Aristoxenos & ET?

I think it is a bit of an exageration to claim that Aristoxenos
invented or proposed equal temperament, though we (including me)
interpret his genera as such and quantize them to 12, 24, 36,72
or 144-tet. What A spoke of are "parts" of tones and he defined the
tone as the difference between the consonances of the 4th and 5th.
A later writer, Cleonides, came up with the 30 parts to the 4th
system we attribute to A -- an enharmonic of 3 + 3 + 24 parts,
a soft chromatic of 4 + 4 + 22 parts, a hemiolic chromatic of 4.5
+ 4.5 + 21 parts, an intense chromatic of 6 + 6 + 18 parts, a soft
diatonic of 6 + 9 + 15, and a diatonic of 6 + 12 + 12. A also
mentioned some other genera somewhat resembling Archytas's tunings,
but he felt that the ones above were best known and fit his
theoretical framework in which the first interval could never
be greater than 6 parts and the second interval larger than the third.

The main reason I think he did not really mean ET is that none of
the other Greek theorists interpreted his tunings that way.
Eratosthenes and Ptolemy understood A's parts as being parts of a
physical string of 120 units. The principal or Dorian mode of each
genus was divided into two tetrachords of 120 ....90 and 80....60 parts.
The string lengths of the upper tetrachords (most of E's own and
Ptolemy's interpretation of A's) had to be divided by 3/2 as there are
only 20 parts between 80 and 60. I think this is pretty good evidence
that equally
tempered intervals were not being considered.

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.

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.

--John


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