source file: mills2.txt Date: Mon, 17 Jun 1996 08:08:23 -0700 Subject: Reply to John Chalmers From: PAULE I think you are confusing the concepts of "consistency" and "articulation" (or uniqueness). For example, 31tet does not uniquely articulate the 11-limit, since 11:10 and 12:11 are both represented by 4 steps. However, 31tet consistently represents the 11-limit, since given any 11-limit triad, the sum of the best approximations to the inner intervals will be the best approximation to the outer interval. In equal tunings with a small number of notes, consistency is usually a less stringent criterion, while for large numbers of notes, consistency will be the more stringent criterion. The point of the consistency criterion is partially that consonance calculations looking at all consonant intervals individually may not tell you anything about consonant chords. Consistency with respect to triads, however, does imply consistency with respect to tetrads, pentads, etc. (By 11-limit triad I mean a triad made up of three just intervals, each of which has a numerator and denominator not higher than 11 (aside from factors of 2).) Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 17 Jun 1996 17:35 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA04811; Mon, 17 Jun 1996 08:35:45 -0700 Date: Mon, 17 Jun 1996 08:35:45 -0700 Message-Id: <9606171535.AA02874@arthaud.saclay.cea.fr> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu