Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 109
tertian ratio, or that of 4 to 3, is found in harmonic middles of
this kind. For let the terms of harmonic proportion be of
such a kind as that the extremes are in a duple ratio, and again
let there be other terms, the extremes of which are in a triple
ratio, viz. let them be
3. 4. 6. and 2. 3. 6.
In the first arrangement, therefore, 6 is double of 3, and in the
second, 6 is triple of 2. If the differences then of these are
collected and compared to each other, a sesquitertian ratio will
be produced, whence the symphony diatessaron is formed. For
the difference between 6 and 3 is 3, and between 6 and 2 is 4,
and 4 compared to 3, produces a sesquitertian ratio.
In the same middle also, the symphony diapente is formed
from the sesquialter ratio. For in the first of the above arrangements,
6 to 4 is sesquialter, and in the second, 3 to 2.
From each of which the symphony diapente is formed. In the
next place, the symphony diapason which consists in a duple
ratio, is formed in the first of these, viz. in the ratio of 6 to 3.
In the second of the arrangements also, the symphonies diapente
and diapason are formed, which preserve a sesquialter
and a duple ratio. For of the terms 2. 3. 6., 3 to 2 is diapente,
and 6 to 3 diapason. And because the extreme terms
in this arrangement, which contains two symphonies, are in a
triple ratio, the differences likewise of the terms will be found
to be triple. For the difference between 3 and 2 is 1, and
between 6 and 3, the difference is 3. But in the first of these
arrangements, viz. 3. 4. 6., the greater term 6 is triple of the
difference between it and the middle term, viz. is triple of 2.
And again, the less term is also triple of the difference between
it and the middle term, viz. 3 is triple of 1. The greatest
symphony likewise which is called bis diapason, or as it were
twice double, because the symphony diapason is collected
from a duple ratio, is found in this harmonic middle. For in