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 . 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 is driven by longitudinal changes in the squared slope of 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 .
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 . The Web version of  includes some illuminating sound examples by Conklin.
- longitudinal modes can be implemented as second-order
resonators (``modal synthesis''), with driving coefficients
- orthogonally projecting  the spatial
derivative of the squared string slope onto the longitudinal mode
shapes (both functions of position ).
- 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 .
- 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).
- As dynamic playing level is increased,
transverse-to-longitudinal coupling becomes audible
- At very high dynamic levels, the model should also include
longitudinal-to-transverse coupling. However, this is usually
- Initial striking force determines the starting regime (1, 2, or 3 above).
- The string model is simplified as it decays.
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 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  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 ; 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].
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?
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 . 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.
High-Accuracy Piano-String Modeling
Stiff Piano Strings