source file: mills3.txt Date: Mon, 27 Oct 1997 22:23:36 +0100 Subject: Inharmonicity (again!) From: "Paul H. Erlich" _Plucked_ strings are slightly inharmonic, because the modes of vibration are affected by the string's nonzero stiffness. _Bowed_ strings, however, if bowed at a constant speed and pressure, are perfectly harmonic, since they undergo exactly the same motion (analysed by Helmholtz) in each slip-stick cycle. As Fourier showed, any periodic motion can be expressed as a sum of sine and cosine components at frequencies that are integer multiples of the fundamental. Of course, the note would have to last an infinitely long time for this to work, since the sine and cosine functions are defined from minus infinity to infinity. But for the purpose of distinguishing between harmonic and inharmonic sounds, it is useful (necessary, in fact, for an unambiguous analysis) to treat the sounds as lasting an infinitely long time. Another popular misconception is that brass instruments are inharmonic because its resonant frequencies are not a perfect overtone series. It is true that the sound produced by slapping the mouthpiece of a brass instrument is slighly inharmonic. But normally, the player's lips are the driving mechanism and the instrument merely determines which components of the lip vibration are amplified more and which are amplified less. The lips vibrate due to the Bernoulli effect, so given a constant tension and airflow, their motion will be exactly periodic, and thus the overtones will form a perfect harmonic series. You can't always believe what the "Physics for Dummies" books tell you; in fact, it was only shortly before receiving my B.S. in Physics from Yale that many of my own misconceptions about acoustics were dispelled. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: RE: harmony and small integer frequency ratios PostedDate: 27-10-97 22:54:29 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $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: 27-10-97 22:53:40-27-10-97 22:53:41,27-10-97 21:54:15-27-10-97 21:54:16 DeliveredDate: 27-10-97 21:54:16 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C125653D.007841BA; Mon, 27 Oct 1997 22:53:31 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29042; Mon, 27 Oct 1997 22:54:29 +0100 Date: Mon, 27 Oct 1997 22:54:29 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA28521 Received: (qmail 15007 invoked from network); 27 Oct 1997 13:54:20 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 27 Oct 1997 13:54:20 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu