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From: nsv+@andrew.cmu.edu (Ned S. VanderVen)
Newsgroups: comp.music
Subject: Re: Musical Frequencies
Message-ID: <sbQe4Uu00WBLM11mYs@andrew.cmu.edu>
Date: 21 Dec 90 23:56:16 GMT
References: <4193@mindlink.UUCP>,
	<506@anaxagoras.ils.nwu.edu>
Organization: Physics, Carnegie Mellon, Pittsburgh, PA
Lines: 45
In-Reply-To: <506@anaxagoras.ils.nwu.edu>

As several recent posts have noted, the stiffness of real strings results
in inharmonicity of the series of normal mode frequencies (characteristic
frequencies of vibration), and to compensate for this the octaves must be
"stretched." But all the explanations posted so far have described the
phenomenon incorrectly.

First of all, the stiffness results in a series of normal mode frequencies
that are sharp, not flat, compared with the harmonic series, and these
departures from the harmonic series become progressively sharper for the
higher modes of the string.

Secondly, a note is not "stretched" in order that its perceived pitch
sound "correct" when played alone.  Octaves are stretch-tuned in order
that the two notes will be relatively free of beats when sounded together.

To illustrate how this works, suppose that because of inharmonicity a
string of fundamental frequency f has a second-mode frequency of 2.02 f
rather than
the harmonic value of exactly 2 f.  The string sounding an octave higher
should be tuned so that its fundamental is equal to the second mode frequency
of the lower note in order that no beats be heard when the notes are
sounded together.  In this example the octave would have to be tuned to a
fundamental frequency of 2.02 f to reduce the beat frequency to zero.
This sharpening of the octave is the phenomenon of "stretching." The
situation is actually more complicated than this because the fourth mode
of the lower string will beat with the second mode of the upper string, and
no tuning will eliminate all the beats. In practice the tuner will adjust
for the most consonant interval, and stretching inevitably results.

The inharmonicity depends on several factors: the diameter, length, tension,
and elastic modulus of the string.  It increases with increasing diameter and
elastic modulus, but decreases with increasing length and tension. In pianos
the inharmonicity is least in the middle of the keyboard, and that is
where the octaves are stretched least. On concert grands the bass strings
are long enough that the inharmonicity is not much greater than in
the middle of the keyboard, but on a small upright, whose bass strings are
too short, the inharmonicity can be severe, and the lower octaves are
widely stretched.

This, incidentally, provides a test of whether the stretching is due to
inharmonicities in the strings or to non-linearities in the hearing
mechanism.  If it were due primarily to the hearing mechanism we would
expect the stretching on all pianos to be about the same. In practice
the stretching varies with the instrument, and experiments by Backus have
shown that string inharmonicity accounts for nearly all of it.