A 440 Hz Standard

[Don A. Gilmore]

----- Original Message ----- 
From: "James Ellis" <claviers@nxs.net>
To: <caut@ptg.org>
Sent: Tuesday, April 13, 2004 4:48 PM
Subject: A 440 Hz Standard


> I'm not sure who said what in this latest exchange.  However, whoever it
> was who said the piano will go flat when the lights come on, is right.
The
> strings of the piano will quickly go slightly flat when the hot light
beams
> hit them, because they warm up rapidly, but the plate does not.  The plate
> is a big heat sink.  Three or four hours later, if the piano stays under
> the lights that long, it will be back up to pitch when the plate warms up.
> The thermal coefficients of cast iron and steel are similar, but not
> identical.  It will take longer for the lights to dry the board out and
the
> piano to go flat again.  Playing against this, you have the fact that wind
> instruments go about 1.7 cents sharp for every degree F temperature rise,
> but the orchestra's strings go flat.  So the orchestra re-tunes, but the
> piano just sits there.  More often than not, the piano gets rolled out on
> stage either after the first number, or after the intermission.  I don't
> need to say more.  You know what happens.
[snip]
> Jim Ellis

Hi Jim:

Having done extensive research into this phenomenon, I just thought I might
tell you guys a little about what I know.  You might find it interesting or
useful.

The effects that produce the change in pitch due to a change in temperature
are all elementary physics/mechanics, but there are a number of these
elementary effects taking place simultaneously, and the piano's design
varies across the gamut, so it's not quite as cut and dried as you might
imagine.

An increase in temperature causes most materials to expand, so as you might
expect, piano strings expand and get flatter in pitch as they get warmer.
But the amount of expansion depends on the length of the string--longer
strings will expand a longer total distance than shorter ones for the same
given rise in temperature.  What remains constant is the *percent* change,
or inches of expansion per inch of string length.  Since the strings in the
piano are many different lengths, they will expand by different amounts.

But hold on; it's still not that simple.  Since the harp is much more rigid
than a string, the length of the string will remain virtually unchanged no
matter how much it would have expanded or contracted if free.  Instead, it
changes in tension.  Let's say we have a string that thermal contraction
would have made shrink by, say, .06" when free.  If we hold the ends rigid
and don't let the string shrink, the tension will increase just as if we
pulled it .06" longer.  In other words, its as if we let it contract and
then pulled it back to its original size.

What is actually more important than the inches of contraction is the
"strain".  Strain is the *percent* elongation per unit length (just like
with the thermal expansion).  Knowing the strain we can determine the
"stress", or the tension per unit crossectional area in psi if we know the
"modulus of elasticity" of the material (which we can look up in a table).
If we know the diameter of a string we can calculate its crossectional area
and thus determine its tension.  Since both thermal expansion and strain
depend on string length, it cancels out, and now we get the same change in
tension for a given change in temperature regardless of the length of the
string.  A residual effect of this is that the segments of string between
the agraffes, tuning pins, string rests and hitch pin will all change in
tension by the same amount, so there will be no differences in tension
across friction members to be equalized.  Piece of cake, right?

Well, now we must remember that the diameter of wire in a piano changes as
we go down.  This won't affect thermal expansion, so it won't affect the
strain, so it won't affect the stress.  But now when we convert from stress
to actual tension, a thicker string will have more tension for a given
stress since it is spread over a larger area.  So temperature affects the
tension more in heavier strings than in lighter ones.  So now we've got it
nailed, huh?

The pitch produced by a given string, assuming its diameter and speaking
length remain constant, is proportional to the square root of its tension.
So the relation is not linear.  For example, increasing the tension of a
string 10% will not increase its vibrating frequency by 10%.  Warmer still
means "flatter", but the two are just not in constant proportion with each
other.  And remember that bass strings are wound with copper, which changes
their mass per unit length, but not their tensile properties, further
complicating matters.

Now, on top of all that, recall that music is based on ratios, not
differences in frequency.  For example to tune a fifth (3:2) above A-110
would require 110 x 3/2 =  165 Hz, or about 55 Hz higher.  But for A-440 a
fifth would be 440 x 3/2 = 660 Hz, or about 220 Hz higher.  That's four
times as much, even though the musical interval sounds the same.  So
different strings require different changes in frequency and thus different
changes in temperature for the same change in perceived pitch (in cents).

I have found that the harp warming up has virtually no effect on the
pitches.  The change in tension due to the expansion/contraction of the harp
is divided among over 200 strings that are under from 100 to 200 lbs of
tension each, so it gets lost in the shuffle.  The direct effect of heat on
the strings themselves far overshadows the miniscule effects of the harp.
Also remember that the effect of stage lights on the strings and harp will
not "even out".  It is not the same as putting a piano in a room of a given
temperature.  The amount that an object will heat as a result of being under
radiative light depends on its absorption and the angle of incident light.
That's why a dull, black, flat object will get hotter than a shiny, silver,
cylindrical one under the same light.

So you can see that a piano shouldn't behave according to a simple "cents
per degree F" formula across the board.

Having said that, through experimentation (and calculation) I have found
that the thermal sensitivity of piano strings doesn't really vary that much.
When all of these factors are added together the strings will all vary by
about 1 or 2 cents per degree farenheit, believe it or not!

Don A. Gilmore
Mechanical Engineer
Kansas City