Sign in
Search Online Books
Free Online Books
Ads
Chapters
See Also
Physical Audio Signal Processing
Virtual Musical Instruments
String Excitation
Pluck Modeling
Feathering
Search Physical Audio Signal Processing
Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?
Feathering
Since the pluck model is linear, the parameters are not signal-dependent. As a result, when the string and spring separate, there is a discontinuous change in the reflection and transmission coefficients. In practice, it is useful to ``feather'' the switch-over from one model to the next [470]. In this instance, one appealing choice is to introduce a nonlinear spring, as is commonly used for piano-hammer models (see §9.3.2 for details).
Let the nonlinear spring model take the form
corresponds to a linear spring. The spring constant
linearized about zero displacement 
is
, approaches zero as 
. In other words, the
spring-constant itself goes to zero with its displacement, instead of
remaining a constant. This behavior serves to ``feather'' contact and
release with the string. We see from Eq.
(9.23) above
that, as displacement goes to zero, the reflectance approaches a
frequency-independent reflection coefficient
,
resulting from the damping 
that remains in the spring model. As
a result, there is still a discontinuity when the spring disengages
from the string.
The foregoing suggests a nonlinear tapering of the damping in
addition to the tapering the stiffness 
as the spring compression
approaches zero. One natural choice would be
approaches zero at the same rate as 
. It
would be interesting to estimate 
for the spring and damper from
measured data. In the absence of such data, 
is easy to compute
(requiring a single multiplication). More generally, an interpolated
lookup of 
values can be used.
In summary, the engagement and disengagement of the plucking system can be ``feathered'' by a nonlinear spring and damper in the plectrum model.
Previous: Digitization of the Damped-Spring Plectrum
Next: Piano Synthesis
Order a Hardcopy of Physical Audio Signal Processing
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
Comments
No comments yet for this page