Physical Modeling: Advanced Models

More advanced physical models can be designed, based on the principles explained in the previous sections.

Resonant Bodies & Coupling

The simple lowpass filter in the example can be replaced by more sophisticated models. For instruments with multiple strings, coupling between strings can be implemented.


Model of a wind instrument with several waveguides, connected with scattering junctions (de Bruin, 1995):




  • Stefan Bilbao, Charlotte Desvages, Michele Ducceschi, Brian Hamilton, Reginald Harrison-Harsley, Alberto Torin, and Craig Webb. Physical modeling, algorithms, and sound synthesis: the ness project. Computer Music Journal, 43(2-3):15–30, 2019.
    [details] [BibTeX▼]


  • Chris Chafe. Case studies of physical models in music composition. In Proceedings of the 18th International Congress on Acoustics. 2004.
    [details] [BibTeX▼]


  • Vesa Välimäki. Discrete-time modeling of acoustic tubes using fractional delay filters. Helsinki University of Technology, 1995.
    [details] [BibTeX▼]
  • Gijs de Bruin and Maarten van Walstijn. Physical models of wind instruments: A generalized excitation coupled with a modular tube simulation platform*. Journal of New Music Research, 24(2):148–163, 1995.
    [details] [BibTeX▼]


  • Matti Karjalainen, Vesa Välimäki, and Zoltán Jánosy. Towards High-Quality Sound Synthesis of the Guitar and String Instruments. In Computer Music Association, 56–63. 1993.
    [details] [BibTeX▼]


  • Julius O Smith. Physical modeling using digital waveguides. Computer music journal, 16(4):74–91, 1992.
    [details] [BibTeX▼]


  • Lejaren Hiller and Pierre Ruiz. Synthesizing musical sounds by solving the wave equation for vibrating objects: part 1. Journal of the Audio Engineering Society, 19(6):462–470, 1971.
    [details] [BibTeX▼]
  • Lejaren Hiller and Pierre Ruiz. Synthesizing musical sounds by solving the wave equation for vibrating objects: part 2. Journal of the Audio Engineering Society, 19(7):542–551, 1971.
    [details] [BibTeX▼]

FM Synthesis: DX7

FM synthesis was not only an outstanding method for experimental music but landed a major commercial success. Although there are many more popular and valuable synthesizers from the 80s, no other device shaped the sound of pop music in that era like the DX7 did. It was not the first ever, but the first affordable FM-capable synth and can generate a wide variety of sounds -- bass, leads, pads, strings, ... -- with extensive (but complicated) editing opportunities. It was also the breakthrough of digital sound synthesis, using the full potential with MIDI.


Yamaha DX7.


  • released in 1983

  • 16 Voices Polyphony

  • 6 sine wave 'operators' per voice

  • velocity sensitive

  • aftertouch

  • LFO

  • MIDI

The DX7 in 80s Pop

Tina Turner - What's Love Got To Do With It

  • 1984

  • blues harp preset

  • starting 2:00

Laura Branigan - Self Control

  • 1984

  • the bells

Harold Faltenmeyer - Axel F

  • 1986

  • marimbas

  • starting 1:40

Kenny Loggins - Danger Zone

  • 1986

  • FM bass

A Comprehensive List

Find a comprenesive list of famous examples, here:

Programming the DX7

The DX7 can be fully programmed using membrane buttons. Alternatively, Sysex messages can be used to work with external programmers, like a laptop, over MIDI. For users new to FM synthesis, it may be confusing not to find any filters. Timbre is solely controlled using the FM parameters, such as operator freuqncy ratios and modulation indices.


The configuration of the six operators, respectively how they are connected, is called algorithm in the Yamaha terminology. In contrast do some of its successors, the DX7 does not allow the free editing of the operator connections but provides a set of 32 pre-defined algorithms, shown in [Fig.2].


Yamaha DX7 manual: algorithm selection.


For generating sounds with evolving timbres, each operator's amplitude can be modulated with an individual ADHSR envelope, shown in [Fig.3]. Depending on the algorithm, this directly influences the modulation index and thus the overtone structure.


Yamaha DX7 manual: envelope editing.


The level of each operator, and therefor modulation indices, can be programmed to depend on velocity. This allows the timbre to depend on the velocity, as in most physical instruments, which is crucial for expressive performances.

FM Synthesis: Pure Data Example

The following Pure Data example shows a 2-operator FM synthesis with two temporal envelopes. This allows the generation of sounds with a dynamic spectrum, for example with a sharp attack and a decrease during the decay, as it is found in many sounds of musical instruments. The patch is derived from the example given by John Chowning in his early FM synthesis publication:


Flow chart for dynamic FM with two operators (Chowning, 1973).

The patch fm_example_adsr.pd can be downloaded from the repository. For the temporal envelopes, the patch relies on the abstraction adsr.pd, which needs to be saved to the same directory as the main patch. This ADSR object is a direct copy of the one used in the examples of the PD help browser.


PD FM Patch.

Faust: A Simple Envelope

Temporal envelopes are essential for many sound synthesis applications. Often, triggered envelopes are used, which can be started by a single trigger value. Faust offers a selection of useful envelopes in the envelopes library. The following example uses an attack-release envelope with exponential trajectories which can be handy for plucked sounds. The output of the sinusoid with fixed frequency is simply multiplied with the en.arfe() function.

Check the envelopes in the library list for more information and other envelopes:

// envelope.dsp
// A fixed frequency sine with
// a trigger and controllable release time.
// - mono (left channel only)
// Henrik von Coler
// 2020-05-07


// a simple trigger button
trigger  = button("Trigger");

// a slider for the release time
release  = hslider("Decay",0.5,0.01,5,0.01);

// generate a single sine and apply the arfe envelope
// the attack time is set to 0.01
process = os.osc(666) * 0.77 * en.arfe(0.01, release, 0,trigger) : _ ;

Additive & Spectral: Parabolic Interpolation

Quadratic Interpolation

The detection of local maxima in a spectrum is limited to the DFT support points without further processing. The following example shows this for a 25 Hz sinusoid at a sampling rate of 100 Hz.

Quadratic or parabolic interpolation can be used to estimate the true peak of the sinusoid. using the detected maximum $a$ and its upper and lower frequency bin.

$p = 0.5 (\alpha-\gamma)/(\alpha-2\beta+\gamma)$

$a^* = \beta-1/4(\alpha-\gamma)$

More details on JOS Website

UX in Spatial Sound Synthesis


The User Experience Questionnaire (UEQ) is a well established tool for measuring the user experience of interactive systems and products (Laugwitz, 2008). Below are two results - one from a communication tool and one from an expressive musical instrument.


UEQ results for WhatsApp (Hinderks, 2019).

The UEQ with a novel DMI


UEQ results for the GLOOO instrument (von Coler, 2021).



  • Henrik von Coler. A System for Expressive Spectro-spatial Sound Synthesis. PhD thesis, TU Berlin, 2021.
    [details] [BibTeX▼]


  • Andreas Hinderks, Anna-Lena Meiners, Francisco José Dom\'ınguez Mayo, and Jörg Thomaschewski. Interpreting the results from the user experience questionnaire (ueq) using importance-performance analysis (ipa). In WEBIST 2019: 15th International Conference on Web Information Systems and Technologies (2019), pp. 388-395. ScitePress Digital Library, 2019.
    [details] [BibTeX▼]



  • Bettina Laugwitz, Theo Held, and Martin Schrepp. Construction and evaluation of a user experience questionnaire. In Proceedings of the 4th Symposium of the Workgroup Human-Computer Interaction and Usability Engineering of the Austrian Computer Society, volume 5298, 63–76. 11 2008. doi:10.1007/978-3-540-89350-9_6.
    [details] [BibTeX▼]

Waveguide with Excitation Input

This example is a first step towards excitation-continuous instruments, such as wind instruments. Instead of initializing the waveguides with a single excitation function, they are fed with an input signal.

// waveguide_input.dsp
// waveguide with excitation by input signal
// - one-pole lowpass termination
// Henrik von Coler
// 2020-11-19


// use '(pm.)l2s' to calculate number of samples
// from length in meters:

segment(maxLength,length) = waveguide(nMax,n)
    nMax = maxLength : l2s;
    n = length : l2s/2;

// one lowpass terminator
fc = hslider("lowpass",1000,10,10000,1);
rt = rTermination(basicBlock,*(-1) : si.smooth(1.0-2*(fc/ma.SR)));

// one gain terminator with control
gain = hslider("gain",0.5,0,1,0.01);
lt = lTermination(*(-1)* gain,basicBlock);

// a simple allpass (Smith Paper)
s = hslider("s",0.9,0,0.9,0.01);
c = hslider("c",0.9,0,0.9,0.01);
allpass = _ <: *(s),(*(c):(+:_)~(*(-s))):_, mem*c:+;

// another allpass
g = hslider("g",0.9, 0,0.9,0.01);
allp = allpass_comb(2,1,g);

scatter = pm.basicBlock(allpass);

idString(length,pos,excite) = endChain(wg)

    nUp   = length*pos;
    nDown = length*(1-pos);

    wg = chain(lt : segment(6,nUp) : out : in(excite) : scatter : segment(6,nDown) :  rt); // waveguide chain

exc = select2(gain>0.9,1,0);

length = hslider("length",1,0.1,10,0.01):si.smoo;

process(in) = idString(length,0.15, in) <: _,_;

Envelopes: Exponential

For percussive, plucked or struck instrument sounds, the envelope needs to model an exponential decay. This is very useful for string-like sounds but most importantly for most electronic musicians, it is the very core of kick drum sounds.

In contrast to the ADSR envelope, the exponential one does not contain a sustain portion for holding a sound. The only parameter is the decay rate, allowing quick adjustment. Alternative to an actual exponential, a modified reciprocal function can be used for easier implementation. The factor $d$ controls the rate of the decay, respectively the decay time:

$$ e = \frac{1}{(1+(d t))} $$

The following example adds a short linear attack before the exponential decay. This minimizes clicks which otherwise occur through the rapid step from $0$ to $1$:

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Attack Time:

Decay Time: