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diapafon to have been confidered as the eflential cha- 
racters of the modes ; and that thefe five modes 
therefore, though they had a difference in pitch, 
were thought to be the fame, as to their effects, with 
thole, after which they were thus named. As to the 
other three, the Hyperphrygian, Hyperasolian, and 
Hyperlydian, their mel'es being refpectively at the 
diftance of an octave from the Hypodorian, Hypo- 
ionian, and Hypophrygian, it was neceiTary to give 
them the fame fptcies with thofe three modes ; for a 
tranfpofition of an octave was always underflood to 
give a return of the fame harmony, as has been al- 
ready fhewn. 
This fecond diagram, and that of the feven ge- 
nuine modes, may be thought fufficient to render the 
lubject intelligible: however, as the fifteen modes 
are parted in the two diagrams, and confined alfo to 
the compafs of a diapafon, for fhewing the fpecies, 
I have added a table, fhewing the tuning of the 
fifteen firings of the lyre for all the modes. The 
letters of Guido’s fcale, placed within the fquares, 
fhew, in their horizontal orders, the tunings of all 
the firings for each mode ; the two broader lines 
include the fpecies of diapafon ; and the mefes are 
diflinguifhed by capital letters, by which means, the 
cafes, where the fame firing is employed for the 
mefes of two modes, appear at the firft view ; and tne 
reafoning, upon which Ptolemey rejedts the eight 
modes, is illuflrated, fo as to need no farther explana- 
tion. 
Having now gone through with what I propofol 
to explain under the three firfl heads, I come, 
Fourthly , 
