Musical Applications of Banded Waveguides

In a companion article found in this issue of Computer Music Journal (Essl et al. 2003), we introduced the theory of banded waveguides, showing the advantages of this synthesis technique that allows efficient simulation of highly inharmonic vibrating structures. In this article, we provide an overview of different musical instruments that have been modeled efficiently using banded waveguides. Additional detail can be found in Cook (2002); Essl and Cook (1999); Essl and Cook (2000); Essl (2002); Kapur et al. (2002); Serafin et al. (2002); and Serafin, Wilkerson, and Smith (2002). Links to software implementations of these models available online can be found in the conclusion of this article.

First, we discuss a banded-waveguide model of bar percussion instruments followed by a model of a musical saw. Next, we show how to use banded waveguides to model bowed glasses and bowls, and we conclude by presenting models of a Tabla and a bowed cymbal.

Bar Percussion Instruments

In this section, we discuss the simulation of bar percussion instruments using banded waveguides. This type of instrument was the first to be modeled using this approach. In fact, banded waveguides were originally invented to model the case of bowed bar percussion instruments. The problem of efficiently modeling this instrument had not been solved, nor had it previously received experimental attention. Both the development of the synthesis method for this case and experimental measurements of bowed bars were reported in detail in Essl and Cook (2000). It was realized that the difficulties that vibrating solid bars pose can be overcome by modeling the resonant modes of bars as spectrally separated closed traveling waves, as described in detail in Essl et al. (2003).

In the past, struck bar percussion instruments have been modeled using resonant modal filters (Wawrzynek 1989; Cook 1997) or additive sinusoidal synthesis (Serra 1986; van den Doel and Pai 1998). Acoustical properties of bar percussion instruments have been studied using finite difference and element methods (Bork 1995; Chaigne and Doutaut 1997; Doutaut, Matignon, and Chaigne 1998; Orduña Bustamante 1991; Bretos, Santa-maría, and Moral 1999; Bork et al. 1999), but these methods are too computationally expensive to run in real time and thus have not been used for interactive performances. Finite element methods have also been used to model combined visual and acoustic simulations of sounding objects, including bar percussion instruments (O'Brien, Cook, and Essl 2001). For summaries and reviews of the research on bar percussion instruments, see Moore (1970), Rossing (1976), and Fletcher and Rossing (1998). Some of the results described in this section have also been presented in Essl and Cook (1999), Essl and Cook (2000), and Essl (2002).

Figure 1.

A banded waveguide structure as proposed in Essl and Cook (1999).

Figure 2.

Sonogram of the simulated bowed bar. Note how many partials appear in the spectrum owing to the nonlinearity of the excitation mechanism.

Modeling Bowed Bars

As explained in the companion article on theory (Essl et al. 2003), banded waveguides are filter structures that consist of a simple band-pass filter and delay line for each significant mode to be modeled. Banded waveguides can be constructed from physical dynamics or from modal measurements. The original banded-waveguide structure as proposed in Essl and Cook (1999) is shown in Figure 1.

Figure 3.

One of the authors playing a saw.

Here and in later applications, we use the modal measurement approach. For details on dynamical derivations and interpretations, refer to Essl (2002). The uniform bar measurement yields the wellknown stretching of the inharmonic partials of a uniform bar (1:2.756:5.404:8.933 and so on) as heard from glockenspiels (Fletcher and Rossing 1998). Marimba, xylophone, and vibraphone bars are undercut, stretching the partials into harmonic ratios of either 1:4:10 or 1:3:6 (Moore 1970).

Using these frequencies, the length of the delays as well as the frequencies of the band-pass filters of all banded...