source file: mills2.txt Date: Tue, 13 Aug 1996 12:16:58 -0700 Subject: RE: TUNING digest 802 From: PAULE Paul R, What symbol did I use multiply? >> Here's a definition of consistency: Given an odd number n, an octave-based >> equal temperament is consistent within the n-limit if, for any odd numbers >> a, b, and c such that 0> a:b plus the number of steps that best approximates b:c is equal to the >> number of steps that best approximates a:c. >Multiple uses of the same symbol make this confusing or actually >impossible to figure out. Thanks, though, for the earlier references. I'd be happy to clear this up, but I don't see what your complaint is directed at. >> Exactly. Here's the implicit derivation of your strict mathematical >> definition: the syntonic comma is (3:2)^4/(5:4) (ignoring octaves); >> therefore the best syntonic comma is 4 times the best perfect fifth minus >> the best major third, mod the number of notes per octave. Another, just as >> musically relevant and just as mathematically strict, definition would be: >> the syntonic comma is (3:2)^3/(5:3) (again ignoring octaves); therefore the >> best syntonic comma is 3 times the best perfect fifth minus the best major >> sixth, mod the number of notes per octave. If the two definitions lead to a >> conflict, I see no reason one should take precedence. Therefore, in such >> cases, there is no "best" syntonic comma. >Perhaps someone else would like to explain the precedence of 1:5 over >3:5. It's fairly clear to me, which does not invalidate attempts to base >a system on 3:5. I don't know whether something could be concocted to >include both in all cases. That's precisely the point. In a case, like 20tet, where the two approaches lead to different results, then the tuning is simply not compatible with a just-intonation view. Defining a "best" size for the syntotic comma is futile in such systems. The precedence of 1:5 over 3:5 seems consistent with an Euler-Fokker, or square lattice, philosophy but not an Erv Wilson, or triangle lattice, philosophy. I have investigated the relationship between the two approaches in terms of lattices and made some geometrical discoveries in the process. Both approaches extend to any number of dimensions -- perhaps I'll write a paper on this someday. Paul, I guess your notation system succeeds within the Euler-Fokker philosophy, but I lean towards a more egalitarian one. > >The method . . . . also allows for an improvement in Blackwood's notations >> in a few cases. >Blackwood's notation for 16, 18, and possibly a few others isn't >consistent in potential use on all steps of the tuning. (I haven't his >work at hand as I write this.) Examining the scales he writes out at the >beginning of each piece in the 12 etudes should show what I am getting >at. Ah, I'll bet this derives from his desire to preserve standard notation of diminished scales. See his PNM article. -Paul E. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 14 Aug 1996 01:19 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03952; Wed, 14 Aug 1996 01:19:20 +0200 Received: from sun4nl.NL.net by ns (smtpxd); id XA03950 Received: from eartha.mills.edu by sun4nl.NL.net with SMTP id AA13286 (5.65b/CWI-3.3); Tue, 13 Aug 1996 18:26:58 +0200 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA14868; Tue, 13 Aug 1996 09:25:37 -0700 Date: Tue, 13 Aug 1996 09:25:37 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu