Poisson Point Process Synthesis — User Guide

What this does

The generator creates two independent stochastic event streams, one for the left channel and one for the right. Each stream is a homogeneous Poisson point process with its own event rate in events per second.

Every event triggers one short sinusoidal grain. Frequency, duration, and amplitude are randomized independently for each event. Grains can overlap freely because event spacing and grain duration are independent variables.

The final stereo image is not created by panning individual grains. The left and right Poisson processes are synthesized independently first; Stereo width then transforms their complete channel sums through a mid/side matrix.

Quick start

1. Run Poisson_Point_Process_Synthesis.praat.
2. Choose Custom or one of the seven presets.
3. Set Duration, Left/Right event rates, Left/Right base frequencies, and Stereo width.
4. Enable Edit details to change sample rate, random seed, frequency spreads, mean grain durations, duration spreads, and nominal amplitudes.
5. Run the script. Independent Poisson event times and grain parameters are generated for both channels, rendered in chunks, combined to stereo, width-adjusted, faded, and optionally peak-protected.

Poisson timing

For a homogeneous Poisson process with rate λ events per second, the waiting time between consecutive events is exponentially distributed:

p(Δt) = λ exp(-λΔt)

expected waiting time = 1 / λ

Therefore the event rate controls average density, not a fixed pulse interval. Individual inter-onset intervals remain random.

The expected event count over a duration T is approximately:

expected events = λ × T

but every finite realization can contain more or fewer events. At very low rates and short durations it is possible to obtain zero events in one or both channels.

Normalized waiting-time QC

The visualization multiplies each realized waiting time by its process rate:

normalized waiting time = λ × Δt

For the theoretical exponential distribution, the expected normalized mean is 1 and the coefficient of variation is also 1. The Info window reports the realized event rate, mean IOI, and IOI CV for each channel.

Grain model

Each Poisson event receives three randomized parameters: centre frequency, duration, and amplitude.

Frequency

For each channel, grain frequency is drawn uniformly from the effective frequency interval:

base frequency - spread/2
        ...
base frequency + spread/2

The requested interval can be truncated by the synthesis frequency guard described below.

Duration

Grain duration is drawn uniformly around the mean:

duration =
  mean duration
  + spread × (U - 0.5)

where U is uniform on [0,1]

The form validation requires the mean to be greater than half the spread so the requested random duration interval remains positive. Internally, duration is also protected by a 10 ms minimum before end-of-Sound clipping.

If a grain begins near the end of the requested Duration, it is shortened so that it does not extend past the final sample. After that clipping, grains of 5 ms or less are not rendered.

Amplitude

Each grain amplitude is uniformly randomized between 70% and 100% of its channel's nominal amplitude:

A = nominalAmplitude × (0.7 + 0.3U)

Hann-windowed sinusoid

The rendered grain is:

g(t) =
  A × sin(2π f t)
    × 0.5[1 - cos(2π(t - tevent)/D)]

inside the grain interval and zero outside it. The Hann envelope reaches zero at both grain boundaries, so overlapping grains sum without a discontinuous rectangular gate.

The carrier uses absolute Sound time t, not a phase reset to zero at each event. The envelope still begins and ends at zero, so this absolute-time phase choice does not create a boundary discontinuity.

Overlap

Overlap is genuine: if a new Poisson event occurs before an earlier grain has finished, both grains are present simultaneously and are added into the same channel.

For implementation stability, the script adds grains to the Sound in groups of 25 formula terms. Chunking changes only how the computation is assembled; it does not change the event realization.

Stereo width

The independent left and right grain sums first form the raw stereo pair. Stereo width then uses an exact mid/side transform:

M = (L + R) / 2
S = (L - R) / 2

L' = M + width × S
R' = M - width × S

The output object remains stereo even when Width = 0.

Controls

Main page

Details page

Presets

Presets override the left/right rates, base frequencies, frequency spreads, grain durations, duration spreads, nominal amplitudes, and Stereo width. Sample rate and Random seed remain controlled by the Details page.

Centre-frequency safety

The engine limits grain centre frequencies to:

0.1 Hz ... 95% of Nyquist

For each channel, the requested interval is first calculated from base frequency ± half the spread. The lower endpoint is clipped upward to 0.1 Hz and the upper endpoint downward to 95% of Nyquist.

This is range truncation, not a common scaling of base frequency and spread. If a requested interval does not intersect the valid synthesis range at all, the script stops.

Output and level

Visualization and QC

The QC block reports target/realized event rates, IOI CV, stored grain means, rendered grain counts, stereo width, centre-frequency guard, sample rate, event counts, final peak/RMS, and whether peak protection changed the level.

Further reading

• Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford University Press. ISBN 978-0-19-853693-2. DOI: 10.1093/oso/9780198536932.001.0001.
• Xenakis, I. (1992). Formalized Music: Thought and Mathematics in Composition, revised edition. Pendragon Press. The book includes chapters on free and Markovian stochastic music and provides the direct compositional context for stochastic-process approaches to musical organization.