[ 721 ] 
quarter ; yet, in all thefe divifions, there was a varL 
ation from the exad truth, even perceptible to the 
ear. All this reafoning is again manifedly levelled 
at the Aridoxenians, who, falfely edeeming the dia- 
pafon to confid exactly of fix tones, or twelve hemi- 
tones, had admitted thirteen modes, at the didance 
of a femitone from each other : and here we fee the 
author dill combats them fairly on the principles of 
the harmonic dodtrine. But, to go on with his ar- 
guments ; having thus fixed the number of the modes 
to feven only, he proceeds to confider the third li- 
mitation. It will be alked, fays he, what are to be 
the excefles chofen for thefe feven modes, fince the 
diapafon cannot be divided into feven equal parts ; 
and , if unequal intervals be ’ admitted, there is 
dill room for variety in the choice? In anfwer to 
this, he propofes, that the intervals in quedion diall 
be taken by means of the fird confonances, diateffa- 
ron and diapente, and their differences or exceffes, 
whatfoever they may be : for he argues, that the 
mutations, by confonant intervals, ought to be fird 
edablifhed; and next to thefe, the mutations, by 
concinnous intervals. Thole, who admitted eight 
tones, he tells uSj had found their pofitions after the 
following manner. The Dorian, Phrygian, and 
Lydian, the three mod antient, being fird fet at the 
didance of a tone from each other, they made the 
next mutation confonant, at a diateffaron from the 
Dorian, towards the acute, and at a limma from the 
Lydian, calling it Mixolydian ; becaufe it lay not at 
the didance of a tone from the Lydian, but only 
at the didance of a limma, or difference between 
ditonus and diateffaron j then this new mode having 
the 
