source file: mills3.txt Date: Wed, 17 Dec 1997 21:35:56 +0100 Subject: Third-tones, 19-tet, Greek Music From: John Chalmers I beg to differ but I think M. Joel Mandelbaum's dissertation is very original, especially his combined error function for evaluating temperaments in terms of the 5-limit consonances and particularly his discovery and compositional use of QEIS scales (moments of symmetry (well-formed scales) where the intervals sizes are 1 and 2 units of the temperament). The cited literature is also very complete, especially for its day (1961). From it, I learned of Novaro's, Brun's, and other researchers' works. Other writers on 19 such as Ariel are still virtually unknown to the xenharmonic community and would have completely so were it not for Mandelbaum's pioneering work. It is a shame that someone has not reprinted this book, though I imagine it is still available through Xerox University Microfilms in Ann Arbor, MI. I'd love to hear a performance of his pieces on a synth rather than the two old pianos he was forced to use. As for the ancient Greek use of 1/3 tones, they are found in Aristoxenos's Soft Chromatic genus (chroma malakon). This tetrachord has the ascending intervals 1/3 1/3 11/6 tones (66.67 66.67 366.67 cents). It fits quite well into 19-tet as 1+1+6 degrees. The other Aristoxenian tetrachord found in 19-tet is the Intense Diatonic as 2+3+3 (permuted as 3+3+2, 3+2+3 also). There is also a non-canonical chromatic tetrachord 2+2+4, which resembles the Byzantine and Islamic genera 16/15 x 15/14 x 7/6 and 14/13 x 13/12 x 8/7. This is the closest approximation possible in 19-tet to the Intense Chromatic 100 100 300 cents. A just interval approximating 1/3 tone is found in Archytas's and Plolemy's tetrachords as 28/27. Ptolemy gives 28/27 x 15/14 x 6/5 (Soft Chromatic) and both he and Archytas offer 28/27 x 8/7 x 9/8 (as Ptolemy's Soft Diatonic, A's Diatonic). Archytas's chromatic 28/27 x 243/224 x 32/27 is similar to Ptolemy's. In 19-tet these tunings would be 1+2+5 and 1+4+3, the first being quite good, the second recognizable. However, there is no evidence that the Greeks used any kind of equal temperament; the ascription of 12, 24, 36, 72...-tet to Aristoxenos is a modern interpretation. If the Greeks had wanted 0 mod 12 ET's, they were more than capable of calculating them. Extracting 19th roots would have been much more difficult, but they could have probably managed by using the mesolabium or some numerical procedure. BTW, a transcription of the Costeley Chanson has been published in 19-tet. Have you ever played and recorded it? --John SMTPOriginator: tuning@eartha.mills.edu From: "Andrew L. Kaye" Subject: Re: TUNING digest 1270 PostedDate: 17-12-97 21:38:47 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 17-12-97 21:36:48-17-12-97 21:36:49,17-12-97 21:36:26-17-12-97 21:36:27 DeliveredDate: 17-12-97 21:36:27 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256570.0071372B; Wed, 17 Dec 1997 21:38:33 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18369; Wed, 17 Dec 1997 21:38:47 +0100 Date: Wed, 17 Dec 1997 21:38:47 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18371 Received: (qmail 14368 invoked from network); 17 Dec 1997 12:38:45 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Dec 1997 12:38:45 -0800 Message-Id: <349837BF.7FCC9EEF@fast.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu