source file: mills2.txt Date: Fri, 17 Jan 1997 10:53:30 -0800 Subject: Consistency From: PAULE Daniel Wolf wrote, >A resolution of 1.56 cents. So that a 3/2 is approximated by an offset of 1 >unit, but a 9/8 by 3 units. So for the TX81Z the best 9/8 is not equal to >the sum of 3/2s. In other words, 768-equal is not consistent within the 9-limit (although 7-limit is fine). A much simpler example is 15-equal. The chord 0 9 18 is fine as a 3/2*3/2, but 0 18 is not the best 9/4 (0 17 is closer). So stacking 3-limit chords poses no difficulties in 15-equal, and the same holds through the 7-limit, but full 9-limit harmony per se is impossible, in the sense of taking the best approximations to just intervals and expecting them to be as related to one another as the just intervals themselves. I wonder, in Brian's scheme of mapping real harmonic series to equal temperaments in the quest for "better" timbres, what happens when you reach a harmonic that is inconsistent with the temperament, e.g. the 11th harmonic in 12-equal? Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 20 Jan 1997 13:15 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07780; Sat, 18 Jan 1997 04:39:40 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA07895 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id TAA26999; Fri, 17 Jan 1997 19:39:38 -0800 Date: Fri, 17 Jan 1997 19:39:38 -0800 Message-Id: <199701180335.TAA21702@well.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu