source file: mills2.txt Date: Thu, 2 Jan 1997 23:09:16 -0800 Subject: integral overtones From: James Kukula Natural systems have all sorts of wierd overtone sets. Nice uniform strings will generate integer overtones. Nonlinear systems - systems with distortion, like overdriven loudspeakers - also seem to generate integer overtones. Systems with just one degree of freedom seem to be constrained to move periodically, which forces integer overtones, as has already been pointed out. One degree of freedom means a two dimensional phase space (the space of combinations of position and velocity). In three or more dimensions one can embed arbitrary graphs, but in two dimensions the possibilities are cut back severely. In a similar way it seems like the trajectory of a simple oscillator just doesn't have the topological freedom to fold around in strange ways. I might well be wrong about this. (I should hunt around in the literature but most of that stuff goes way over my head!) But the tendency of nonlinearity to create integral overtones seems to go beyond one dimensional systems. In a relatively calm ocean, waves of all different frequencies run over each other freely. But once waves start cresting, then distinct shapes start to appear and move through space together. This sort of phase coherence is again a periodic motion that implies integer overtones. It does seem that nature has a certain prejudice in favor of integers! Actually our common use of integers to count distinct objects seems related to the kind of phase coherence created by nonlinearity. Nonlinearity makes things clump together and split apart, and that's why there are things to count. (And to do the counting!) Huh, and that's really the root of the distinction between digital and analog circuitry - digital circuits use distortion to make nice distinct pulses. (It's getting too late at night!) Jim Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 3 Jan 1997 09:11 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01241; Fri, 3 Jan 1997 09:14:34 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA01239 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id AAA18907; Fri, 3 Jan 1997 00:14:25 -0800 Date: Fri, 3 Jan 1997 00:14:25 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu