While I cannot verify Johan Liljencrants' figures, I would like to give
some basic information about piano wire and show everyone why I said it
is a fixed and very reliable constant. As far as there being only 4
tons on a typical grand piano, I would say this is greatly in error,
but no problem because it is the principles here that are important.
Go by the principles.
Hooke's law states that in a material like piano wire, the stress is
proportional to the strain. Stress is just F/A, force divided by the
area. Strain is E/L, or elongation (not extrusion) divided by the
normal length. So Young's modulus YM=Stress/Strain, or F/A|E/L =
FL/AE. In a word, YM is Hooke's Law. A physical law means it is
absolute and there are no loopholes or exceptions. It works the same
way every time it's measured. You can't say, "Well, maybe, but when
you're tuning a piano, there still is this eensy-weensy little bit of
elongation that happens just before Hooke's law sets in, and that's
what us tuners know that no physicist ever heard of."
Yes, I know it's silly to put it that way, but that's exactly what the
PTG says when they teach that all piano wire has to elongate like
chewing gum before it reaches "stability." Actually, piano wire becomes
stable a few months after it is coiled, so, it isn't sold until then.
Simple as that. They want it to normalize. New wire does have a time
to "settle" and no tuner will ever have to deal with it. It's all part
of manufacturing specifications.
There is a stress that can be applied to a wire great enough so that
the wire will not return to its original length. That is exceeding
its modulus, and at that point it usually snaps instantly. But were
you to catch it just before it does snap and you measured it exactly,
you would find that you stretched it permanently. You would have also
ruined it, most likely. It will break. But that point where the wire
has to permanently stretch first is not at the beginning of the
tension. So Hooke's law says that no piano tuner can claim to approach
the limit of that wire's stress factor by tuning it repeatedly until
everything stabilizes. This is why I say they are totally in error.
Young's modulus for piano wire is about 2x10^7 lbs/sq.in, or 1.92 x
10^12 dynes/cm sq. Remembering that there are 3 strings per treble note
down to the bass wound strings, In an equi-tempered scale having an
average tension of 150 lbs/string (most large grands have more), the
treble strings alone, before you get to the bass exert a force of about
28,400 lbs. or 14 tons. The bass section is better figured by weighing
the strings, but in a 25 note bass section having 15 duplexes, it will
probably be another 8000 lbs, since the bass is usually strung at a
higher tension and crosses the treble at an angle. This vectors into
the overall plate stress to actually strengthen it and prevent bowing.
So figure 200 lbs/bass string. That makes this little baby grand we
speak of a rough estimate of 18 tons. However the concert grands
usually exceed 30 tons, if I heard correctly. The total tonnage is
immaterial. The rules that get you there are not.
If a 7' piano is strung at an average of 165 lbs/string in the treble,
that's 16 tons plus a likely 4.5 tons more in the bass for a total of
21.5 tons. That sounds to me like most medium-sized grands. By about
1930, there were only 22 piano scales computed individually, and all
the rest were built on these by making some adjustments to the original
scales. The shape of the bridges and the width of connection to the
soundboard was also an important to the scale by changing the loudness
and the partials of any particular portion of it, relative to the rest.
Regarding plate compression, cast iron and cast steel are vastly
different, yet both are used for piano plates. Cast iron formulas have
also changed considerably since the first plates were cast for pianos.
There was a time when the plates had to rust outdoors for 3-5 years to
normalize them so they would settle down. Recently, we have new cast
iron that doesn't require weathering. It would be wrong to say that the
plate doesn't compress some, theoretically. It would be a waste of time
however to calculate the compression of the plate to predict the
supposed stability of the tuning. Only if the plate is allowed to
bend, as in a portion of shear, would there be definite changes in the
tuning over its lifetime. Grey cast iron and cast steel should have no
significant compressibility in a piano -- certainly not under only two
or three dozen tons, practically speaking.
I'm sure that a piano maker who has done the math and studied these
things in detail could tell us all a lot more, but this should help to
set the record straight regarding the physics of music wire and the
forces in a piano created by it. Everything is based on percentages
of change when it comes to instability of the framing and
compressibilities. So if you want to worry about compressive forces,
worry about the pin plank, the soundboard, and the bridges. The bass
bridge is also usually cantilevered about 3 inches, too, allowing it to
bend down like a torsion spring. Proportionally these wooden parts are
by far many orders of magnitude greater than the change you will have
in a piano plate. Let's not strain the gnats and swallow the camels.