The banded digital waveguide mesh Stefania Serafin, Patty Huang, and Julius O. Smith III CCRMA, Department of Music Stanford University Stanford, CA, 94309 USA http://www-ccrma.stanford.edu serafin, pph, jos@ccrma.stanford.edu Abstract In this paper we propose a new technique to model complex resonators, which uses a combination of digital waveguides and waveguide meshes banded in frequency. An application for simulating a bowed cymbal is discussed. 1 Introduction Digital waveguides ([8]) are a widely used synthesis tech- nique for creating physical models of musical instru- ments. While one-dimensional digital waveguides are very efficient for modeling musical instruments whose spectrum is quasi-harmonic, such as strings and tubes, other synthesis techniques are more suitable to model vi- brating systems like bars and plates, where the spectrum is more complex. In order to extend the classical one-dimensional waveguide and use it as a tool to model also complex res- onators, different solutions have been proposed. In this paper we first describe some existing model- ing techniques based on digital waveguides, and then we propose a further extension motivated by the limitations of previous results. 2 The one-dimensional waveguide Figure 1 shows a one-dimensional digital waveguide. A lossless digital waveguide is a bidirectional delay line at some wave impedance, and each delay-line element con- tains a sampled traveling-wave component. z -1 z z z z z z z -1 -1 -1 -1 -1 -1 -1 Figure 1: A digital waveguide formed of two delay lines with opposite wave propagation directions. Efficient physical models of vibrating strings, wind instruments and other quasi-harmonic systems have been implememented using the digital waveguide principle. For an overview on the current status of physical models using digital waveguides, see [8]. 2.1 Advantages and disadvantages of one- dimensional waveguides As stated before, the principle behind digital waveguides makes them a very efficient synthesis tool for systems with negligible or weak stiffness. In situations where stiffness is noticeble but not high, such as piano strings, allpass filters can be used to model the inharmonicity of overtones ([6]). The role of allpass filters is to create a frequency-dependent propagation velocity, which results in partials stretched in frequency. For very stiff systems such as rigid bars, however, a combination of waveguides and allpass filters provides a very inefficient structure not suitable for real-time sound synthesis. In these cases, usually other synthesis tech- niques, such as spectral modeling ([7]) or modal synthesis ([1]), have been used. To cope with this problem, i.e. to be able to use waveguides to also model stiff resonators, Essl and Cook ([2]) proposed a new data structure called banded waveguides, as described in the following section. 3 Banded digital waveguides Figure 2 shows the block diagram structure of a com- bination of banded waveguides as described in [2]. As the name suggests, in banded waveguides the spectrum of a vibrating system is divided into frequency bands,