Coupled strings and pitch drop

[stewart@maxwell.com]

Hello to all those on the list.

First, I'm not a piano tech nor a professional musician, rather a fumble
fingered amatuer who loves the instruments and enjoys both the music and
the art/science of making instruments perform their best.  I've enjoyed
immensely browsing through the archives and truly appreciate your community
of experts.

An earlier thread, one of many that branched from the historical tuning
discussions, identified the phenonmenon of double or triple strings
sounding lower than a single string.  No one jumped in to explain, so
thought I'd try.  May be old hat to Del or other experts in scale design,
so if this is baby talk, please excuse.

The drop in pitch or frequency is real.  There's a nice set of coupled
partial diferential equations that describe the behavior, but the simple
explaination is that each of the individual strings "sees" a small fraction
of the other string's mass  through the coupling coefficient.   Each string
in the coupled set vibrates as if it were slightly more massive, so a shift
down in pitch. As an extreme case, think of all the strings of a double or
triple glued together (really strong coupling!).  The frequency would drop
by 1/(sqroot 2) ro 1/(sqroot 3).  In a piano, the coupling is rather weak,
mostly through the air, bridge and soundboard so the pitch shift is small.
Each piano, probably each note may have differences in the coupling so the
shift may sometimes be too small to notice.

Hope this satisfies the cat's curiousity.

Rich Stewart