa- 
tonic kind, more intenle or acute than that other, which, inllead 
of two full tones for/^,yo/, la, afligns what we now call a greater 
and a lefc tone; which by the more nice mulicians of this and 
the laft age feems to be more embraced 3 affigning to fa, fol, the 
ratio of 9 to 8, which they call the greater tone, and to fol, la^ 
that of 10 to 9, which they call the leffer tone 3 and therefore to 
/^, la, the Ditone or greater third, that of 5 to 4- for \~ x f 
=^ ■*■§:= 5:, and conlequently to mi, fa, which is the remainder 
of the fourth, that of i5 to 1 5 5 for i) f ( Jf> that is, if out of 
the ratio of 4 to 9 we take that of 5 to 4, there remains that of 
IC) to 15 : Many other ways there are of dividing the fourth or 
"Dia-teffaron, or the ratio of 4 to 3 into three parts, anfwering 
to what in a looler way of expreflion, we call an half-note, and 
two whole notes, but this of 7^ >< t ^ y- — 7, is that which is 
now 
