of tensions @ pitch

[Ron Nossaman]

A while back, someone asked about total string tension and bearing changes
with pitch changes. I'm sorry, I don't remember who. I didn't have anything
set up specifically for this at the time, but I got curious and finally got
around to chasing it down the other evening. Here's what I found.

Using:
 Freq=fork*0.0625*2^((U-1)/12)
 Tension=((Freq*L*d)/20833)^2*(1+W*(D^2/d^2-1))+Ns

where fork=pitch at A4, U=unison #, L=speaking length in inches, d=core
diameter in mils, D=overall diameter in mils, Ns= number of strings in
unison, and W=.89 for copper.

In an unspecified 6 foot piano with bearing ranging from 1/2 degree in the
bass, to a little over 1.5 degree in the treble, here are some of the
results at different pitches.

pitch            total tension            total bearing
@445               40,487                   655
@440               39,583                   640
@435               38,688                   626
@430               37,804                   612
@425               36,930                   597

The bearing angles would decrease as the tension was raised and the
soundboard deflected, so the total bearing load deviation should be less
than these figures would indicate, centered around the 640 @ A440. It would
take way too much time and effort to set the bearing load and angle routines
up to reflect the change unless it was actually good for something besides
being really neat, so this will have to do for now. The tension figures
should be pretty close unless I messed something up. 

This should give you and me a rough idea of what a piano has to accommodate
in load change with a pitch correction. It also indicates to me that I
overestimated the effect of tension changes on soundboard load, and
therefore deflection, and that Dr. Sanderson was most likely correct in
saying that plate flex is a bigger factor in pitch drop during a pitch
raise. A bit more evidence toward obtaining part of a complete education. If
anyone cares, you're welcome. If anyone has corrections on my method, I'm
interested. This is the sort of thing it's hard to get answers to.

 Ron