Impulse and response

[Sarah Fox]

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Hi all,

All this discussion of hammers, strings, and SB is quite interesting.  I =
thought I would offer my impressions of the respective contributions of =
these components, as I understand them.     =20

Hammer:  The hammer delivers a broadband impulse to the string.  If you =
were to take a hammer and "whack" it against an unstrung soundboard, the =
sound you would hear would be the "impulse" delivered into the system.

String:  The string is an oscillator -- or a filter.  It oscillates to =
the spectral components of the impulse corresponding to the fundamental =
and all of the string's partials.  Energy from the other frequencies is =
quickly dissipated.  Assuming it is responding in a linear fashion, it =
will create no spectral energy of its own.  If it is overdriven (i.e. =
nonlinear), distortion products will be created.  This will take the =
form of energy from lower frequencies being distributed to higher =
harmonics (partials).  Perceptually, this will be a "brightening" of the =
sound.  The most important function of the string is that it *stores* =
energy from its fundamental and partials, for measured dissipation =
through the soundboard.  Without this storage, there would be no =
sustain, and the piano would sound more like a weird sort of drum.

Soundboard:  The soudboard bleeds off acoustic energy from the string to =
the air.  If the board were made of concrete, the string would vibrate =
forever (assuming 0 inelasticity and 0 friction).  If the soundboard =
didn't exist, the string would also vibrate forever.  When the board =
moves in response to vibration, it is at a cost to continued vibration.  =
(Consider that the soundboard's movement is always driven by force from =
the string, so it is a braking influence.  It's a bit like running in =
sand, which can get rather fatiguing rather quickly.)  The board creates =
no spectral content of its own, assuming it behaves in a linear fasion.  =
Rather, it bleeds off the energy of the string.  It may vibrate more =
easily at one frequency than another, and so it may bleed off energy at =
different rates for the different partials, giving them different =
sustain.  That is, the brightness, darkness, and/or tonal properties of =
the note could change throughout the sustain, as a function of the =
soundboard's response properties.

Given all this, the largest impact on the spectral content of the sound =
is obviously going to be in the hammer itself, since that is the source =
of all spectral energy that goes into the system.  Assuming linearity, =
of course, the string and the soundboard merely store and dissipate the =
energy from the impulse created by the hammer.  Here are a few thoughts =
on the impulse.  (BTW, I apologize for trying to describe functions =
verbally.  It seemed preferable to drawing a bunch of graphs):

The impulse can be represented by graphing force over time.  When the =
hammer hits the string, force increases to a point, and then it =
decreases.  For simplicity, I'll discuss what happens when it increases. =
 The decreasing phase is the same in reverse.  (Well, really not, since =
there's hysteresis, hammer deceleration, and string acceleration, =
but...)...

The perfect broadband impulse would be a "step" function:  Force is =
zero, until some point in time, at which force and is suddenly changed =
to some fixed value.  This impulse would have have a flat spectrum, from =
frequencies of zero to infinity.  It would of course cause displacement =
of whatever it is applied to, and with this displacement, there would be =
energy delivered into the system, with uniform energy distribution (per =
Hz) from zero to infinity.  This step function is approximated by the =
"pop" that one might hear when plugging a microphone or guitar into an =
amp (to the ability of the speakers to reproduce it).  A broadband =
impulse might be delivered to a string, approximately, by striking the =
string with a glass marble.  Obviously the sound will be very bright.  =
Assuming the piano's response is flat (which it won't be), there will be =
equal spectral energy at the fundamental and each partial.

The perfect narrowband impulse wouldn't be an "impulse" at all.  It =
would be a continual sinusoidal variation in force at the fundamental =
frequency of the string, fording the string into sympathetic motion.  =
Assuming perfect linearity, there would be no partials.

Then there are gradations inbetween.  The slower the ramping of force, =
the darker the tone will be.  Considering this, I would think these =
properties of the hammer would be important, and I would be very =
interested in the impressions of those who have experience with voicing =
issues (which I do not):

The "spring constant" of the hammer felt.  How much force results from =
how much compression of the felt.  The higher the spring constant, the =
harder the hammer, the faster the ramping of force, the brighter the =
sound.

The shape of the hammer:  As the tip of the hammer is compressed against =
a flat surface, more and more felt mates with the surface, and so the =
force is distributed across a larger area.  This results in a change in =
spring constant of the hammer during compression.  For a given hammer =
position, mass, felt composition, etc., a larger radius of crown (or a =
flatter crown) should result in a brighter sound, as the mating area and =
applied force would ramp more rapidly during the collision.

Mass of the hammer:  As Bernhard alluded, a heavier hammer will move =
more slowly than a lighter hammer, given the same energy delivered at =
the key.  The collision will therefore be slower, the felt compression =
will be slower, and so the ramping of force will be slower.  A heavier =
hammer should result, therefore, in a darker sound.

Variation in felt consistency with depth into the hammer:  If the deeper =
felt is more tense, and the surface felt is well "sugar-coated," the =
initial felt compression will not yield nearly so much ramping of force =
as the deeper compression that follows.  This arrangement will probably =
produce fewer overtones than a tense surface with spongier felt =
underlying it.  The narrowest-band, darkest sounding collision would =
probably occur with a slow ramp of force, gently accelerating, and then =
gently falling off as the string builds velocity and the hammer loses =
velocity.

Of course the latter factor would also impact the "linearity" of the =
hammer.  With a harder blow, felt deformation will go deeper into the =
hammer and wider over the crown.  The more quickly the felt tension =
increases with increasing hammer depth, the more the sound would =
brighten with a hard blow.  (Is this the "distortion" to which people =
refer?  If so, I submit that it's not distortion at all; rather, it's =
merely a difference in the impulse spectrum.)  =20

Of course all this is quite complicated, and I suppose that's why it =
comes down to more of an art than a science.  Perhaps it's a bit like =
cooking, in that a taste test will demand a bit more oregano or a pinch =
more salt.  Unlike cooking, there are undoubtedly tradeoffs in voicing.

Anyway, as I said, I've merely scratched my head about this stuff.  If =
any of it resonates with the experienced voicing techs amongst you, I'd =
really enjoy hearing your thoughts.

Peace,
Sarah

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