source file: mills3.txt Date: Tue, 12 Aug 1997 22:44:12 +0200 Subject: Natural Harmony From: James Kukula >> Furthermore, a system with one >>degree of freedom is pretty much constrained to move in a periodic fashion, >>at least for bounded trajectories. > >This would be true for autonomous systems -- that is, where the >state of the system depends entirely upon its previous state, >and not explicitly on time. The wave equation is a 2-D >autonomous equation -- it depends upon amplitude and its >derivative with respect to time. An autonomous system with 2 >continuous variables can have a stable periodic motion, but >cannot have prolonged aperiodic motion. You're right that periodicity would depend on the system being autonomous. I suspect we're having some terminological confusion here. I studied physics once upon a time but that was a long time ago. Somehow we need to distinguish between ordinary differential equations and partial differential equations for starters. What I mean by a system with one degree of freedom is something like a simple pendulum. Once we get into the wave equation, to me that's a partial differential equation. That describes a system like a vibrating string, that has roughly an infinite number of degrees of freedom - i.e. each little section of the string can be budged up or down a smidgeon independently of how much other little sections are budged, though of course the elastic forces acting on the various little sections depend precisely on all those budgements. Another factor coming into play could be the order of the various differential equations. E.g. harmonic oscillators are second order differential equations, because acceleration is the second time derivative of position. >>Ideal one-dimensional systems of coupled linear oscillators, like vibrating >>strings or columns of air, will have an integer overtone series. > >I'm not at all sure what this means. To clarify, though: > >A one dimensional continous autonomous system cannot have >periodic behaviour: that would mean moving in different >directions from the same state. So it's my turn to be confused. An ideal string can certainly have periodic motion. Actually any vibrating body will have a whole set of modes of oscillation, and if you get the body to vibrate in just one mode, then its motion will be periodic. That's what modes of oscillation are, roughly speaking. (A little more precisely, the periodic motion will be follow a nice sinewave shape). I suspect we're just getting caught by words like "one dimensional continuous". I'm not sure just how much physics is really appropriate in this forum. But it seems worthwhile to try to clarify: >>The article points out that there's a more common class of seismic >>signals called harmonic tremors with a nice overtone series. > > That strikes me as surprising (although I'm not suggesting that they're >lying of course). It seems surprising because of the answer to your second >question: > >>Strikes me a tuning FAQ ought to address the question, are integer overtone >>series natural or artificial, where do they come from? > > Harmonic partials come from vibrating systems with uniform and elastic >vibrating media bound by two immobile "nodes". Integer ratio overtones can come from other mechanisms besides uniform and elastic etc. If you take a nice clean laboratory sinewave oscillator and run it through an amplifier and through a loud speaker and CRANK THE VOLUME WAY UP it won't be a nice clean sinewave any more. The distortion of the loudspeak, amplifier, etc. will generate nice integer ratio overtones! No vibrating strings around. I'm not sure if this is related to the one-degree-of-freedom business or not. But periodic motion necessarily generates integer-ratio overtones, and systems with one degree of freedom are more or less forced to be periodic - as long as they're autonomous and probably some other restrictions, this gets over my head pretty fast. Anyway it seems like integer-ratio overtones are important to tuning theory so maybe not too much of a waste of bandwidth to try to get clear. Jim SMTPOriginator: tuning@eartha.mills.edu From: Denis.Atadan@mvs.udel.edu Subject: Great Tape Swap Deadline PostedDate: 13-08-97 01:51:49 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 13-08-97 01:52:03-13-08-97 01:52:04,13-08-97 01:50:25-13-08-97 01:50:26 DeliveredDate: 13-08-97 01:50:26 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564F1.00831A99; Wed, 13 Aug 1997 01:52:09 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00624; Wed, 13 Aug 1997 01:51:49 +0200 Date: Wed, 13 Aug 1997 01:51:49 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00622 Received: (qmail 28317 invoked from network); 12 Aug 1997 23:49:21 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 12 Aug 1997 23:49:21 -0000 Message-Id: <199708122349.TAA09820@copland.udel.edu> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu