source file: mills2.txt Date: Fri, 28 Feb 1997 19:01:12 -0800 Subject: Tetrachords From: John Chalmers Eratosthenes: The simplest explanation of E's enharmonic and chromatic tunings is that he took an open string of 120 parts and subtracted Aristoxenos's parts in order. For the enharmonic, this procedure results in 120 117 114 90 and 114/90 is 19/15. In the case of the chromatic, 120 114 108 90, and 108/90 is 6/5. His diatonic is the old Sumero- Babylonian "Pythagorean" tuning. I think E chose 120 for the open string because astronomers were used to using base-60 notation (actually a form of decimally-coded sexagesimal). Ptolemy's string lengths are all in base-60 as it would have been the most familiar notation for expressing fractions, which are cumbersome in the Greek and Roman numeral systems. As for computing Aristo's parts as 30th roots of 4/3, I agree that it would be difficult to extract the 5th root of 4/3 and I think you may have the solution to why no known Greek theorist recorded A's genera in ET. Still, use might have been made of the Mesolabium, an instrument invented by Eratosthenes himself (Heath) (or Archimedes, elsewhere) for just this type of problem. After the 5th roots were approximated, it would have been relatively easy to compute the square and cube roots of the segments. Barbera also thinks that Aristoxenos meant 4/3 for the tetrachord span and listed the roots for Aristo's main genera. However, if one simply tries to approximate the divisions, it is possible to do so with just square roots. The enharmonic may be described as two successive intervals of the square root of 256/243 followed by a ditone of 81/64 and the Intense chromatic as two square roots of 9/8 and 32/27. One further thought, I am inclined to agree with you about Ptolemy having observed 1:2 divisions in practice. His enharmonic still looks artificial to me, mostly because the enharmonic was supposed to be extinct in practice by his time. I suspect it was supplied by analogy with the Soft Chromatic, which greatly resembles Archytas's chromatic, and the Soft Diatonic and Intense Chromatic, both of which may be descended from Aristoxenos's Soft Diatonic (100 150 250 cents) versus 22/21 x 12/11 x 7/6 (81 151 267) and 21/20 x 10/9 x 8/7 (85 182 231). (I believe Winnington-Ingram had similar thoughts, but I don't have his paper on Aristoxenos and musical intervals at hand). In 500 years musical taste could have changed sufficiently that the 1:2 pyknon was to be prefered in the non-diatonic genera (the stereotype soft diatonic is just on the border between Chromatic and diatonic and sounds chromatic to me most of the time). Polychronios: Thanks for the source of the 68-tet division formerly used in the Greek Orthodox liturgical music. I hadn't heard of Archbishop Chrysanthos before. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 1 Mar 1997 05:49 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01064; Sat, 1 Mar 1997 05:49:34 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01118 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id UAA19781; Fri, 28 Feb 1997 20:48:00 -0800 Date: Fri, 28 Feb 1997 20:48:00 -0800 Message-Id: <970228234545_851117499@emout03.mail.aol.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu