source file: mills2.txt Date: Fri, 14 Jun 1996 14:18:02 -0700 Subject: Reply to Paul Erlich From: John Chalmers Paul: Your recent post on the extent to which various ET's articulate the harmonic series is an interesting and potentially useful set of data. The data are very relevant to the process of mapping JI constructs in ET's. To expand a bit, 22-tet expresses the 4:5:6:7 dominant 7th chord as 7:6:5:4 degrees of 22-tet and thus distinguishing each interval of a second or 3rd (8/7, 7/6, 6/5 and 5/4) with a unique span roughly proportional to its JI width. Fifteen-tet distinguishes the 16/15, 10/9 and 9/8 as 1, 2 and 3 degrees though it wildly distorts the the differences between the intervals. Still the major scale is recognizable as a "nearly just" scale (As "nearly just" is an oxymoron in English, I proposed the term "dicaeoid," or "dikaioid," which means the same thing and sounds more "academic," if nothing else.) William Stoney did something similar with function called HarFac in the paper below in which he scooped Erv and me. Back in the late 60's we had no idea where we could publish an article on computer generated tuning tables and the analysis of various ET's. (We didn't include a HarFac, just the raw error of each harmonic and subharmonic interval, alas.) Stoney, W. "Theoretical Possibilities for Equally Tempered Musical Systems", pp. 162-171 in The Computer and Music, H.B. Lincoln ed., Cornell University Press, Ithaca, 1970. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 15 Jun 1996 03:26 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id SAA24468; Fri, 14 Jun 1996 18:26:33 -0700 Date: Fri, 14 Jun 1996 18:26:33 -0700 Message-Id: <960614212617_328650216@emout10.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu