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From: sandell@ils.nwu.edu (Greg Sandell)
Newsgroups: comp.music
Subject: Re: Musical Frequencies
Message-ID: <506@anaxagoras.ils.nwu.edu>
Date: 21 Dec 90 17:51:16 GMT
References: <4193@mindlink.UUCP>
Sender: news@ils.nwu.edu
Reply-To: sandell@ils.nwu.edu (Greg Sandell)
Organization: The Institute for the Learning Sciences
Lines: 45

In article <4193@mindlink.UUCP>, a577@mindlink.UUCP (Curt Sampson) writes:
> > mercuri@grad1.cis.upenn.edu writes:
> > 
> > I'm sure that at least a half dozen people have informed you that although
> > your formula is "correct", it will sound like hell. Most instruments are
> > "stretch tuned" (higher in the high octaves, lower in the low octaves) to
> > accommodate for the nonlinearity in the hearing process. String players do
> > this "automatically" as they are playing.
> 
> Actually, from what I have heard this nonlinearity is not in the hearing
> process but in the string instrument itself.  I've been told that as
the string
> gets shorter the harmonics get flatter, thus necessitating a "sharpening" of
> the string so that it sounds in tune.  I've never heard this applied to
> --
> Curt_Sampson@mindlink.UUCP

Yes, the phenomenon of stretch-tuning is covered by Benade in his
HORNS, STRINGS & HARMONY quite nicely.  Here's what I remember:
not any old string stretched taught creates harmonic sounds (i.e.
harmonics having frequencies in integer ratios to one another); 
the string has to have specific physical properties.  The most
important properties are thickness, evenness of thickness across
the length, and stiffness.  Piano strings have the problem of being
too stiff, and the result is flat harmonics in portions of the
spectrum which are important for determining the pitch of the tone
(according to Ritsma, the 3rd through fifth harmonics when the
fundamental is between 100-400Hz; yes Virginia, pitch perception is
not just a matter of the frequency of the fundamental).  As Curt says,
those flat harmonics make the pitch sound flat, so piano tuners
adjust the tuning to make those strings sharper.  They are sharp
in fundamental frequency (which is an objective phenomenon) but *not*
in pitch, which is a psychoacoustic phenomenon.

A related matter:  guitars with old, worn-out strings sound perpetually
out of tune, no matter how hard you try tuning them.  Reason:  the
thickness of the string is not even across its length anymore, so
the harmonics have gone sour.

- Greg

****************************************************************
* Greg Sandell (sandell@ils.nwu.edu)          Evanston, IL USA *
* Institute for the Learning Sciences, Northwestern University *
****************************************************************