Seasonal pitch change:

[RicB]

Hi Folks

Well, I used a little time thinking about this latest post of Ron Ns on 
the subject, and droped back a couple weeks on the previous related 
thread to find a relevant claim in more specifics then in his last. I 
ran into this quote, from 
http://ptg.org/pipermail/pianotech.php/2007-March/203148.html

        "A swelling bridge cap moving the string up the pins 0.2mm
        (0.0079", which is less than I have observed with controlled
        tests) will change the length around 0.025mm, or ten times as
        much as soundboard rise."

An interesting tidbit for back engineering to see what he did and how he 
came up with it. First, let me say that its not the speaking length we 
are talking about here. The significant increase in the overall string 
length this causes is on the surface of the bridge itself. The speaking 
length would increase, if ever so slightly on long strings... and the 
back length a bit more until they get close to the shorter speaking 
lengths.. but we'll put aside any additions to those lengths for now and 
perhaps deal with them later.

By my figuring, you end up with somewhat different results then Rons.

Heres what I come up with for the following assumptions.

    - Bridge width 20 mm
    - Angle of string displacment 10 ¤
    - pins slanted at 10 degrees.

To begin with, a 0.2 mm rise up a 10 ¤ slanted pin will be displaced 
sideways by

    Sin 10 * (0.2 / Cos 10).

Since there are 2 pins we have to take this times to resulting in a 
figure of 0.07053079228 mm.

    2 * Sin 10 * (0.2 / Cos 10)

To find out how much this lengthens the string across the bridge surface 
you need to add this added sideways displacement to the existing 
displacement. Then simply square the result, add it to the square of 20 
and take the root of the sum (finds the new string length), finnally 
subtracting the origional length (given by (20 / Cos 10)

The string length before the added displacment is given by

    20 / Cos 10

The existing sideways displacement is this figure time Sin 10

    Sin 10 * 20 / Cos 10

Adding the additional sideways displacement to this you have

    2 * Sin 10 * (0.2 / Cos 10) + Sin 10 * 20 / Cos 10

Finding the new string length using the old  a^2 + b^2 = c^2 you have

    SQRT( (2 * Sin 10 * 0.2 / Cos 10 + Sin 10 * 20 / Cos 10)^2 + 20^2)

The origional string length is simply

    20 / Cos 10

Which is subtracted from the new length.  The whole operation is then.

    SQRT( (2 * Sin 10 * 0.2 / Cos 10 + Sin 10 * 20 / Cos 10)^2  + 
    20^2)  - (20 / Cos 10)

And the result of this is about half what Ron comes up with. Presumably 
because he figured on a 20 degree pin slant.  A 0.2 mm rise in this 
example only causes a 0.012366 mm increase in string length.  In 
addition however will be a very slight increase in the speaking length 
and likewise for back lengths.  In long strings the increase to the 
speaking length is nearly nothing.  For the 406 mm long string in Rons 
example using my 10 degree pin slant, (see his whole post at the address 
above) with a 100 mm back length and a 200 mm length from the front 
termination to the tuning pin thrown in... and assuming a 0.85 mm Ø  
would result in a roughly 5 cent change in pitch. A 50 mm long string 
otherwise similiar with a 0.8 mm Ø  would see a 22 cent change, and a 
1000 mm long string with a 0.925 mm Ø would only see around a 3 cent change.

It should be noted that this increase in string length will affect the 
pitch of a string just like an increase in vertical deflection would. 
And as such a 0.124 mm increase will affect shorter strings much more 
then longer strings. Assuming then such a 0.2 mm rise in the bridge 
surface is consistant over the entire bridge... then with nothing else 
working on the system the effect of this would be to cause an increased 
pitch rise inversely proportional in some sense to the length of the 
strings.  I.e. the shorter the string the a larger the pitch change.  
Another interesting point is that this increased side displacment will 
leave its footprint in the indentation... causing it to be wider then it 
would be had side displacement remained constant. In Rons example this 
would be an extra 0.15 mm width.... quite visable to the eye.  In my 
example 0.07 mm.

Seems to me like the consequences of this idea dont really fit well with 
what we see in pianos. This kind of an effect would dominant greatly the 
pitch picture in the high treble, and there are some questions about 
what must compensate then for the abrubt change in pitch change that is 
seen between the tenor / treble break and again at the treble / 
diskant.  And then why doesnt the very top just fly off the chart ?  

More thoughts for troubled minds... :)

Cheers
RicB