DIDS Group (group 1)

Physical Modeling

Digital Waveguide - Israel

Smith used extensions of Karplus-Strong algorithm to create Digital Waveguide Theory; most popular synthesis by physical models.

Smith, Julius O. Physical Modeling Using Digital Waveguides.
Stanford University, 1992.

Smith, Julius O. Digital Waveguide Model. Digital Waveguide Model. September 13, 2008, from

Aird, Mark. Extending Digital Waveguides to include Material Modeling. EDWMM September 13, 2008, from

Digital Waveguide Synthesis. Wikipedia September 13, 2008, from

Karplus-Strong – Daniel

Alex Strong and Kevin Karplus

Parameters available for control: pitch, amplitude, decay time.
Pitch: integer approximately the period of the sound in samples (periodicity parameter p)

Amplitude: initial peak amplitude A

Decay time: determined by pitch and decay stretch factor S

Wavetable Synthesis:
- repeating # of samples producing periodic signal

Yt = Yt – p

p = wavetable length or periodic parameter

- simple waveform calculated and loaded into wavetable before note is played. Sampling frequency of fs, frequency of tone = fs / p

- Modification to wavetable done by changing its value only.

Plucked-String Algorithm:
- Simplest modfication invented by Alex Strong 1978:

Yt = ½ (Yt – p + Yt –p –1)

- Averaging process produces slow decay of waveform
- Pitch corresponds to period of p + ½ samples
- Frequency = fs / p + ½
- Produce realistic string sound start with high harmonics
- accomplished by filling random values at beginning of each new note
- If p is small, variation between different initial conditions à poor control of amplitude
- Values p<32 undesirable; restriction requires sampling rates of at least 28.6 KHz
o Adequate performance can be achieved for sampling rates around 20 KHz

http://cnx.org/content/m15489/latest/#video-build <— This is an interesting video of a professor building the algorithm in a program called Labview.

Roads, ed. 1989. The music machine: selected readings from Computer Music Journal. Cambridge, Mass: MIT Press. ML1093.M88 1989. History and criticism. pp. 467 - 477

Exciter/Resonator - Guitarza
i found a few things, but they were all basically the same thing…



Before the synthesis of excitation can occur, it is necessary to determine the initial condition of the exciter. At the most fundamental level there exist only two types of initial conditions: those in which only one state of equilibrium exists (percussion), and those in which the exciter begins a new cycle of excitation from a variable equilibrium (bowed and wind instruments). Direct generation Modeling is a black-box technique for those instruments that are persistently excited. This technique may include any system that can generate an excitation signal. The most commonly used example is the table-lookup generator. Direct generation is usually incorporated into feed-forward coupling structures (see below). Memoryless Nonlinear Modeling is a black-box technique often used to model the exciter of wind instruments. This is because it generates an excitation signal that is derived from an "external" input signal that normally incorporates the excitation actions of the performer (FIGURE 4)(Borin 34). This model is also capable of using information that is coming from the resonator. Thus, the resonator's reaction to an excitation influences the excitations that follow.
Mechanical Modeling is a white-box technique where the exciter is described using springs, masses and dampers (FIGURE 5). Generally, excitations of this type are represented by a series of differential equations that govern the dynamic behavior of these elements (Borin 34). These models can be used to model almost any instrument.


"The description of a resonator, without serious loss of generality, is reducible to that of a causal, linear, and time-varying dynamical system" (Borin 35). As with exciters, there are both black- and white-box techniques for modeling resonators. Both techniques can produce surprisingly musical results. Transfer-Function Modeling is a simple, black-box technique that ignores the physical structure of the resonator. The transfer-function model usually implements a transformation of pairs of dual variables, such as pressure and flow or velocity and force (Borin 35). Because this resonator is such a generic device, it is not the most musically useful resonator model. Mechanical Modeling of the resonator is very similar to mechanical modeling of the exciter. A series of differential equations are used to simulate the dynamic behavior of virtual masses, springs, and dampers. Waveguide Modeling is an efficient technique that is based on the analytic solution of the equation that describes the propagation of waves in a medium. For example, the waveguide model of the reed of a wind instrument requires only one multiply, two additions, and one table lookup per sample of synthesized sound (Smith 275). Because of the small number of simulations that it requires, this technique was the first to be incorporated into commercially available synthesizers.

Modal Models vs. Mass-spring models - David

Cook. 2002. Real sound synthesis for interactive applications. Natick, Mass: AK Peters. TK7881.4 .C666 2002. Synthesis, physical modeling.

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