Richard Brekne wrote:
> I am curious.... just how many of you have read Gabriel Weinreichs
> paper "The coupled motion of piano strings" ?
This seems to be three papers, due editing - the JASA version being the
most technical (JASA v.62 nr.6 [Dec. '77] - p.1474-1484), and the SciAm
apparently the least scientific. The phenomenon of coupling he describes
also has been applied to single strings (Sankey) and to account for the
presence of theoretically missing partials (Legge and Fletcher).
The latter ("Nonlinear generation of missing modes on a vibrating
string". JASA v.76 nr.1 [Jul. '84] - p.5-12) might account for Ron's
observation of backscale influence to pitch drop. Their model uses
bridge compliance to explain a partial failure of Young's law, where
partials corresponding to striking points are predicted to be absent
(C.V. Raman observes its complete failure for vina and tanpura due their
unique bridges). Legge and Fletcher predict the coupling magnitude for
the indirectly generated missing partial (of a single string) will be
proportional to the backscale length and its bearing height at the
bridge (they write [sin(w)], where [w] is the bearing angle), and
behaves as the square of the initial amplitude, arising somewhat later
than the characteristic partials.
The mechanical properties of the bridge assembly (not always uniform
even within a unison!) may be modified by the strings themselves - by
speaking lengths, bearing, number of strings in motion and backscale
lengths. It should be that the behavior of multiple strings will differ
from that of a single - the basis for Weinreich's theories - and
obviously backscales can be classed as coupled vibrating strings.
Clark
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