Key Ratio ?

[Richard Brekne]

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:) found a typo that needed correcting. Changed word in red below.

Richard Brekne wrote:

> 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 rise 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 still 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
>

--
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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