source file: mills2.txt Date: Sat, 12 Jul 1997 18:49:14 +0200 Subject: ET, Limits From: John Chalmers Gary: I think the context makes it clear that I was talking about equal temperaments and contrasting closed cyclic systems to open, infinite ones. Partch defined limit implicitly and used it in the titles of two chapters (Chapter Seven, Analysis of the 5 Limit, and Chapter Six, Application of the 11 limit). However, I can find no place where he explicitly defines it, though it is clear from context (to me, at least) that the Prime Limit of a tuning is determined by the highest prime number used to define ratios in that tuning. Partch does use 9 and mentions 15 as odd numbers which define ratios, but does not use them to define Limits. He appears to consider ratios such as 9/8, 81/64, 27/16 and their inversions to be at the 3 limit. One of the problems with HP's exposition is his technical vocabulary. "Identity" as in the definitions of ratios, e.g., "Ratios of 9: those ratios with identities no larger than 9, in which 9 is present: 9/8, 16/9, 9/5, 10/9, 9/7, 14/9" is not very meaningful as few people would consider 9 as identical to 1 or any other number (except under the modulo N operation). What he means is "odd multiple" and he carefully distinguishes his concept of identity from that of partial or _"ingredient of Harmonic Content"_. Calling the identities "correlatives" does not make the concept appreciably clearer, and he defines the correlatives as the set 1, 3, 5, 7, 9, 11 .... He further defines Odentity and Udentity according to whether the odd number appears in the numerator (over numbers) or denominator. At least in Genesis of a Music, HP seems to have restricted the term Limit to prime numbers, though odd numbers serve as "Identities" in defining ratios. The definition of a Ratio of N is analogous to that of 9 given above. Replace 9 by N in the first portion and list all ratios in the tuning containing N, but no larger prime, as a factor. It is not necessary to assume octave equivalence, though Partch does as he uses ratios both as a labels for tones of his scale and for the relation to a 1/1. Two is thus not considered a ratio defining number in his theory. I can only say that with exposure, his technical vocabulary becomes clearer. Since the LCM is a function of all prime factors, including 2, and their powers, the concept of LCM and Prime Limit are very different. All inversions of both the major and minor triads are at the 5 Limit as 5 is the largest Prime Number that appears in any of them (1:3:5, 4:5:6, 5:6:8, 6:8:10, 10:12:15, 12:15:20, 15:20:24, /1:/3:/5 etc.). --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 12 Jul 1997 21:27 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32325; Sat, 12 Jul 1997 21:28:24 +0200 Date: Sat, 12 Jul 1997 21:28:24 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA32293 Received: (qmail 28887 invoked from network); 12 Jul 1997 19:28:18 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 12 Jul 1997 19:28:18 -0000 Message-Id: <199707121525_MC2-1AB9-D028@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu