Wavefolding Example

The following example calculates the spectrum of a sinusoidal function, folded with a sinusoidal transfer function:



Time Domain:

Frequency Domain:

Envelopes: ADSR

Envelopes are an essential part of control in electronic music and computer music. They are used to shape the characteristics of sound or other processes over time and are an integral part of synthesizers. Since they are that basic and versatile, they will be introduced in this early section.

One of the most common envelopes, already featured in early synthesizers and in prominent examples as the MiniMoog, is the ADSR envelope (Hiyoshi, 1979). It is comprised of four segments:

  • Attack
  • Decay
  • Sustain
  • Release

Attack time, decay time and release time can usually be controlled by the user via dials or sliders, whereas the sustain time depends on the duration a key is pressed and the sustain level may depend on the stroke velocity. Depending on the settings, the ADSR model can generate amplitude and timbral envelopes for slowly evolving sounds like strings or sounds with sharp attacks and release:

Your browser does not support the HTML5 canvas tag

Attack Time:

Decay Time:

Sustain Level:

Sustain Time:

Release Time:

When used in synthesizers, this envelope can be used to control the overall level or the timbre - for example through the cutoff frequency of a filter or by means of partial amplitudes.


  • Teruo Hiyoshi, Akira Nakada, Tsutomu Suzuki, Eiichiro Aoki, and Eiichi Yamaga. Envelope generator. December 18 1979. US Patent 4,178,826.
  • Working with Groups

    Creating Groups

    Groups - or group nodes - can be a very useful concept for keeping track of the signal flow. Without any further actions, all nodes are placed in the default Group 1. Additional groups can be arranged regarding the order of execution. A new group can be added from sclang as follows:

    ~g1 = Group.new();

    Relative Group Positions

    As with nodes, further groups can be added in relation to existing groups. The following action makes sure that a new group will be placed after the previously defined group:

    ~g2 = Group.after(~g1);

    Nested Groups

    Groups can contain other groups, allowing a hierarchical structure of nodes:

    ~g3 = Group.head(~g2);

    More on Groups

    The group object allows many more actions. They are listed in the SC documentation on groups. After adding another group before the third one

    ~g4 = Group.before(~g3);

    the server node structure looks as follows:


    The server does not know the groups by their variable names in sclang. Hence they are numerated. Node indices - or IDs - of groups can be assessed from the language:




    Use groups to sort the nodes from the example in the section on Combining Nodes

    Pure Data: Installing Externals with Deken

    The basic install of PD is referred to as Vanilla. Although many things are possible with this plain version, some additional libraries are very helpful and there is a handful which can be considered standard.

    Find and Install Extensions

    PD comes with Deken, a builtin tool for installing external libraries. Deken can be opened from the menu of the PD GUI. On Linux installs it is located under Help->Find Externals. Deken lets you search for externals by name. The best match is usually found at the top of the results. cyclone is an example for a library with many useful objects:


    Deken lets you select where to install externals in its Preferences menu. Everything will be located in the specified directory afterwards.

    Add Libraries to Search Paths

    Once installed, it may be necessary to add the individual libraries to the search paths. This is done in an extra step. On Linux installs, this can be found under Edit->Preferences->Path:


    Digital Waveguides: Ideal String without Losses

    Ideal String with two Delay Lines

    Based on the previously introduced discrete solution for the wave equation, an ideal, lossless string can be implemented using two delay lines with direct coupling. The left-traveling and right-traveling wave are connected end-to-end. Output samples of each delay line are direcly inserted to the input of the counterpart:

    The Excitation Function

    The waveguides can be initiated - or excited - with any arbitrary function. This may vary, depending on the excitation principle of the instrument to be modeled. For a plucked string, the excitation can be a triangular function with a maximum at the plucking point $p$. Both waveguides (left/right travelling) are initiated with the same function:

    $$y[i] = \begin{cases} \frac{i}{p} & \mbox{for } i \leq p \\ 1-\frac{i-p}{N-p} & \mbox{for } i > p \\ \end{cases}$$


    When both waveguides are shifted by one sample each $\frac{1}{f_s}$ seconds, the ideal string is oscillating with a frequency of $f_0 = \frac{f_s}{N}$. It will oscillate continuously. In fact it is a superimposition of two oscillators with the waveform defined by the excitation function.

    Once Loop Reflect