source file: mills2.txt Date: Thu, 3 Oct 1996 14:14:41 -0700 Subject: Reply to Daniel Wolf From: PAULE Daniel, The notion of psychoacousitc "stability" that I eluded to is clearly only a part of what constitutes "stability" in a musical context. I'd be happy to use a different word if you could suggest a better one -- "rootedness?" Nevertheless, it has some explanatory power, as I have shown. The masking will of course be far worse in the subharmonic case, where the combination tones will tend to fill up all of frequency space, while in the harmonic case, the combination tones will be restricted to integer multiples of the fundamental. For root-position triads in the same register, the combination tones will be spaced 2.5 times more widely for major than for minor; as the harmonic limit increases past 5, this number grows very rapidly. If I misunderstood you here, could you give a concrete example of what you were talking about? >The intuitionist (like an algorithm maker, or a student of musical >cognition) proceeds in strictly chronological steps. If from the unity 1:1, >the twoity is observed, and from the sum of 1+1, 2 is constructed, yielding >the first harmonic interval, only then may the first subharmonic (the >inversion) be constructed. What if you're dealing with string lengths instead of frequencies? Certainly string lengths are a more "intuitive" quantity than unobservably fast frequencies! (This is intended to be a reductio ad absurdum). >Your examples of string lengths >etc. I found curious, as the (pre-wave length) music theory using this >instrumentality was inversionally flexible in the extreme, and ran counter >to Pauls strong harmonic series approach. Actually, looking at string lengths would lead you to believe that subharmonic relationships are even simpler that harmonic ones. An a priori numerological analysis will not favor harmonics or subharmonics in any way, despite your claim that there are differences. It is only by looking at the psychophysical phenomena that the inequivalency manifests itself. >There is >indeed a (mathematical and physical) boundary problem for the infinitesimal >(with null or the vacuum) that the infinite does not have. Please explain this in as technical terms as you like. I have a degree in physics. Taking your stamements at face value, why does this "mathematical boundary problem" affect frequencies and not string lengths, oscillatory periods, or wavelengths? If you were correct, then these three quantities would be happier producing subharmonic series (in which these quantities march unimpeded up to infinity) than harmonic series (in which they plummet down to the depths of the infinitesimal). Again, note that I am being facetious and my intent is to demonstrate the absurdity of your arguments. -Paul E. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 4 Oct 1996 11:52 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02686; Fri, 4 Oct 1996 10:54:06 +0100 Received: from sun4nl.NL.net by ns (smtpxd); id XA02681 Received: from eartha.mills.edu by sun4nl.NL.net (5.65b/NLnet-3.4) id AA10997; Thu, 3 Oct 1996 17:40:26 +0200 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA15850; Thu, 3 Oct 1996 08:38:55 -0700 Date: Thu, 3 Oct 1996 08:38:55 -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