RE: Aristoxenos & ET?

>Finally, computing string lengths for ET's was well within the
>capabilities of the Greek mathematicians. Archytas himself was known
>for a 3-D construction (using more than a compass and straight edge)
>for the cube root of 2 and the extraction of square roots was long known.
>Hence computing string lengths to reasonable preceision for the 0 mod
>12 ET's would have been easy, had the Greeks ever wanted to.

That presumes a certain conceptual leap that the Greeks never took.

>I think Aristoxenos was trying to rigorously describe the perceived
>sizes of the intervals of the various genera without recourse to
>ratios, string lengths or other extra-musical factors. In this
>respect, he is our earlist psychoacoustician.

This was exactly by point. Aristoxenos _heard_ twelve equal parts to the
octave. Neither he nor his contemporaries knew how to describe what he heard
in mathematical terms. But considering the close agreement between
Pythagorean tuning and 12-tet, it seems fair to say that Aristoxenus was
hearing logarithmically. The octave divided into twelve parts arithmetically
or harmonically gets you really horrible "Pythagorean" modes, and the fact
that later theorists tried to interpret Aristoxenus this way shows that the
proscription against irrationals was still strongly influencing scientific
thought.

-Paul


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