Unison coupling

[Ron Nossaman]

>The frequency of vibrating piano strings is not stable, but tends to 
>lower as the string continues to sound.

* Excursion amplitude while the string is sounding stretches it sharp. The
greater the amplitude the higher the attack pitch for any given diameter,
length, pitch?


> In other words, the flatness observed when 
>strings sound together may just be because the strings get to the flat 
>part of the envelope faster, fast enough that the sharper part of the 
>envelope goes by fast enough that it can be missed.

* This sounds real plausible to me. More on a possible "why" further down.


>For what it is worth, an explanation for why this phenomenon was not 
>observed and described until just a few years ago might be that, if the 
>phenomenon occurs throughout the scale, as I believe it does, then, if 
>the temperament throughout the scale of the piano is tuned with single 
>strings, then the temperament of the piano with all strings of the 
>unisons sounding will simply be "shifted" slightly flat with few ill 
>effects to the temperament.

* Also quite reasonable and plausible.


>It also is possible that the apparent frequency shift caused by unison 
>coupling is as small or smaller than the normal frequency shift caused by 
>loud and soft playing, perhaps making the effect negligible in practice.
>
>Kent Swafford

* Those who habitually stretch octaves a bit on the "sparkly" side may be
compensating for the effect, without being aware of the connection. They
have just found that they like the sound of the piano better that way and
aren't really concerned about the details. All quite reasonable.

Now for a possible "why": 
Set up a quick experiment in energy transference. Stretch a string taut
between two solid supports a couple of feet apart, like a door frame. A
third of the way in from each end of the string, hang a weight (a pound or
so), on another string about a foot long. Make the two weights and their
string lengths as close to the same as you can. Then, start one weight
swinging perpendicular to the common support string and see what happens.
At first, the second weight doesn't move much, but soon, the second weight
will be swinging and the first will stop. This will oscillate back and
forth until both weights quit moving. It happens because both weights are
anchored on a shared, but compliant base. The base compliance determines
the energy feedback rate and power. Make the string across the doorway sag
more and repeat the experiment. The more slack string makes the
transference faster, and shortens the total swing time of the system. The
impedance (and the resonant frequency) of the suspension lowers as the
string is made less taut. Now try it with two slightly different string
lengths for the weights, and it doesn't look nearly as organized.  

In a piano, the bridge and soundboard are, VERY roughly, the horizontal
string in the above experiment. As the compliance of the bridge and board
assembly act as the intermediary for two or more strings trying to pull one
another into phase with each other. The closer the strings are to being in
tune with each other, the more organized and efficient the energy
transfers, and the shorter the attack pitch lingers. I agree that this
probably happens throughout the piano, but why is it most obvious in the
killer octave area? I think it's because the soundboard impedance is
usually too low in that area (compressing the envelope by allowing too high
an energy transfer rate back and forth between strings and soundboard
assembly), and there are usually plenty of tuned duplexes and short
backscales in that area bleeding energy from the speaking lengths and
further compressing the attack phase of the envelope. It probably happens
just as much in the upper treble, but there usually isn't a discernable
attack phase in a "dink", so no one much notices. In the lowest bichords,
The envelope is expanded so much that it's hard to tell if the phenomenon
exists.

I hope this is at least marginally comprehensible. I can mentally picture a
lot of what I think I know about piano physics connecting here very easily
and naturally, but I'm not even sure how to explain it all (which the above
failed miserably to do), much less attempting to prove anything. One thing
I'm pretty sure about though, there's no magic varnish involved. I believe
you're thinking in the right direction though, FWIW.

Ron N