C 273 ] 
Arijloxenus and others often mention the 
Tone as divided into four Parts, and the Semitone 
into two ; thereby making ten Divifions or 'Diefes 
in the fourth. And this is true, if we confider thefe 
Sounds in one Tenfion $ that is, cither afcending or 
defcending: But, accurately fpeaking, when we con- 
fider all the *Diefes or Divifions of the fourth, both 
afcending and defcending, we fhall find thirteen; five 
to each Tone, and three to the Semitone Major . But 
then it is to be obferved, that fome of thefe Divifions 
will be Iefs than the enharmonic Hiefis : For, if we 
divide the Semitone Major into the Semitone Minor, 
and enharmonic ‘Diejls, afcending, for Inftance, E , 
9feE, F, and then divide in like manner defcending, 
F, h F, E 7 we fhall have the Semitone Major divided 
into three Parts thus, E , ^F y )&E, F s where the 
Interval between h F and SfeE is lefs than the enhar- 
monic <r DieJis between E and h F> and between 
and F , as is eafily proved. 
Now, if we fuppofe thefe fmall Intervals equal, 
by incrcafing the lead Divifion, and diminifning the 
true enharmonic c Diefis y we fhall then have a fourth 
divided into thirteen equal Parts s and, confequently, 
the Octave divided into three fuch equal Parts ; 
which gives us the celebrated Temperature of 
Huygens , the mod perfect of all. 
From this it appears, that the Divifion of the 
Octave into 3 x Parts, was neceffarily implied in the 
Doctrine of the Ancients. The firft of the Moderns 
who mentioned fuch a Divifion was Don Vincentino , 
in his Book intitled L’Antica Mufica ridotta alia 
moderna Trattica , printed at Rome , 1555- folio. An 
Indrument had been made according to his Notion ; 
which was condemned by Zar lino and $alinas y with- 
M m z out 
