Key Ratio ?

[Richard Brekne]

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Hi folks

I keep getting amazed in conversations I have with different folks about
how to correctly measure the arms of the key (or any other lever for
that matter) to arrive at the correct key ratio (lever ratio). So heres
a little overstated example. I hear the following three procedures,  and
some folks are quite adamant about the correctness of the one they
adhere to.

1 ) Measure straight across the top (makes most sense for our purposes
to me btw).

2 ) Measure down to the point where the key contacts its support
identifying this point as the fulcrum.

3 ) Still others say it is this second but each length is multiplied by
the cosine of the angle formed from the horizontal and the line taken
down to this fulcrum. (in the below example  < A and < B ) which really
is so close to the first method its not worth noting the difference.

It seems customary in our work to divide the short arm by the long arm
so in each case thats what I'll do using the exagerated example below.

The first method yeilds a ratio of 0.5, the second, 0.79,  and the third
0.5

Now I constructed this exact lever at the shop, and hung it on a swivel
for minumum frictions to check out what weights would balance the lever
horizontally.  Using 10 grams lead on the short side I needed 5 grams on
the long side.  Speaks for a 0.5 ratio eh ?? If it was anything like the
second method I would have needed more like 7.9 grams to balance 10...
and this wasnt even close.

Then I did some measuring of movement distances and found as close as I
could measure that from the exact  horizontal position shown below, a
vertical drop of 20 mm on the long end resulted in a 9 mm drop on the
short end. This works out to 0.45 ratio.  Again.. if the second method
of measureing the arms of the key is correct this 20 mm of vertical drop
on the long end should have resulted in something much closer to 15,8 mm
rise on the short end.

As it turns out, I believe I could extend those arms down that same
dotted line forever and not essentially change the ratio relative to
weight. Movement wise things are a bit different as the point that lies
on the normal to the horizontal intersecting the fulcrum also moves more
the lower the fulcrum is. But it stll conforms closer by far to the 0.5
figure then the 0.79 figure.

The only real difference in a real piano key is that we are not dealing
with anywhere near so extreme angles as in the example below.

So... I ask you...  why we are supposed to measure down to the balance
rail, and up to the capstan and at the same time  not take into
consideration the horizontal deflection of their angles before figuring
their ratio ?

[Image]
Cheers
RicB

--
Richard Brekne
RPT, N.P.T.F.
UiB, Bergen, Norway
mailto:rbrekne@broadpark.no
http://home.broadpark.no/~rbrekne/ricmain.html
http://www.hf.uib.no/grieg/personer/cv_RB.html


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