source file: mills3.txt Date: Wed, 31 Dec 1997 21:38:05 +0100 Subject: Greek triads, Classification of temperaments From: John Chalmers I agree with Bill Alves's statements re Greek Music. Some ancient authors admit that there are intervals which somewhat resemble the consonances of the 4th, 5th and Octave, but are different. These intervals are the 3rds, 6ths and the tritone. The 7/6 may have been an important interval in the enharmonic genus, at least at the time of Archytas. While the Greeks had a form of heterophonic accompaniment, I think it's fair to say that there is no good evidence for triadic harmony or even texture in the ancient world. The recent discussion on the classification of temperaments, tunings and scales presupposes that 12-tet is the standard, that the Fifth is the generator of the scale and that we are interested in diatonic (7-tone) and chromatic (12-tone) scales. If say 19-tet and the 12-tone MOS generated by 11 degrees were the norm, the classification would be somewhat different. Similarly, one might use a 5-tone subset of 7-tet (Colin Brown's) or a 5-tone cycle (one of Wilson's patented keyboards). See his articles in XH2 and XH3, and, of course Bosanquet's original work on the general theory of the octave and the generalized keyboard. In the Bosanquetian scheme, temperaments are defined as positive or negative to various "orders" according to the number of degrees separating B# from C (or in other words, is the Pythagorean comma positive or negative and by how many degrees of the temperament?). Bosanquet calls this parameter "r." The neutral system is 12-tet where the fifth has 700 cents and the PC (DC) has 0 degrees (r=0). Another parameter, the number of units separating C from C# defines the system as "singulary," binary, ternary, etc. according to Wilson's terminology. This parameter is equivalent to selecting the diatonic scale as the basic scale in the temperament. For some reason, it appears to have no accepted name as far as I can tell; one might call it "s" from "Seven-Fifths' semitone." (Seven-tet would be the "zero-ary" system where C=C#.) Relative to 19, the first positive series is 12, 31, 50; relative to 5, it is 3, 8, 13; and relative to 7, 5, 12, 19. If for some reason one took 13-tet, 8 degrees and the 8-tone MOS as the norm, 8, 21, 34 and 47 would be the first positive series and 5,18 and 31 the first negative. In each case, the role of the PC is taken by a chain generators equal to the number of the neutral system (19, 5,7, 13 etc.) and the These two parameters are sufficient to classify all temperaments in a 2-D array from which one can deduce a usable notation and keyboard variety. Thus, this theory is not entirely abstract. --John SMTPOriginator: tuning@eartha.mills.edu From: John Chalmers Subject: phonons, speculation PostedDate: 31-12-97 22:14:45 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: 31-12-97 22:12:24-31-12-97 22:12:25,31-12-97 22:11:47-31-12-97 22:11:49 DeliveredDate: 31-12-97 22:11:49 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 C125657E.00747AA2; Wed, 31 Dec 1997 22:14:12 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA10086; Wed, 31 Dec 1997 22:14:45 +0100 Date: Wed, 31 Dec 1997 22:14:45 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA10155 Received: (qmail 1811 invoked from network); 31 Dec 1997 13:14:36 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 31 Dec 1997 13:14:36 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu