Killer Octave Question

[A440A@aol.com]

Ron writes: 
<< Soundboard crown still isn't supported like a buttressed arch, Ed, and rim 
or belly rail spreading the few thousandth's difference between the length 
dimension measured straight from rib end to rib end, and that measured from 
rib end to rib end over the crown won't make the crown go flat. It just 
won't.<< 

  I know that someone has done the math on this.  How much "spread" is 
required of a rib that is 20 inches long, curved on a radius of say 40' to 
allow the center to drop .020"???  It has been too long since I dragged my 
kids through high school geometry to remember arc and such, but I know 
somebody out there has the answer at hand. 

>>Flexibility in the belly rail and rim will certainly damp the 
soundboard and affect tone production adversely, but the crown is formed 
and supported by rib and panel, independent of the rim.  >>

   There is evidence for debate on the arch theory. The observed change in 
the Mason & Hamilin centripedal resonator when the strings are removed.  In 
every case I have seen, (more than several), destringing a Mason changes the 
tension on the bars, ( judged by rattles that appear when destrung).  
   Perhaps the strongest indication I know of that the arched board attempts 
to spread the case is the Chickering grands with the inner rim.  All of these 
pianos,(I have had 4 of them), when destrung, show the inner rim to come 
inwards.  A wedge that is very lightly placed between the inner and outer rim 
before stringing will be extremely tight when the strings are pulled up to 
pitch and the screws that pass from inner to outer around the perimeter will 
also be looser.  There is definitely outward movement of the relatively 
flexible inner rim under downbearing pressure.  If the board simply presses 
downward on the shelf, this particular construction, with its cantilevered  
support, would bend inwards, no?  These Chickerings also have really nice 
sound with virtually no bearing, even though without any strings on them, 
there is a boatload of bearing  They simply flatten out and I have, in the 
past, attempted to add bearing by lowering nosebolts,but the board just keeps 
on flattening out as I do.  
    The unsupported span of the belly rail on grands often has a slight curve 
outwards between the horn and their attachment to the treble case side.  I 
don't think this was built into the pianos originally, so  what other force 
but soundboard spread would do this?  
Ed Foote RPT 
www.uk-piano.org/edfoote/
www.uk-piano.org/edfoote/well_tempered_piano.html