source file: mills2.txt Date: Sun, 16 Feb 1997 15:28:33 -0800 Subject: Re: can anyone explain these "ghosttones"? From: Matt Nathan ----------begin crosspost from alt.sci.physics.acoustics---------- Subject: Re: Proof of Harmonic Series? Date: 16 Feb 1997 22:25:51 GMT From: J.Wolfe@unsw.edu.au (JW) Organization: University of New South Wales Newsgroups: alt.sci.physics.acoustics References: 1 , 2 In article , clucy@cix.compulink.co.uk ("Charles Lucy") says: >Can anyone point me to conclusive physical proof that harmonics >(i,e, the sounds that you hear) are only to be found at >integer frequency ratios. >I am seriously questioning this "hallowed" rule, and would >appreciate pointers from the acousticians. >(My experience and intuition tells me that harmonics beat) >Does anyone have firm experimental data to validate the >integer ratio mapping of harmonics on vibrating strings? Vibrating strings: The overtones of plucked strings are in general not harmonically related. For very long, thin strings they come close, but normally the higher partials are slightly sharper than the harmonic ratios. This is due (mainly) to the finite bending stiffness of the string at the ends. This is particularly noticeable in small pianos. In such pianos the 'octaves' are usually tuned a little sharp so that the first overtone of one string does not beat with the string that is (approximately) one octave higher. People say that the effect is smaller on grand pianos, because the strings are much longer, but I have never done any experiments. On the few harpsichords I have looked at, the overtones were very close to being harmonic. Bowed strings are quite different. The stick-slip action of the bow is usually (very close to) periodic, and as Fourier's theorem tells you, the components of a periodic function are harmonic. One can take a very inharmonic string (e.g. attach a small weight to it) and then bow it carefully and produce a spectrum whose components are as harmonic as you can measure. In wind instruments, the reed, the player's lips or the air-jet take the role of the bow and produce a periodic function. So in steady state the components of these instruments are harmonic, even though the resonances of the instrument may not be. Ask a beginning flute player to play two notes an octave apart, and you'll be surprised at the interval that results. But look at or listen to the components of the lower note, and you'll find that they are as harmonic as you can measure. Note the point about steady state: only a periodic wave that is of infinite duration has exactly periodic components. In practice, most string and wind instruments only play notes for say 10s or so, so you cannot measure the frequencies more accurately than 0-.1 Hz. And usually the players will add some vibrato as well. So, as in every other branch of science, you can never say that the results accord exactly with the theory, because you can never make measurements with infinite accuracy. But, to the accuracy of your ears, or of a spectrum analyser running over a few seconds, the components of a steadily bowed string or steadily blown wind instrument are harmonic. Plucked strings are not harmonic, and noticeably so in many cases such as pianos. Percussion instruments (you mention bells) have components that depart even further from harmonic ratios. Joe Wolfe, Physics, UNSW, Sydney ------------end crosspost from alt.sci.physics.acoustics---------- Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 17 Feb 1997 00:52 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08138; Mon, 17 Feb 1997 00:52:08 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA08131 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id PAA17349; Sun, 16 Feb 1997 15:50:10 -0800 Date: Sun, 16 Feb 1997 15:50:10 -0800 Message-Id: <33079B04.2A64@ix.netcom.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu