source file: mills2.txt Date: Fri, 8 Nov 1996 15:11:52 -0800 Subject: RE: Aristoxenos & ET? From: PAULE >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 Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 9 Nov 1996 04:42 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02410; Sat, 9 Nov 1996 04:43:49 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA02403 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id TAA18488; Fri, 8 Nov 1996 19:43:46 -0800 Date: Fri, 8 Nov 1996 19:43:46 -0800 Message-Id: <199611082241_MC1-BCA-4765@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu