[CAUT] calculating change in pitch for change in lengths

[Richard Brekne]

I thought I should add that in all this discussion, aside from the side 
spring along the BPP tangent, most of the discussion was confined to 
looking at the some of the consequences of attributing seasonal pitch 
changes to a strictly vertical deflection of the string by the rise and 
fall of the soundboard.  Ron N threw out an example which my own 
spreadsheet confirms.

He threw out an A4 of with a speaking length of 406 mm, back length of 
130 mm (1 mm Ø ?) and a deflection of 1 degree (a 1,7 mm vertical 
deflection) and in subsequent posts provided some interesting thoughts.  
One point was that a 0.1" (2.54 mm) strictly vertical rise because of a 
soundboard pushing upwards would result in a 23 cent rise in pitch.... 
but the cost was a jump to a 2.5 degree down bearing angle.  I'd add to 
that such a push upwards would have to fight off a total of 7.7 lbs of 
down bearing per string.  I think he's right in stating it seems highly 
unlikely a soundboard can do this.

His other interesting thoughts go along the line of what happens if the 
bridge surface swells and pushes the string vertically. If one looks at 
the strictly vertical forces again he points out that more then likely a 
bridge swelling more then very slightly would in actuality just force 
the board downwards. Again... because a soundboard can only push upwards 
so much.  Figure out what 7.7 pounds per string does for downbearing on 
the whole board... :)

Then there was the sideways deflection the bridge pins cause. Any change 
in vertical position of a string in relation to the pins, changes the 
length of the string segment over the bridge... in addition to 
deflecting (or at least trying to) the string strictly vertically.  He 
states that a 0.2 mm climb of the pins produces about a 0.018 mm change 
in length for the string segment.  He doesn't give the offset angle, 
bridge width or pin angle... but if one takes a 12 degree offset and pin 
angle with a 18 mm wide bridge surface I get that result.  There are two 
things here two think about.  First... since this change in length 
affects only the segment on the bridge, its only significance to pitch 
change is the net increase in tension this change has on the entire 
length of the string... from hitch pin to tuning pin. That increase in 
tension gets applied to the speaking length... which isn't really 
changed much at all here.  Second... the speaking length is altered if 
ever so slightly by  a 0.2 mm vertical rise at the front bridge pin. 
That change in speaking length also carries with it a small increase in 
tension... and the same thing happens at the back length.  The combined 
change in tension gets applied to this very slightly longer speaking 
length to arrive at the new pitch, which is about a 6.8 cent change in 
this example. Here the increase in down bearing is quite minimal... only 
about 0.4 lbs per string.

All of this assumes that nothing else is going on... purposely... so as 
to shed light on what looks plausible or not.  To me, I agree right off 
that it does not seem plausible that the soundboard can actually rise or 
fall enough strictly vertically to effect pitch directly in any 
significant way.  I find the <<string climb>> up the pin interesting 
enough... as it is certainly looks doable from the perspective of what 
load the soundboard can bear... but if you apply the results to all 
strings... you end up with a steep climbing curve that is very much more 
related to speaking length then anything else. To be sure... lower BPP's 
will be affected more then higher ones... but if you run the numbers it 
ends up looking like the shorter the string the bigger the effect. Since 
we don't see pitch changes in that follow this pattern in real life 
pianos... this tells me something else must be a far more dominant factor.

Many folks have made points as to the strengths of the forces the 
soundboard attempts to apply upwards against the strings. Many have made 
measurements of crown that they say change largely... and at the same 
time it seems to be common to hear that down bearing angles don't change 
much.  If both these are true... then perhaps what we are looking at is 
an overall swelling of both bridge and soundboard as a whole... and 
seeing this unable to push the strings much upwards, but rather distort 
the assembly downwards.  With these kinds of forces going on it is also 
likely to think along the lines that the bridge itself is not going to 
remain in the exact same horizontal position. If it tilts even slightly 
one way or the other... then all kinds of interesting things happen to 
the speaking length... along with the strictly vertical deflection 
results that in the end are present whatever those are.  The fact that 
climate control efforts in all their variants have a stabilizing effect 
attests that wood is moving and changing dimensions.

This is why I say I don't think we really have a good answer at 
present.  We have only shed a little light on what can not happen... and 
what can not account for what we observe.  Strikes me that the little 
math involved here, and the subject matter at large should be of 
interest to just about any tech.  Its a fascinating subject and a 
journey down a constructive road to boot.

Perhaps if we looked at this from a reverse perspective... we might find 
some enlightenment.  If one measures the pitch of each string before 
tuning on a few pianos one tunes often enough to know where they were 
pitch-wise a season before... and calculate what kind of change in 
string tensions these changes in pitch imply, one might be able to guess 
better at what length changes would be needed to affect those tension 
changes.  Just a thought.

Cheers
RicB