Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

the least. Hence a tone is the difference of the symphonies
diatessaron and diapente;" just as the epogdous alone is the
difference between the sesquitertian and sesquialter ratio.
Moreover, the less harmony is when in an arrangement of
solid numbers, two proportionalities only are assumed, as the
arithmetical and geometrical, or the geometrical and harmonic,
or lastly, the arithmetical and harmonic; as in the terms 5. 15.
25. 45. For in these numbers, the arithmetical and geometrical
proportionalities alone are found; the geometrical in 5. 15.
and 45, since 5 : 15 :: 15 : 45; and the arithmetical in 5. 25.
and 45, since there is the same difference between 25 and 5, as
between 45 and 25. There is a less harmony also in the numbers
40. 25. 16. 10. For 40. 16 and 10 form an harmonic proportion,
since as 40 is to 10, so is the difference between 40
and 16, to the difference between 16 and 10. But 40. 25. and
10 form an arithmetical proportionality, since the difference
between 40 and 25, is the same as the difference between 25
and 10. And lastly, there is a less harmony in the terms 80.
40. 32. 20, which comprehend in themselves geometric and
harmonic proportionalities. For 80. 40. 20. are in geometrical;
but 80. 32. and 20 in harmonic proportion. The following
scheme however will facilitate the comprehension of all that
has been said in this chapter.
* For 8 to 12 which is diapente, is compounded of 8 to 9, and 9 to 12; so
likewise $-=++$.