Aristoxenus

Paul Erlich (message sent via Manuel Op de Coul) wrote:

> Amazing how my posts are continually misunderstood. I wrote,
>
> >> In particular, would some mathematician not have interpreted
> >>Aristoxenus' ideas in terms of the twelfth root of two?
>
> Why does that lead people to argue that Aristoxenus did not advocate
> equal temperament? Of course he did not. The idea of irrational
> numbers was too repugnant to the ancient Greeks. Aristoxenus just
> described what he heard, and a different mathematical mindset may
> have led to different interpretations of his descriptions. As these
> descriptions were self-contradictory, it seems useless to try to
> figure out what Aristoxenus "actually meant" in mathematical terms;
> however, this didn't prevent ancient Greeks or 16th century
> theorists from doing just that.

I see what Paul means here, but I misinterpreted him to the extent
that I thought he himself subscribed to the view that Aristoxenus
implicitly described 12TET. However my misinterpretation ends there,
and there was much more to my reply. I'll explain why I'm not
satisfied with the above:

First, as Dan Wolf was quick to point out, the Greeks, from Pythagoras
onwards certainly did not despise/ignore/try to remain in ignorance of
irrational numbers. Prior to Pythagoras's theorem, they did indeed
hold to a theory of proportion which required, for example, that the
three sides of a right-angled triangle must always be rationally
related. Pythagoras showed that this was not a necessary truth, by
proving that the square root of 2 was irrational; since, for example,
if take each of the other two sides of an isoceles right-angled
triangle to be one unit in length, then by the theorem, then 2^(1/2)
is the length of the hypoteneuse, the proof that the latter bore no
rational relation to the other sides demolished the older Greek theory
of proportions. Here is the proof traditionally ascribed to Pythagoras
(I'll flesh it out considerably):

If 2^(1/2) is rational then we can express it in the form a/b, where a
and b are relatively prime integers (i.e. they have no common
denominator higher than 1). Every ratio of two integers can be reduced
to this canonical lowest form, so if 2^(1/2) is rational, it must also
be expressible in such a form. Thus since a/b 2^(1/2)

a (2^(1/2) * b

Squaring each side, we obtain

a^2 2(b^2)

If 2^(1/2) is indeed rational, then this equation must be soluble for
relatively prime integers a and b.

Now from the equation we see that since a^2 equals a multiple of 2,
a^2 must be even, and therefore a is itself even, i.e. there is a c
such that a 2c. Hence

4(c^2) 2(b^2).

Dividing both sides by two we obtain

2(c^2) b^2

But then b is also even, and so a and b are both divisible by 2. But
this is a contradiction, since a and b are relatively prime, and thus
have no common divisor higher than 1. Since there are no solutions to
the equation a^2 2(b^2) for any relatively prime integers a and b,
2^(1/2) cannot be expressed as such a ratio, a/b, and therefore
2^(1/2) is an irrational number.

This, of course, was only the beginning of the Greek exploration of
irrational numbers, and Dan has listed the most important instances of
later Greek mathematicians employing them. But it certainly marked the
end of the older Greek theory of proportions. Now if this proof of the
irrationality was provided by Pythagoras, or even if it originated
from among his followers, its provenance is considerably earlier than
Aristoxenus, and centuries earlier than Ptolemy.

It would seem, therefore that Paul cannot credibly call upon any
supposed Greek repugnance for irrational numbers in order to explain
why Greek mathematicians failed to interpret Aristoxenus in terms of
an equal division of the octave. If any such repugnance had existed,
it had been swept away before Aristoxenus, and ironically, by the very
Pythagorean school to which Paul would ascribe much of the blame.

Secondly, even given that the Greeks, post-Pythagoras, were perfectly
well able to conduct their mathematics in areas requiring irrational
numbers (I trust Paul is by now convinced of this), I would still
contend, contra Paul, that 2^(1/12) is not the irrational number that
would have suggested itself in any Greek attempt to revise
Aristoxenus. As I said yesterday, Aristoxenus's tunings are presented
as divisions specifically of the tetrachord, and not of the octave. If
Greek mathematicians or music theorists had wished to find an
equal-tempered re-interpretation of Aristoxenus, they would surely
have attempted to work out a division of the fourth, and not of the
octave. It was only because of the special status that Ptolemy
accorded the 2/1, five centuries after Aristoxenus, and his
characteristic presentation of Aristoxenus's divisions in the form of
an octave spanning the diezeugmenon and meson tetrachords, that we,
and Renaissance theorists before us, have forgotten the earlier
emphasis (in addition, of course, to our own notions of the primacy of
the octave). What any Greek revision of Aristoxenus would require then
is something like the model Cleonides derived (uncontroversially) from
Aristoxenus, of a fourth divided into 30 parts; our Greek
mathematician would then have to extract the 30th root of 4/3, which
would mean (for them) the cubic root of the fifth root of 2, divided
by the fifth root of the cubic root of the square root of 3, and then
try to measure this out for a monochord. Not a very attractive
proposition, especially when you have no method for extracting a fifth
root!

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/


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