A common feature of stringed musical instruments is the presence of a
*resonator*, usually made of wood. For the violin family and
guitars, this resonator is called the *body* of the instrument.
For pianos, harpsichords, and the like, the principal resonator is a
large wooden plate called the *sound board*. The number of
resonant modes in the range of human hearing depends strongly on the
size of the resonator.

Signal-processing models of physical resonators typically employ a
two-pole digital filter to model each resonant mode. Such a
second-order filter is commonly called a *biquad*, because its
transfer function is a ratio of quadratic polynomials in . The
resonance frequency, bandwidth, phase, and gain can all be adjusted to
match a resonant peak in measured frequency-response data
[88].
A general biquad section requires five multiplications for each sample
of sound computed.
Even with today's high-speed computers, it is prohibitively expensive
to model resonators mode for mode.
Modal synthesis [4], [161, Appendix E]^{12}refers to models of resonators consisting of a sum of second-order
resonators. Modal synthesis is effective when there is a small number
of long-ringing modes, such as in the marimba or xylophone.

For larger systems, such as stringed instrument bodies, the number of resonant modes in the range of human hearing is prohibitively large. In volumes, the number of resonances below some frequency limit increases, asymptotically, as frequency cubed, while in membranes (and thin plates) it grows as frequency squared [118, p. 293].

It is not always necessary to preserve the precise tuning and
bandwidth of all the modes in a body resonator model. As an example,
it is known that between 2 and 25 sinusoids (``undamped resonances'')
having equal amplitude and random phases can sound equivalent to noise
in approximately one critical band of hearing
[63,109,205]. (Below 5 kHz, around 15 suffice.)
Also, the required number of sinusoids decreases in the presence of
reverberation [63]. Applying this finding to resonant
modes, we expect to be able to model the modes *statistically* when
there are many per critical band of hearing. Moreover, when the
bandwidths are relatively large, we may revert to a statistically
matched model when there are several modes per bandwidth.

One should note, however, that psychoacoustic results for a specific
type of stimulus do not often generalize to every type of stimulus.
Thus, for example, the specifications for statistically matched
high-frequency modes in a body-resonator may be one thing for
*impulse responses*, and quite another for *periodic
excitations with vibrato*. As J. Woodhouse puts it [201]:

``... somewhere in the subtleties of resonance patterns must be found the answer to the whole mysterious question of what makes one violin more expensive than another. ... when it comes to ultimate judgements of musical quality there may be years of research still to be done. It is an interesting question whether virtual violins will become good enough that players start making `strad-level' judgements about them.''

There are several ways to potentially take advantage of the ear's
relative insensitivity to exact modal frequencies and bandwidths at
high frequencies. One is to superimpose several different harmonic
sums of modes, implemented efficiently using 1D digital waveguides.
Another is to use the
*digital waveguide mesh* for high-frequency statistical mode distribution models;
see §9.2 for some pointers into the waveguide mesh
literature.

Essl's 2002 thesis [56] introduces the topic of ``banded waveguides.'' The basic modeling idea is to use a filtered delay-line loop to model a ``closed wave-train path'' corresponding to an individual mode in extended objects such as cymbals and bars. The loop contains a bandpass filter which eliminates energy at frequencies other than the desired mode. The result is a modeling element somewhere in between a digital waveguide loop (yielding many quasi-harmonic modes) and a second-order resonator which most efficiently models a single mode. For certain classes of physical objects, improved transient responses are obtained relative to ordinary modal synthesis. Banded waveguides efficiently implement a large number of quickly decaying modes for each long-ringing mode retained in the model.

In summary, there are many ways to avoid the expense of modeling each resonant mode using a second-order digital filter, and this section has mentioned some of them.

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