C 713 ] 
towards the acute, at the diftances of a fourth re- 
flectively from the five firft mentioned, and taking 
their denominations from them alfo, but compounded 
with the prepofition hyper 3 which would complete 
them to fifteen, divided into five grave, five middle, 
and five acute. But the Ariftoxenians limiting the 
pofitions of the mefe to the com pals of an oCtave, 
had eflablilhed only three out of thefe lall five, viz. 
the old Mixolydian, whofe mefe we have fhewn to 
Be in d , at a fourth from the Dorian, and which 
was therefore afterwards called Hyperdorian ; the 
Hyperionian, whofe mefe was in e flat, at a fourth 
from the Ionian y and the Hypermixolydian (for fo it 
had been originally named, the prepofition being taken 
in a different fenfe), the mefe of which was in e, at 
a fourth from the Phrygian, and which was there- 
fore afterwards called the Hyperphrygian. To make 
thefe acute modes therefore five in number, as well 
as the grave and middle ones, two new modes were 
added, viz. the Hyperasolian, whofe mefe was i nf y 
and the Hyperlydian, whofe mefe was in f # 3 at a 
fourth refpeCtively from the Aiolian and Lydian. 
The modes being thus augmented to fifteen, upon 
the Angle principle of the harmonic doCtrine, their 
mefes will be found to ftand all together in the fol- 
lowing order.. 
r 
Hyperlydian - - - - - in f# 
Hyperasolian - - - - - - - f 
Acute ( Hyperphrygian, or Hypermixolydian - e 
Hyperionian - - b 
Hyperdorian, or Mixolydian - - - d 
Middle 
3 
