[ 712 3 
Six of thefe modes, with the feven Ptolemey al- 
lowed, made up the thirteen allowed by the Arifto- 
xenians : the other two feem to have been added 
afterwards, more with a view to regularity in the 
names and pofitions of the modes, than to any par- 
ticular ufe, as will appear in the courfe of the ex- 
planation. For the lettling thefe eight modes,. I 
fhall have no occafion to go farther than the fixing 
the pofition of the mefe for each, according to the 
harmonic dodtrine : the refult of applying the mufL 
cal dodtrine to them will fufficiently appear, when 
I come to explain the reafons, which Ptolemey has 
afligned for rejedting tnem. 
The old Dorian, Phrygian, and Lydian modes, 
having, as we have feen, their mefes refpedtively in 
a , by c# , at the diflance of a tone from each other, 
thefe tonic fpaces were afterwards divided, to make 
room for the Ionian mefe in b flat, and the M - olian in 
c natural. To thefe five modes, whofe mefes fuc- 
ceeded each other at the interval of a femitone, and 
which came, in time, to be called the middle modes, 
five others were made to correfpond, that . lay re- 
fpedtively graver by a fourth, and took their deno- 
minations from them, but compounded with the 
prepofition hypo , to diftinguifh them. Of thefe, we 
have already fhewn the Hypodorian mefe to have 
been in e t the Hypophrygian inj ^ 3 and the Hypo- 
lydian in g# , Now, the two tonic fpaces between 
e, j# > and g#, being likewife divisible, the Hypo- 
ionian mefe was inferted in f natural, and the Hypo- 
32olian in g natural, at a fourth refpedtively from the 
Ionian and ALolian. To thefe ten modes, it was con- 
ceived that five more might properly be ranged 
towards 
