Ancient Geeks and irrationals

This is a few days late, but there was some question of how much some
Greeks *really* minded irrational quantities...


"Repugnance" towards irrationals was strictly limited to disciplines
regarded as subordinate to mathematics. Look at the Greek quadrivium
to see how music fits in:

MATHEMATICS GEOMETRY
(number) (magnitude)

MUSIC ASTRONOMY
(number as embodied (magnitude as embodied
in sound) in celestial motion)

The schism between the sciences dealing with the discrete vs those
dealing with the continuous (number vs magnitude) was a huge factor
in the development of mathematical methods. It was precisely because
Greek geometry *could* deal with the continuous and the irrational
that it flourished, rather than Greek mathematics, which dealt only
with the discrete and the rational. So, for example, a proof of what
we think of today as a simple algebraic identity like


(a-b)^2 (a-b)(a+b)

had to be ingeniously couched in purely geometrical terms. Eudoxus
(a contemporary of Plato) put forward a brilliant theory of
irrational proportions that could have incorporated them into Greek
mathematics, but unfortunately no-one really took him up on it till
the late 19th century. (By the way, Plato shouldn't be mentioned in
connection with hiding the fact that there are irrational numbers --
somewhere he says that any man unaware of their existence is no
better than a swine...)

I guess if Pythagoreans could have construed music as *magnitude*
embodied in sound (i.e. subordinate to geometry) rather than *number*
embodied in sound, there would never have been a problem with
irrational proportions in music. Don't jump to any conclusions and
think that Aristoxenus favoured such a switch in music's allegiance,
however: as mentioned already in this thread he places music
subordinate to nothing save perception, as exemplified in his
do-it-yourself investigation (he never called it a "proof") of
whether the fourth contains two and a half tones, which you can find
at the end of book II of the Elements of Harmonics.


Anyway, despite Greek geometry's perfectly adequate treatment of
irrational proportions, repugnance towards them in areas where
*number* was thought to matter was very real, and is evidenced in
many writings dealing with "mathematical" disciplines. One fantastic
example, mentioned by Aristides Quintilianus and also found in the
Hippocratic *Endemics*, tells physicians that diseases whose symptoms
appear in "concordant" ratios (like one day for every two they are
absent) are not dangerous, while those whose symptoms appear in
irrational or continuous proportions are "deadly, and to be feared".
Heck, even Aristotle thought that pleasant colours resulted from
elementary particles of black and white mixed in simple ratios,
whereas irrational proportions gave rise to unpleasant colours (this
is somewhere in *De Sensu*). And as far as musical thinking goes,
here's Barker's translation of Adrastus:

Under irrational relations noises are irrational
and unmelodic, and should not strictly even be called
notes, but only sounds; but under relations that place
them in certain relations to one another, they are
[...] strictly and properly notes.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Wed, 5 Mar 1997 05:17 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06286; Wed, 5 Mar 1997 05:17:10 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA06248
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id UAA16996; Tue, 4 Mar 1997 20:15:41 -0800
Date: Tue, 4 Mar 1997 20:15:41 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

-------------------- Begin Original Message --------------------

Message text written by INTERNET:tuning@ella.mills.edu

"Now, surely everybody who reads this list is aware that
C# and Db are physically identical on a piano. Since Beethoven
probably also knew this, what did he mean when he said that
they were different ?"


-------------------- End Original Message --------------------

Beethoven may not have been commenting on how they sounded on a piano,
but on instruments of more flexible pitch.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sun, 9 Mar 1997 19:05 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA10249; Sun, 9 Mar 1997 19:05:04 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA10288
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id KAA29842; Sun, 9 Mar 1997 10:03:20 -0800
Date: Sun, 9 Mar 1997 10:03:20 -0800
Message-Id: <3322FAD2.587B@dial.pipex.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu