Nonlinear Piano Strings

It turns out that piano strings exhibit audible nonlinear effects, especially in the first three octaves of its pitch range at fortissimo playing levels and beyond [26]. As a result, for highest quality piano synthesis, we need more than what is obtainable from a linearized wave equation such as Eq. (9.30).

As can be seen from a derivation of the wave equation for an ideal
string vibrating in 3D space (§B.6), there is
fundamentally *nonlinear coupling* between transverse and
longitudinal string vibrations. It turns out that the coupling from
transverse-to-longitudinal is much stronger than vice versa, so that
piano synthesis models can get by with one-way coupling at normal
dynamic playing levels [30,164]. As
elaborated in §B.6 and the references cited there, the
longitudinal displacement
is driven by transverse changes in the
*squared slope* of the string:

where

Since longitudinal waves travel an order of magnitude faster than transverse waves, this coupling gives rise to

In addition to the excitation of longitudinal modes, the nonlinear
transverse-to-longitudinal coupling results in a powerful
*longitudinal attack pulse*, which is the leading component of the initial ``shock
noise'' audible in a piano tone. This longitudinal attack pulse hits
the bridge well before the first transverse wave and is therefore
quite significant perceptually. A detailed simulation of both
longitudinal and transverse waves in an ideal string excited by a
Gaussian pulse is given in [394].

Another important (*i.e.*, audible) effect due to nonlinear
transverse-to-longitudinal coupling is so-called *phantom
partials*. Phantom partials are
ongoing *intermodulation products* from the transverse partials
as they transduce (nonlinearly) into longitudinal waves. The term
``phantom partial'' was coined by Conklin [85]. The Web
version of [18] includes some illuminating sound
examples by Conklin.

- Nonlinear Piano-String Synthesis
- Regimes of Piano-String Vibration
- Efficient Waveguide Synthesis of Nonlinear Piano Strings
- Checking the Approximations

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University