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Empirically Calibrated Stulov Model

Under normal playing conditions the following approximate Stulov model suffices:

$\displaystyle f_h(t) = Q_0\left[u^p + \alpha \frac{d(u^p)}{dt}\right]
$

where

\begin{eqnarray*}
\alpha &=& 248 + 1.83\,n - 5.5 \cdot 10^{-2}n^2\\
Q_0 &=& 183\,e^{0.045\,n}\\
p &=& 3.7 + 0.015\,n\\
n &=& \mbox{ Piano key number $n\in[1,88]$}\\
u &=& \mbox{ hammer-felt compression}\\
f_h &=& \mbox{ force of hammer felt against string}
\end{eqnarray*}

with

Additionally, the piano-hammer mass may be approximated across the keyboard by

$\displaystyle m = 11.074 - 0.074\,n + 0.0001\,n^2
$


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Download PianoHammer.pdf
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``Piano-Hammer Modeling'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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