source file: mills3.txt Date: Mon, 11 Aug 1997 20:46:10 +0200 Subject: Re: Natural Harmony From: gbreed@cix.compulink.co.uk (Graham Breed) It seems nobody has picked up on this, so I'll offer some comments. Most of this is pretty involved, so you have my permission to skip to the end of this post. James Kukula wrote: >The sound generated by a periodic motion will have integer overtones, by some- >classical theorem of Fourier or whoever. The Fourier series is a way of turning a periodic signal into a sum of sinusoidal waves of integer frequency multiples. I've never heard of this stated as a theorem, but it is. > 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. With 3 variables, chaos can arise, and a system with 1 degree of freedom can have a 3 dimensional equation of motion. I can explain more of this off-list. >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. A perfect, infinitely thin string with perfectly clamped endpoints _can_ have non-integer overtones. This will always be the case where the speed of sound in the string depends upon its frequency. I'm sure two coupled oscillators can produce chaotic motion. As regards this FAQ, can we have some kind of historical and geographical coverage of standard tuning practice? When was meantone / well / equal temperament in use is a question that seems to come up fairly frequently. SMTPOriginator: tuning@eartha.mills.edu From: mr88cet@texas.net Subject: Re: Natural Harmony PostedDate: 11-08-97 20:49:19 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: 11-08-97 20:49:20-11-08-97 20:49:21,11-08-97 20:47:45-11-08-97 20:47:45 DeliveredDate: 11-08-97 20:47:45 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564F0.00676143; Mon, 11 Aug 1997 20:49:11 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08071; Mon, 11 Aug 1997 20:49:19 +0200 Date: Mon, 11 Aug 1997 20:49:19 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA08066 Received: (qmail 21133 invoked from network); 11 Aug 1997 18:45:25 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Aug 1997 18:45:25 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu