Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

The third proportion among the four, but the ninth in order
is, when three terms being given, as is the middle to the least
term, so is the difference of the extremes to the difference of
the less terms; as in 4. 6. 7. For 6 to 4 is sesquialter, the
difference of which is 2. But 7 differs from 4 by 3, which is
sesquialter to 2.
And the fourth proportion, but which is in order the tenth,
is when in three terms, as is the middle to the least term, so is
the difference of the extremes to the difference of the greater
terms; as in 3. 5. 8. For 5 the middle term is to 3 in a superbipartient
ratio. But the difference of the extremes is 5, which
to the difference of the greater terms 8 and 5, i.e. 3, is also
superbipartient."
CHAPTER XXXII
On the greatest and most perfect symphony, which is extended
in three intervals;-and also on the less symphony.
IT now remains for us to speak of the greatest and most
perfect harmony, which subsisting in three intervals, obtains
great power in the temperaments of musical modulation, and in
the speculation of natural questions. For nothing can be found
more perfect than a middle of this kind, which being extended
in three intervals, is allotted the nature and essence of the
most perfect body. For thus we have shown that a cube,
which possesses three dimensions, is a full and perfect harmony.
This, however, may be discovered, if two terms being
given, which have themselves increased by three intervals, viz.
+ Jordanus Brunus has added an eleventh to these proportions; and this is when
in three terms, as is the greatest to the middle term, so is the difference of the
extremes to the difference of the grater terms; as in 6. 4. 3. For as 6 is to 4;
so is the difference of 6 and 3, i.e. 3, to the difference of 6 and 4, i.c. 2.