Inharmonicity (again!)

_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
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