Impedance Matching

[John Hartman]

Richard Brekne wrote:
> 
> John Hartman wrote:
> 
> > Richard,
> >
> > In the "Piano Technicians Journal" August to November 1995 there is a
> > series of articles I wrote on how down bearing effects piano tone. I
> > discussed the relationship between down bearing and impedance. There is
> > also a little on how the sound board modes may be involved. You may want
> > to check these out since what you are talking about here is quite
> > similar.
> >
> > John Hartman
> 
> Thanks for the reply John, grin I was hoping to get a more to the point
> answer, even if it was short and sweet, but I guess I am going to have to
> rely most on my own digging. In anycase, thanks for the tip and I will order
> these issues. If you know of anymore articles relating to this please let me
> know. June, of this year touches on the borders of this in a real
> interesting series on bridge repair.
> 
> Richard Brekne
> I.C.P.T.G.  N.P.T.F.
> Bergen, Norway

Richard,

You wrote:

<Ok.. let me see if I understand this correctly so far. If we are going
to try and match strings to the soundboard impedance, we first have to
undestand a few things about soundboards. In the most general of terms
this means understanding how soundboard stiffness relates to sustain,
and how soundboard reasonant frequencies can be manipulated.>

OK so far. I don't think that anyone is going to try to match  string
impedance to soundboard impedance directly or with any amount of
accuracy. I believe that generally that the soundboard and attached
assemblies, bridges and rim, should have a higher impedance than the
strings. Otherwise the tone would be loud and short. The piano is a
percussion instrument that sounds melodic. More specifically the lower
sounding strings sound best if connected to a bridge-soundboard-rim that
has a little lower impedance. The higher sounding strings sound better
if attached to a bridge-soundboard-rim that is somewhat more stiff and
or massive. As for as the soundboard resonate frequencies are concerned
they are almost impossible to control. In general it would be better if
they were higher in frequency. For example if the lowest mode were tuned
to 100 Hz instead of around the 50 Hz more commonly seen. Unfortunately
to construct such a soundboard would make it too heavy and it's
impedance would be too high. I believe that down bearing is one of the
ways to work around this limitation. There is a price to pay, as always,
the additional stress from the string bearing load will sooner or latter
damage the sound board.

<My reading so far tells me that in a stiff soundboard, the  lowest
resonate frequencies (those under 500Hz)  of the soundboard will be
higher and the peaks and valleys of those resonances will be less
defined then a less stiff soundboard. The less defined those resonances
are, the fewer problems  incured with string frequencies that coincide
with those, being sucked up by the board (ie, no sustain at board
resonances) Since there is an upper limit with regards to the stiffness
of the board (because too much sustain is also = to thin crashy sound ?)
we are left with a trade off situation.>

This sounds right on track to me.

<We can effect change in these resonance frequencies by increasing or
decreasing stiffness to the board by

1: increasing / decreasing the height of the ribs, particularily in the
middel area of the board.
2: increasing / decreasing the height of the bridge.
3: increasing / decreasing the downward pressure exerted by the strings
on the bridge
4: adding / subtracting mass at various points on the board.
5: changing the tension of the strings.

(anything else ?)>

6: changing the material used in the ribs, panel and bridges. What
species of wood.
7: Tapering the edges of the panel and scalloping the ribs.
7-1/2: the general thickness of the panel.
8: Amount of soundboard crown and how formed.
9: The length of the back string scale.
10: The construction and shape of the rim and belly rail. The size of
the board.
11: the grain direction of the panel and the angle and spacing of the
ribs.

<The basic idea is to create a situation where the board is in general
stiff enough to radiate sound adequatly, and at the same time manipulate
the resonate frequencies so that they (as much as posible) do not
coincide with string frequencies at concert pitch ?? And at the same
time incorporate a stringing scale with decent inharmonicity
characteristics ??>

I don't think you are going to manipulate the frequencies all that
closely. As far as I know the inhrmonicity of the scale has little to do
with soundboard resonate frequencies. String scales impact the situation
because there are scales with higher and lower string impedance's. Long,
higher tension scales have more string impedance and require a heavier
and stiffer soundboard assemble.


<I realize that this is just a begining and far from complete, but are
there any mistakes in my understanding so far ??>

I think you are on your way with a very good start.

<A couple questions here.. what is meant by the "phase angle" of the
impedance and how is this (if it is at all) manipulated. And secondly, I
am wondering if these resonant frequencies can have anything at all to
do with this buisness of para inharmonicity. Strikes me as odd that
negative inharmonicity is found primarilly in the area under 500 Hz and
only then at a few (non adjacent ??) strings,  which is the same area
where these resonanances are of most concern. I may be jumping to
conclusions there but it does strike me as a posibility.>

I am far from an expert on this but I have run into the concept of
impedance phase angle in my reading. I pretty much discounted it as an
issue in the sound board design. It refers to the variation in a bodies
impedance in relation to it's resonate frequencies. Once a body begins
to vibrate it's impedance fluctuates with the phase of the oscillations.
An analogy could be made of a swing. I haven't been on a swing lately
but if memory serves there is a optimum moment in he swings motion to
kick ones feet to put more energy into the system. Understanding the
impedance phase angle is more important to designing bridges than
pianos. 

John Hartman