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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 [30,163]. As elaborated in §B.6 and the references cited there, the longitudinal displacement $ \xi$ is driven by longitudinal changes in the squared slope of the string:

$\displaystyle \epsilon\ddot{\xi} = ES\xi'' + \frac{1}{2}ES[(y')^2]' $

where

$\displaystyle [(y')^2]' \isdefs \frac{\partial}{\partial x} \left[\frac{\partial}{\partial x}\, y(t,x)\right]^2. $

Since longitudinal waves travel an order of magnitude faster than transverse waves, this coupling gives rise to high-frequency inharmonic overtones (corresponding to longitudinal modes of vibration) in the sound. Since the nonlinear coupling is distributed along the length of the string, the longitudinal modes are continuously being excited by the transverse vibration across time $ t$ and position $ x$ along the string.

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 [391].

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

When one-way transverse-to-longitudinal coupling is sufficiently accurate, we can model its effects separately based on observations of transverse waves. For example [30,28],

  • longitudinal modes can be implemented as second-order resonators (``modal synthesis''), with driving coefficients given by

  • orthogonally projecting [451] the spatial derivative of the squared string slope onto the longitudinal mode shapes (both functions of position $ x$).

  • If tension variations along the string are neglected (reasonable since longitudinal waves are so much faster than transverse waves), then the longitudinal force on the bridge can be derived from the estimated instantaneous tension in the string, and efficient methods for this have been developed for guitar-string synthesis, particularly by Tolonen (§9.1.6).

An excellent review of nonlinear piano-string synthesis (emphasizing the modal synthesis approach) is given in [30].

Regimes of Piano-String Vibration

In general, more complex synthesis models are needed at higher dynamic playing levels. The main three regimes of vibration are as follows [27]:

  1. The simplest piano-string vibration regime is characterized by linear superposition in which transverse and longitudinal waves decouple into separate modes, as implied by Eq.$ \,$(9.30). In this case, transverse and longitudinal waves can be simulated in separate digital waveguides (Ch. 6). The longitudinal waveguide is of course an order of magnitude shorter than the transverse waveguide(s).

  2. As dynamic playing level is increased, transverse-to-longitudinal coupling becomes audible [26].

  3. At very high dynamic levels, the model should also include longitudinal-to-transverse coupling. However, this is usually neglected.

Other regimes arise depending on whether tension can be assumed to be uniform along the string [27].

Efficient Waveguide Synthesis of Nonlinear Piano Strings

The following nonlinear waveguide piano-synthesis model is under current consideration [427]:

  • Initial striking force determines the starting regime (1, 2, or 3 above).
  • The string model is simplified as it decays.
At pianissimo playing levels, for example, a linear string model (§9.4.1) can suffice. At higher playing levels (e.g., fortissimo for the first three octaves of the piano keyboard [26]), the string model starts in regime 2.

Because the energy stored in a piano string decays monotonically after a hammer strike (neglecting coupling from other strings), we may switch to progressively simpler models as the energy of vibration falls below corresponding thresholds. Since the purpose of model-switching is merely to save computation, it need not happen immediately, so it may be triggered by a string-energy estimate based on observing the string at a single point $ x$ over the past period or so of vibration. Perhaps most simply, the model-regime classifier can be based on the maximum magnitude of the bridge force over at least one period. If the regime 2 model includes an instantaneous string-length (tension) estimate, one may simply compare that to a threshold to determine when the simpler model can be used. If the longitudinal components and/or phantom partials are not completely inaudible when the model switches, then standard cross-fading techniques should be applied so that inharmonic partials are faded out rather than abruptly and audibly cut off.

To obtain a realistic initial shock noise in the tone for regime 1 (and for any model that does not compute it automatically), the appropriate shock pulse, computed as a function of hammer striking force (or velocity, displacement, etc.), can be summed into the longitudinal waveguide during the hammer contact period.

The longitudinal bridge force may be generated from the estimated string length (§9.1.6). This force should be exerted on the bridge in the direction coaxial with the string at the bridge (a direction available from the two transverse displacements one sample away from the bridge).

Phantom partials may be generated in the longitudinal waveguide as explicit intermodulation products based on the transverse-wave overtones known to be most contributing; for example, the Goetzel algorithm [451] could be used to track relevant partial amplitudes for this purpose. Such explicit synthesis of phantom partials, however, makes modal synthesis more compelling for the longitudinal component [30]; in a modal synthesis model, the longitudinal attack pulse can be replaced by a one-shot (per hammer strike) table playback, scaled and perhaps filtered as a function of the hammer striking velocity.

On the high end for regime 2 modeling, a full nonlinear coupling may be implemented along the three waveguides (two transverse and one longitudinal). At this level of complexity, a wide variety of finite-difference schemes should also be considered (§7.3.1) [53,555].

Checking the Approximations

The preceding discussion considered several possible approximations for nonlinear piano-string synthesis. Other neglected terms in the stiff-string wave equation were not even discussed, such as terms due to shear deformation and rotary inertia that are included in the (highly accurate) Timoshenko beam theory formulation [261,169]. The following questions naturally arise:

  • How do we know for sure our approximations are inaudible?
  • We can listen, but could we miss an audible effect?
  • Could a difference become audible after more listening?
To answer these questions, it is helpful to have a truth reference--a ``perceptually exact'' model.

Note that there are software tools (e.g., from the world of perceptual audio coding [62,472]) that can be used to measure the audible equivalence of two sounds [456]. For an audio coder, these tools predict the audibility of the difference between original and compressed sounds. For sound synthesis applications, we want to compare our ``exact'' and ``computationally efficient'' synthesis models.

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High-Accuracy Piano-String Modeling
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Stiff Piano Strings