source file: mills3.txt Date: Sat, 30 Aug 1997 05:00:34 +0200 Subject: Indian 5-limit, Symmetry From: John Chalmers I think we are in agreement that because of the skhisma, a sufficiently long Pythagorean series approximates 5 limit intervals. While Greek music is a lost art at the present time, it was not during the Hellenistic and Roman periods. Islamic music theory is heavily dependent upon Greek theory, even if some writers admitted that certain scales and tetrachords weren't played any more and proceeded to discuss the 14 and 17 tone Pythagorean cycles. As for "broken symmetry, one might consider a static analog, the symmetrical division of an octave into repeating modules as in Messiaen's modes of limited transposition (2 1 1 2 1 1 2 1 1, etc.) and certain types of scales generated by cycles of intervals relatively prime to the octave. These "well-formed" or MOS (moment of symmetry) scales have 2 or more instances of a repeated block of intervals and a truncated series. For example, the diatonic scale in 12-tet is 2 2 1 2 2 1 2 (as the Greek Hypophrygian mode) where the block is 2 2 1 and the broken set is 2. In 19 tet, the homologous 12 tone mode is 2 2 1 2 1 2 2 1 2 1 2 1 where the repeating block is 2 2 1 2 1. In 13-tet, 5 or 8 degrees generates 2 2 1 2 2 1 2 1, and there are many more examples in other ET's where instead of a symmetrical division of the octave, there is an almost symmetrical division. (The symmetrical divisions are in general of a type called "maximally even" by John Clough and his students and collaborators, who have also developed the theory of well-formed scales.) --John SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk Subject: Re: 5-limit in India? PostedDate: 30-08-97 14:11:13 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $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: 30-08-97 14:11:24-30-08-97 14:11:25,30-08-97 14:11:11-30-08-97 14:11:11 DeliveredDate: 30-08-97 14:11:12 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C1256503.0042F1A3; Sat, 30 Aug 1997 14:11:12 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09295; Sat, 30 Aug 1997 14:11:13 +0200 Date: Sat, 30 Aug 1997 14:11:13 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA09288 Received: (qmail 5567 invoked from network); 30 Aug 1997 12:11:10 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 30 Aug 1997 12:11:10 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu