source file: mills2.txt Date: Tue, 4 Mar 1997 17:09:23 -0800 Subject: Ancient Geeks and irrationals From: Jonathan Wild 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