Zeitraum Diagram (2014)

◊ Zeitraum Diagram is an interactive graphical formulation* of Zeitraum. It is an abstraction from the primary formulation of the piece as an installation. The diagram focuses on the geometrical relationships between the sound sources and the listener and it allows to sonify the temporal manifestations of these relationships. Experiencing the visualisation and the sonification shares important features with the experience one may have with the installation. The alterations in the timing of the sound events according to changes of the listening position are of the same kind in the sonification as in the installation. The visualisation and the sonification can also be understood as virtual installation, exposing the temporal relationships of the sound events in a planar Zeitraum.

The positions of the listener and the sound sources are represented as points in a plane. The sources and the listener can be moved and sources can be deleted or new ones can be created. All elements of the diagram and all aspects of the sonification change according to the interactive modifications of any of the positions. Changing the listening position (black point) allows to navigate the Zeitraum and changing the source positions (grey points) permits to modify the spatio-temporal composition of the piece. The latter is not possible in the installation, making the complementarity of the two formulations appear.

The default configuration of the sources is isomorphic to the loudspeaker constellation in the installation, but reduced to two dimensions and ignoring the orientation of the loudspeakers (i.e. the sources are omnidirectional and arranged in a plane). It corresponds to a top view of the loudspeaker constellation (cf. figure 1). The distances between each sound source and the listener are represented by straight lines from the source to the listener (cf. figure 2) and optionally by circles centred on the listener and passing through the sources (i.e. the radius of the circle represents the distance). This exposes the patterns of ratios between the distances (cf. figures 3, 5, and 6).

In the installation formulation the different propagation delays of the sounds projected by the loudspeakers are equalised such that the arrival times are aligned for one fixed position (the temporal sweet spot or reference position, which is marked by a light spot in the instalation). This alignment, which is the central compositional operation of the piece, can be performed in the diagrammatic formulation for any listening position interactively. The reference position is marked in the diagram as yet another movable point (red). The alignment is visualised as a projection of the source positions onto a circle centred around the reference position, expressing the condition of equal distances of these virtual sources (small red circles) from the temporal sweet spot. The radius of this circle corresponds to the distance of the speaker farthest away from the temporal sweet spot. The additional delays introduced by the alignment are represented by extending the black lines to the speakers with a red lines to the virtual speakers (cf. figure 4).

Once the alignment has been performed (hit key A in the diagram window to align for listening position) and the virtual sources are displayed (can be toggled with key L), the listening position coincides with the reference position (red point). Although this operation hides the listener (black point), it can still be moved away from the reference position, as it is selected first upon point-and-drag. Once listening and reference position do not coincide any longer, the reference position may also be moved, in turn changing the time alignment of all sources.

If the listener is moved away from the reference position, the distances to the virtual sources start to diverge, which can be observed by the circles assuming different radii, representing different propagation times (cf. figure 5). In order to keep the diagram geometrically consistent, the virtual sources are positioned on their circles such that the lines connecting them to the listener pass through the position of their corresponding (real) sources. The movement of the virtual sources as a function of the listening position has no equivalent in the installation formulation, for which only the radii of the circles are relevant, which represent the total propagation delays. This geometrical representation was chosen because it exposes the fact that the alignment at an off-centre position amplifies the effect of temporal dispersion, which is a key feature of the piece (cf. figure 6).


The sonification can be switched on and off in the diagram using the P key. The same noise bursts than in the installation are used as source signals, but the sound result is completely different (cf. ◊ auralisation), as the sounds do not propagate through an acoustic environment. The real propagation delays due to the source positions and the virtual ones due to the alignment are simulated in the sonification and result in the same rhythmical situation as in the installation.


*) It is currently implemented as a ◊ webpage controllable by mouse and keyboard. A future version with an improved graphical user interface will also run on touch screens. You may want to run the application in another browser window while reading this text. Zeitraum Diagram is implemented in JavaScript using ◊ p5.js and the ◊ Web Audio API. It has been tested under Mac OS X Yosemite with the browsers Safari, Firefox and Chrome. The source code can be found ◊ here.

Figure 1: Default sources (grey) and listener (black) constellation

Figure 2: Three sources (grey) and listener (black)

Figure 3: Distances represented as circles

Figure 4: Aligned sources (virtual sources)

Figure 5: Listener slighly off sweet spot

Figure 6: Listener far off sweet spot (default constellation)

Zeitraum Diagram allows to explore the compositional decisions that led to Zeitraum and also to expand on the idea of the piece through a number of explorative functions implemented in the diagram. Use the H key to toggle the help screen listing all options, which are be briefly explained in the following.

Event frequency: plus (+) and minus (-) keys


With the two keys the event frequency can be changed in 25% steps (+ for faster and - for slower). If the display options (key D) is on the frequency is displayed in Hz (events per second) .

Store/recall constellation: number keys 1 to 9


Constellations of sources, listener and reference position can be stored by hitting one of the keys 1 to 9 while holding down the alternate key. A stored constellation may be recalled by hitting the corresponding key. This allows for an easy comparison of constellations. Only the positions are recorded, no other parameters are stored.

Align: key A


Hitting the A key moves the reference position for the time alignment to the current listening position. If the time alignment is displayed (to be switched on or off with L key), the reference position is displayed as a filled red circle and the aligned (i.e. virtual) speakers are displayed as open red circles.

Display symmetry axes: key B


With the B key the display of the symmetry axes between each pair of sources can be switched on or off. Each point on the axis represented as green line has the same distance to the two corresponding sources.

Centre diagram: key C


The C key can be used to centre the diagram about the listening position. This is useful for keeping the whole diagram displayed inside the window. The scaling can be changed with the up and down keys while holding shift.

Display parameters: key D


With the D key the numerical display of some parameters at the lower end of the window can be switched on and off. Displayed are the speed of sound (in m/s) which can be changed  with the left and right cursor keys while holding down the shift key. The default value is the nominal speed of sound in the air of 343 m/s.

Display control options: key H


The display of all options can be switched on or off with the H key.

Display alignment: Key L

The L key toggles the display of the aligned virtual sources as open red circles, the alignment delays as red lines and the reference position as filled red circle. The red lines drawn from the corresponding source represent the additional delay added to the propagation delay to realise the alignment, making sure that it takes the same time for each source signal to arrive at the reference position. The reference position can also be changed dynamically by clicking and dragging it with the mouse.

Display distance circles: key O


With the O key the representation of the distances from the source to listener position as red circles centred on the listener and passing through the real or virtual source can be switched on or off.

Play sonification: key P

Hitting the P key will switch on or off the sonification. To adjust the sound level use the volume of your computer. Make sure that your sound output is not muted.

Reset constellation: key R


With the R key the current constellation of sources can be erased, i.e. all sources are removed and only the listening and the reference position (if switched on with the L key) remain. All other parameters remain unchanged.

Source sequence: key S


The S key alternates the sequence type between a random sequence with repetition control and a fixed sequence based on the source number, which are assigned consecutively when creating sources with shift clicks or the M key.

Source type: key X


Hitting the X key alternates between two source signal types, the exponentially decaying (T60 = 300 ms) pink noise burst used in the installation and an exponentially decaying sine wave. The frequency intervals of the sine waves are tuned to semitones for subsequent sources. The first source is tuned to 440 Hz. The different pitches allow for an easier identification of patterns in the source sequence depending on the S key. For the signals to be heard, the sonification has to be activated with the P key.