A waveguide model for slapbass synthesis

Starting from the waveguide model for plucked strings, a new digital signal processing model for the slapping technique on electric bassguitars is derived. The model includes amplitude limitations for the string at the frets and/or the ngerboard. These highly nonlinear elements are realized by conditional re ections which depend on the local string displacement. A model of the string dynamics for the two slapbass techniques | knocking the string with the thumb knuckle and plucking very strong with the index or middle nger | has been implemented both as MATLAB and C simulations and synthesizes sounds close to the natural instrument. 1. MODELING PLUCKED STRINGS One of the simplest musical instruments amenable to physical modeling is the plucked string. The string is modeled as a linear waveguide: two waves are traveling along the string, one in either direction. As a digital representation, simple delay lines can be used. Rigid terminations yield inverted re ections of the waves (for displacement, velocity, and acceleration as state variables of the waveguide model) at the ends of the string. In a more general setting [1], these re ections are implemented by lters which can handle dispersion at the re ection and along the string, too. The representation of a waveguide as a discrete-time delay line corresponds to spatial sampling. The waveguide section index m represents the distance x along the string. Obviously, the best choice for the spatial sampling distance is xs = c=fs, with the (phase) velocity c of the waves on the string and the temporal sampling rate fs in the model, chosen according to the highest frequency of interest. Then, a wave travels the distance xs on the waveguide in the time of one sampling interval. The frequency of the synthesized tone depends on the number of delay units in the delay line N as follows: f = fs=2N . The `Karplus{Strong' algorithm [2] uses random initial conditions for the delay lines, corresponding to random initial displacement and velocity of the string, whereas more recent plucked-string models [1, 3, 4, 5] start with a triangular initial displacement of the string and zero velocity. The re ection lters have low-pass characteristic whether they are FIR like in the `Karplus{Strong' algorithm or IIR Presently also with  OFAI (Austrian Research Inst. for Articial Intelligence), Vienna, Austria. lters like in the plucked-string model [1]. According to physics, the DC gain of the re ection lter at either end of the string must be 1. The lter transfer function then controls the decay of the tone and its harmonics. For electric guitars, a (magnetic) pickup transforms the velocity of the string in an electric signal that feeds the ampli er. For any choice of the state variable in the waveguide model (displacement, velocity, or acceleration), the pickup signal can be generated by taking the sum of the two delay line signals at the pickup position and, unless velocity is the state variable, converting it to a velocity by derivation or integration. 2. PHYSICS OF THE SLAPBASS With the slapping technique, a brilliant and percussive sound can be produced on electric bassguitars. There are two di erent slapping techniques: the string is either struck (`slapped') with the knuckle of the thumb or it is pulled strongly away from the guitar body with the index or middle nger. In both cases, the string hits the frets during the rst fundamental periods of the tone. The strongly pulled string can be modeled in the same way as the plucked string above using a triangular initial displacement. Slapping with the knuckle is very similar to striking a piano string with a hammer [6] and can be modeled as an initial velocity impulse at the striking position. When the string excitation is so strong that it hits the frets, this results in a nonlinear limitation of the string amplitude to the space above the frets and the ngerboard. A model of this amplitude limitation requires to test which part of the string actually hits the frets or the ngerboard. The test can be implemented in a straight-forward manner if the string displacement is chosen as the state variable of the waveguides. To realize this amplitude limitation, the samples between two delay elements are re ected into the delay line that travels in the opposite direction if the sum exceeds the free distance to the fret. When the samples are inverted and the distance of the fret is added, like in gure 1, the amplitude y of the string displacement at a certain position x is limited to yfret(x), which describes the geometry of ngerboard and frets (the ngerboard is below the string: yfret(x) < 0). For the discrete-time simulation the continuous space variable x is replaced by the waveguide section index m. To obtain the pickup signal, the time derivative of the string displacement (i.e., the sum of the samples in the dey+[n;m]+y [n;m] > yfret