The Piano String

Piano strings are audibly stiff [197,196]. Generic stiff-string modeling was introduced in §4.8. In this section, we discuss further details specific to piano strings.

The main effect of string stiffness is to stretch the partial overtone series, so that piano tones are not precisely harmonic. As a result, piano tuners typically stretch the tuning of the piano slightly. For example, the total amount of tuning stretch from the lowest to highest note has been measured to be approximately 35 cents on a Kurzweil PC88, 45 cents on a Steinway Model M, and 60 cents on a Steinway Model D.^6.1

A wave equation suitable for modeling piano strings is given by [74,43,297,487]

$$f(t,x) = \epsilon{\ddot y}- K y''+ EIy''''+ R_0{\dot y}+ R_2 {\ddot y'} \protect$$ (6.1)

where the partial derivative notation $y'$ and ${\dot y}$ are defined on page , and

$$\begin{eqnarray*} f(t,x) &=& \mbox{driving force density (N/m) at position $x$\ ... ... I &=& \mbox{radius of gyration of the string cross-section (m)} \end{eqnarray*}$$

Young's modulus and the radius of gyration are defined in Appendix E.

The first two terms on the right-hand side come from the ideal string wave equation (see Eq. (G.1)), and they model transverse acceleration and the transverse restoring force due to tension, respectively. The term $EIy''''$ is the transverse restoring force exerted by a stiff string when it is bent. (In an ideal string with zero diameter, this force is zero.) The final two terms provide damping. The damping associated with $R_0$ is frequency-independent, while the damping due to the $R_2$ term increases with frequency.

In [], the damping in real piano strings was modeled using a length 17 FIR filter for the lowest strings, and a length 9 FIR filter for the remaining strings.