Formula

The sawtooth is an asymmetric waveform with a sharp timbre. The related Fourier series is described by the following characteristics:

• odd and even harmonics

• alternating sign

• slow decrease towards higher partials

\(X(t) = \frac{2}{\pi} \sum\limits_{k=1}^{N} (-1)^i \frac{\sin(2 \pi i f\ t)}{i}\)

Interactive Example

Pitch (Hz):

Number of Harmonics:

Output Gain:

In contrast to the triangular wave, the interactive example shows the occurrence of ripples at the steep edges of the waveform. The higher the number of partials, the denser the ripples. This is referred to as the Gibbs phenomenon.

Faust Code

The Faust project features a rich set of properly documented examples. Nevertheless, this class has an accompanying Git Repository for code snippets and small tutorials.

Links

There are a couple of links, which reappear as references in this lecture or can be used as additional resources.

• CCRMA website of Julius Smith

• Faust Tutorials by Romain Michon

Installing the Library

On Ubuntu systems, as the ones used in this class, the liblo library is installed with the following command:

$ sudo apt-get install liblo-dev

Including the Library

The liblo comes with additional C++11 wrappers to offer an object-oriented workflow. This feature is also used in the examples of this class. The following lines include both headers:

#include <lo/lo.h>
#include <lo/lo_cpp.h>

The GainExample

The GainExample is based on the ThroughExample, adding the capability to control the gain of the passed through signal with OSC messages.

$ ./gain_example -p 6666

// create new server
st = new lo::ServerThread ( p );

// / Add the example handler to the server !
st->add_method("/gain", "f", gain_callback, this);

st -> start ();

statCast->gain = argv[0]->f;

// get the recent gain value from the OSC manager
double gain = oscman->get_gain();

out[chanCNT][sampCNT] = in[chanCNT][sampCNT] * gain;

Compiling

When compiling with g++, the liblo library needs to be linked in addition to the JACK library:

$ g++ -Wall -std=c++11 src/main.cpp src/gain_example.cpp src/oscman.cpp -ljack -llo -o gain_example

Compiling a Program

Examples and projects in this modules can be compiled with g++ from the GNU Compiler Collection. With the proper libraries installed, g++ can be called directly for small to medium sized projects. The first example, just passing through the audio, is compiled with the following command:

g++ -Wall src/gain_example.cpp src/oscman.cpp -ljack -o gain_example

The compiler gets the extra argument Wall to print all warnings. All source (cpp) files are passed to the compiler, followed by all libraries which need to be linked (linker arguments). The name of the binary or executable is specified after the -o flag.

Build Scripts

Ready Instruments

For a quick start, fully functional physical modeling instruments can be used from the physmodels.lib library. These *_ui_MIDI functions just need to be called in the process function:

import("all.lib");

process = nylonGuitar_ui_MIDI : _;

The same algortithms can also be used on a slightly lower level, combining them with custom control and embedding them into larger models:

import("all.lib");

process = nylonGuitarModel(3,1,button("trigger")) : _;

Ready Elements

The physmodels.lib library comes with many building blocks for physical modeling, which can be used to compose instruments. These blocks are instrument-specific, as for example:

• (pm.)nylonString

• (pm.)violinBridge

• (pm.)fluteHead

Bidirectional Utilities & Basic Elements

The bidirectional utitlities and basic elements in Faust's physical modeling library offer a more direct way of assembling physical models. This includes waveguides, terminations, excitation and others:

• (pm.)chain

• (pm.)waveguide

• (pm.)lTermination

• (pm.)rTermination

• (pm.)in

From Scratch

Taking a look at the physmodels.lib library, even the bidirectional utilities and basic elements are made of standard faust functions:

https://github.com/grame-cncm/faustlibraries/blob/master/physmodels.lib

chain(A:As) = ((ro.crossnn(1),_',_ : _,A : ro.crossnn(1),_,_ : _,chain(As) : ro.crossnn(1),_,_)) ~ _ : !,_,_,_;
chain(A) = A;

Pupitre D'Espace

Pierre Schaeffer developed the Pupitre D'Espace, a device for spatializing tape music in real time, in 1951. This special case of spatialization for acousmatic music is also referred do as diffusion. The device works with three induction coils for detecting the position of the hand-held transponder in space.

Image from Nicolau Centola's PhD thesis.

Poème électronique

The Philips Pavilion (from medienkunstnetz )

The Philips Pavilion, built for the 1958 World's Fair in Brussels, featured a multi-channel sound system with an unconventional loudspeaker arrangement. It was used for Edgard Varèse's Poème électronique to move sound along paths defined by the loudspeaker positions.

Source movements in 'Poème électronique'

Rotationstisch

Starting in 1959, Stockhausen used a rotating table (Rotationstisch) for creating sound movements in quadraphonic tape compositions. A loudspeaker in the center is rotated with the table, captured by four fixed microphones surrounding it. The directivity but also the inherent Doppler effect creates the image of a rotating sound source, when played back on a quadraphonic setup.

Loudspeaker Orchestras

A loudspeaker orchestra uses loudspeakers themselves as musical instruments, rather than as means for reproducing sound. A typical setup uses models with very different, distinct characteristics, placed at individual positions, rather than in a geometric shape. During a performance, pre-composed music, often stereo, can be sent to the different speakers or speaker groups. This process, referred to as diffusion, is a standard technique in Acousmatic Music.

The Acousmonium

The Acousmonium, launched by French GRM (Groupe de Recherches Musicales) in 1974, is the original and most prominent loudspeaker orchestra.

Francois Bayle with the Acousmonium from "Our Research for Lost Route to Root" (Jérôme Barthélemy, 2008)

BEAST

The BEAST (Birmingham ElectroAcoustic Sound Theatre) is a younger system, following the principles of the Acousmonium. It was brought to Berlin for the 2010 edition of the festival Inventionen, when Jonty Harrison was guest professor at TU Berlin.

The BEAST at Elisabeth-Kirche, Berlin

HaLaPhon

Principle

The HaLaPhon, developed by Hans Peter Haller at SWR in the 70s and 80s, is a device for spatialized performances of mixed music, and live electronics. The first version was a fully analog design, whereas the following ones used analog signal processing with digital control.

The HaLaPhon principle is based on digitally controlled amplifiers (DCA), which are placed between a source signal and loudspeakers. It is thus a channel-based-panning paradigm. Source signals can be tape or microphones:

DCA (called 'Gate') in the HaLaPhon.

---

Each DCA can be used with an individual characteristic curve for different applications:

DCA: Different characteristic curves.

Quadraphonic Rotation

A simple example shows how the DCAs can be used to realize a rotation in a quadraphonic setup:

Circular movement with four speakers.

---

Quadraphonic setup with four DCAs.

Envelopes

The digital process control of the HaLaPhon generates control signals, referred to as envelopes by Haller. Envelopes are generated through LFOs with the following waveforms:

Circular movement with four speakers.

---

Envelopes for each loudspeaker gain are synchronized in the control unit, resulting in movement patterns. These can be stored on the device and triggered by the sound director or by signal analysis:

Quadraphonic setup with four DCAs.

---

References

Exercise

Where to put SC Classes

SuperCollider classes are defined in .sc files with a specific structure. For compiling a class when booting the interpreter, it needs to be located in a directory which is scanned by SC. For this reason, an installation of SC creates a directory for user-defined content. Inside sclang, this directory can be shown with the following command:

Platform.userExtensionDir

On Linux systems, this is usually:

/home/someusername/.local/share/SuperCollider/Extensions

For more information, read the SC documentation on extensions.

Structure of SC Classes

The following explanations are based on the example in the repository. A class is defined inside brackets, with the class name:

SimpleSynth
{
  ...
}

var dur;

// constructor
*new { | p |
        ^super.new.init(p)
}

// initialize method
init { | p |
        dur    = 1;
}

play
      { | f |
    ...
  }

Creating Help Files

In SC, help files are integrated into the SCIde for quick access. Help files for classes are also created during compilation. They need to be placed in a directory relative to the .sc file with the extension .schelp:

HelpSource/Classes/SimpleSynth.schelp

Read the SC documentation on help files for more information.