C 7> 6 ] 
T. As to the diftance, or ratio, as he expreffes if, 
of the extreme tones ; for the Pythagoreans, whole 
method he adopts with fome improvements of his 
own, meafured intervals by the ratios of the com- 
prehending founds. 
2. As to the number of tones to be admitted be- 
tween thefe extremes. And, 
3. As to the intervals, at which they were to hand 
in fuccefiion, which he calls their exceffes.. 
Thus in the diatelfaron confonance, which he in- 
fiances, thefe three circumftances are obfervabie; 
firft, That the ratio of the extreme founds is fefqui- 
tertian ; fecondly, That the component intervals, or 
ratios, are three j and thirdly, That fuch and fuch. 
are the differences of thofe ratios,, meaning the inter- 
vals in fuccefiion. But here he obferves, that, in 
the confonances, thefe limitations have each their 
diftindt caufe ; whereas in the tones, the firft being: 
determined, the other two neceffarily followed, as 
being dependent on the fame conditions. This re- 
mark will hardly be intelligible, without fome ex- 
planation. The interval, or ratio of the extreme 
founds in each confonance, though differently treated 
by the Ariftoxenian and Pythagorean fchools, were 
yet determined, both by the one and the other, 
upon principles, which concerned not their inter- 
mediate divilion : their intermediate divifion again 
was fettled by a dodtrine, that required, in the com- 
polition of intervals,, either that every fourth found 
Ihould complete the diatelfaron, or every filth the 
diapente ; without one or other of which cir- 
cumftances, the compofition was held inconcin- 
nous 
