In general, strings are in great shape. In most cases, parity with real strings is possible at low cost (e.g., several voices in real time on a single processor). Strings are relatively easy to model efficiently because in the real world they are generally uniform, tightly stretched, and nearly rigidly terminated. As a result, they are highly linear under normal playing conditions. The digital waveguide approach to string modeling therefore works very well for strings typically used in musical instruments. In these models, the wave propagation delay along the string is implemented using an ordinary delay line, while damping and dispersion characteristics associated with propagation on the string are lumped into low-order digital filters.
The first-order effect of nonlinearity in strings is normally to sharpen the fundamental frequency slightly at the beginning of a hard pluck or strike, particularly on a low tension string such as a banjo string. Since variable pitch is routinely implemented for purposes of vibrato anyway, it is quite easy to add this main effect of nonlinearity to the extent it is there.
Another complicating factor for strings is the important coupling that exists in most instruments, e.g., in the piano . At CCRMA, we have almost never seen real string measurements that do not exhibit beating or otherwise modulated decay rates due to coupling, and these effects add an important quality to the sound. As an extreme example, even a solid-body electric guitar (a 1969 Les Paul Deluxe) shows pronounced beats by the seventh partial. In this case, the coupling is primarily between the vertical and horizontal planes of vibration on a single string. In principle, the two transverse vibration planes are always a little out of tune with each other because the bridge in most instruments moves more easily in the direction normal to the body than in the horizontal direction.
To simulate both vertical and horizontal planes of transverse string vibration, two digital waveguides are needed, slightly out of tune. It is generally sufficient to implement coupling only at the bridge, although in principle they are coupled along the entire length of the string [22, see appendix]. To simulate longitudinal compression waves as well, which are quite audible in the low piano strings, a third waveguide is needed which is much shorter than the two transverse waveguides because compression waves generally travel much faster in strings than transverse waves. In bowed strings , it is argued that torsional waves are also important, thus adding yet a fourth digital waveguide per string.
In the piano, for key ranges in which the hammer strikes three strings simultaneously, nine coupled waveguides are required per key for a complete simulation (not including torsional waves); however, in a practical, high-quality, virtual piano, one waveguide per coupled string (modeling only the vertical, transverse plane) suffices quite well. It is difficult to get by with less than the correct number of strings, however, because their detuning determines the entire amplitude envelope as well as beating and aftersound effects . Efficient implementation of coupled strings is discussed in . The best existing digital waveguide piano implementation appears to be a SynthBuilder patch  based on commuted waveguide synthesis . In this technique, applicable to all linear instruments excited over a short duration, the soundboard is commuted with the string and hammer so that the soundboard model--otherwise a giant digital filter or waveguide network--can be replaced by a simple recording of its impulse response; in other words, the soundboard impulse response can be ``played into the string'' via a small digital filter (representing the hammer) which adjusts the brightness according to striking velocity.
The most cost-effective guitars to date also appear to be based on commuted waveguide synthesis. Matti Karjalainen's excellent flamenco guitar, played at his ICMC-93 talk in Tokyo , implemented six virtual strings in real time on the TI TMSC30 signal processing chip, controlled from a Common Lisp/CLOS environment. The Sondius SynthBuilder classical guitar patch has been ported to a MHz Pentium where each real-time voice occupies less than two percent of the processor. The SynthBuilder distortion-guitar patch sounds authentic to most listeners, and it is possible to get six strings running in real time on a single DSP56002 clocked at MHz, with room left over for effects. (A MHz DSP56001 can run five strings and a flanger in real time.) The distortion-feedback model employs a saturating ``virtual amplifier'' whose output feeds back to the string after a propagation delay , and is a good example of how important nonlinear extensions become straightforward when the building blocks have physical interpretations.
The best bowed strings so far seem also to be based on commuted wageuide synthesis . (In this case, the bowed string is modeled as a periodically plucked string.) Due to the great expense of implementing explicit models for resonating structures such as cello bodies or piano soundboards, commuted models, which replace the resonator model by its recorded impulse response, have an enormous cost advantage over non-commuted physical models. As a result, it's probably safe to say that all acoustic stringed instruments are best synthesized today using commuted waveguide models, as long as the rich resonating body is deemed important. Electric instruments, on the other hand, such as a Zeta violin or solid-body electric guitar, can be modeled as nothing but a string and a pick-up, so there's little to commute, and direct waveguide models are appropriate.
The commuting of body and string is only an exact model when the string is held steady (i.e., when the overall system is linear and time-invariant). As a result, artifacts are encountered in the simplest implementations of bowed strings during highly expressive legato playing. The solutions of these difficulties lead away from true physical modeling, but the results so far are quite promising.
There are several areas for future development in string modeling. For example, transverse and longitudinal waves are really nonlinearly coupled to each other [75,25,22]. A related phenomenon is that bridge geometry typically causes nonlinear frequency doubling , even though all elements meeting at the bridge, including the bridge suspension itself, may be linear. A notable exception to the general absence of nonlinear string research is the model for the Finnish Kantele . Further remarks on nonlinearity are given below under ``Outstanding Problems.''
In the simplified ``commuted synthesis'' models, difficulties must be overcome to provide correct behavior during note-to-note transitions, a situation that arises not only in legato performance, but also in any melodic context other than isolated notes separated by rests. Even the most commonly used plucked string model [37,31], arguably the easiest modeling task of all, has problems on legato transitions. Legato problems arise when a new note begins on a string that is already sounding, or when the string length is changed suddenly while sounding. The reason is that really the model itself should be changed during an excitation from that of an isolated string to that of a string with a excitation or ``finger'' attached. While the ``pick,'' ``finger,'' etc., is in contact with the string, the string is divided into two sections joined by a time-varying, damped scattering junction [51,52]. There is also new signal energy injected in both directions on the string in superposition with the scattering (partial wave reflection and transmission). This physically accurate model of string excitation/partial-termination has been used for bowed strings , and it applies equally well to plucked or struck strings. It is very difficult to get high quality legato performance using only a single delay line.
A reduced-cost, approximate solution for obtaining good sounding note transitions in a basic string model was proposed in . In this technique, the string delay line is ``branched'' during the transition, i.e., a second feedback loop is formed at the new loop delay, thus forming two delay lines sharing the same memory, one corresponding to the old pitch and the other corresponding to the new pitch. A cross-fade from the old-pitch delay to the new-pitch delay sounds good if the cross-fade time and duration are carefully chosen. Another way to look at this algorithm is in terms of ``read pointers'' and ``write pointers.'' A normal delay line consists of a single write pointer followed by a single read pointer, delayed by one-period. During a legato transition, we simply cross-fade from a read-pointer at the old-pitch delay to a read-pointer at the new-pitch delay. In this type of implementation, the write-pointer always traverses the full delay memory corresponding to the minimum supported pitch in order that read-pointers may be instantiated at any pitch-period delay at any time. Conceptually, this simplified model of note transitions can be derived from the more rigorous model by replacing the scattering junction at the excitation or finger by a single reflection coefficient.
Comparatively little work has appeared on calibrating string model parameters to recorded measurements [49,35,65,67]. Working from the opposite direction, it is somewhat difficult to find information on the fundamental physical properties of the strings used in real musical instruments [11,41]. In general, as more physical parameters are pinned down by a priori knowledge, techniques for automatic calibration become more successful. Finally, there seems to be no end in sight to future research in the area of distortion algorithms for electric guitar simulations (although strictly speaking, the string is not normally where the nonlinearity lies, but rather somewhere else in the processing, such as in the virtual preamp or speaker).
While digital models for vibrating strings are ``ready for prime time,'' the various ways of exciting them are still under active development. The simplest cases are plucked and struck excitations, and sufficiently good models exist for these [55,35,57,67,69]. Bowed strings, however, are not yet quite considered to be ``parity ready'' due to the problems mentioned earlier.