The same written A can come from a violin, flute or piano.
SOUND LAB · 01–08
Eight experiments: from one note to timbre, pitch, rhythm, noise and space.
Each lab changes one thing at a time and lets you hear what changed.
Controlled synthetic sounds · headphones recommendedBEFORE THE SLIDERS
A written note is an instruction. Sound is what reaches the ear.
The score tells a player what to do. The instrument turns that instruction into changing air pressure. We hear the result as pitch, colour, rhythm and space.
440 Hz means 440 cycles each second.
It often follows repetition, but it is not one spectral line.
Harmonics, attack, noise and resonance all matter.
A SHORT ROUTE TO THIS LAB
Music moved from organising notes to exploring sound itself.
From Bach through the Classical and Romantic periods, pitch relations are the main grammar.
Schoenberg, Berg and Webern used atonality and twelve-tone methods.
This shift becomes central after 1945.
Scelsi, Grisey and Murail focus on beating, spectra and changing colour.
Two points of accuracy
Bach: “well-tempered” does not mean today's exact equal temperament. The tuning remains debated.
Spectral music: the French movement formed around L'Itinéraire in the 1970s. Composers had explored the inside of sound before and outside that group.
Oxford · Bach's Well-Tempered Clavier ↗ · IRCAM · Spectralisms ↗
HARMONICS → TIMBRE
Build one note from its harmonics.
Set the lowest component to 220 Hz. Then add 440, 660, 880 Hz and so on at different strengths. The musical pitch can stay recognisably A while the colour changes.
RELATIVE SPECTRUM
A3
Each line is one sinusoidal component at n f0. This is a controlled model of spectral balance, not a full instrument model.
Plain-language explanationthen the compact formula
Think of one musical note as a recipe rather than one ingredient. If the fundamental is 220 Hz, an ideal harmonic recipe can also contain 440, 660, 880 Hz and higher multiples.
The sliders do not change the note name. They change how much of each harmonic is present. More upper harmonics usually make the synthetic tone sound brighter; fewer can make it sound darker or purer.
x(t) = Σ An sin(2π n f0 t + φn)
Read the formula as: add many little sine waves together. f0 is the fundamental, nf0 gives the harmonic frequencies, and An says how strong each one is.
Real instruments are richer again: attack, decay, noise, resonance, inharmonicity and changes through time also matter.
PITCH IS NOT ONE SPECTRAL LINE
You can hear 110 Hz even when 110 Hz is missing.
Play 220, 330, 440 and 550 Hz together. There is no 110-Hz component, but the four frequencies are 2×, 3×, 4× and 5×110. Many listeners still hear a low pitch near 110 Hz.
A · FUNDAMENTAL PRESENT
B · FUNDAMENTAL REMOVED
LISTENING QUESTION
When the 110 Hz line disappears, do you still hear roughly the same low pitch?
Why can 220 + 330 + 440 + 550 Hz sound like 110 Hz?the missing fundamental
Start with the numbers. 220, 330, 440 and 550 Hz are not random:
220 = 2×110 · 330 = 3×110 · 440 = 4×110 · 550 = 5×110
So every component belongs to the same 110-Hz harmonic family. The gaps are also 110 Hz. When these components are added, the combined waveform contains a repeating timing pattern associated with 110 repetitions per second, even though there is no physical 110-Hz spectral line.
The auditory system can group the harmonics and infer their common fundamental. That is why many listeners still hear a pitch near 110 Hz.
The important lesson: perceived pitch is not the same thing as “which frequency has the tallest FFT peak”.
The exact strength of the effect depends on the stimulus and listener; the experiment shows the principle, not a universal pitch detector.
TWO CLOSE FREQUENCIES → BEATING
Two nearby frequencies make a slow pulse.
660 Hz and 664.7 Hz are both steady tones. Put them together and the loudness seems to swell about 4.7 times per second. Nothing is turning a volume knob: the pulse comes from interference.
TONE 1440.0 Hz
TONE 2442.0 Hz
BEAT RATE2.0 / s
BEAT PERIOD0.50 s
INTERFERENCE IN TIME
440 + 442 Hz
TWO NEARBY SPECTRAL LINES
Δf = |f₂ − f₁|
Why do two steady tones make a pulse?interference / beating
Imagine 660 Hz and 664.7 Hz. Sometimes their wave peaks line up and reinforce each other. A little later one has slipped out of alignment and they partly cancel. This repeats over and over.
The difference is 4.7 Hz, so the loudness envelope rises and falls about 4.7 times per second.
beat rate = |f2 − f1|
Show the trigonometric identity
sin(2πf1t) + sin(2πf2t) = 2 cos(π(f2−f1)t) · sin(2π((f1+f2)/2)t)
A nominally stable pitch can therefore contain slow internal motion without any explicit amplitude modulation.
ONE REPETITION RATE · DIFFERENT TIME SCALES
Speed up a rhythm until it starts to sound like pitch.
At 2 pulses per second you hear separate events. Around a few tens of pulses per second they fuse into flutter or buzz. Faster still, the repetition can become pitch-like. The exact boundary depends on the sound and the listener.
separate events
T = 125 ms
The perceptual labels are approximate, not universal thresholds. Headphones recommended at a moderate level.
TIME DOMAIN · FIRST 0.5 s
8 pulses / second
IDEALISED COMB STRUCTURE
spacing = repetition rate
How can a rhythm turn into pitch?same periodic idea, faster time scale
There is no magic switch between “rhythm” and “pitch”. Both can come from repetition.
At 2 Hz, a pulse arrives every half second, so you hear separate events. At 12 Hz, events arrive too quickly to count comfortably and sound like flutter. Around tens of repetitions per second they fuse into buzz; at still higher rates, periodicity can support a pitch-like percept.
frep = 1 / T
If the period T gets smaller, the repetition frequency gets larger. A pulse train also creates a comb-like spectrum whose line spacing equals that repetition rate.
Why this matters musically: rhythm, roughness, timbre and pitch can be different perceptual readings of periodic structure at different time scales.
HARMONIC → INHARMONIC
Move the overtones away from exact integer multiples.
A harmonic sound has components near f₀, 2f₀, 3f₀, 4f₀… Move the upper components away from those exact multiples and the sound stops fusing in the same familiar way.
harmonic
fₙ = f₀ · nα
This power-law deformation is a didactic model created for this site. It is not a reconstruction of a Grisey or Murail compositional algorithm.
PARTIAL FREQUENCIES
α = 1.00 · harmonic reference
What changes when the partials stop being exact multiples?harmonicity / inharmonicity
For an ideal harmonic sound, the upper components sit at 2×, 3×, 4× … the fundamental. That regular spacing helps the ear group them into one pitched object.
This lab keeps the bottom frequency fixed but moves the upper partials. As their spacing becomes less like the harmonic series, the colour changes and the components may fuse less strongly.
fn = f0 · nα
Here α=1 gives exact harmonics. α<1 compresses the upper spacing; α>1 stretches it. This power law is only a teaching device, not a model of a Grisey or Murail score.
LINEAR → NONLINEAR
Distortion can create frequencies that were not in the input.
A linear system can make a sine louder, quieter or phase-shifted, but it cannot invent harmonics. A nonlinear transfer curve reshapes the waveform, so new frequency components appear.
near-linearsaturated
MODELy = tanh(gx) / tanh(g)
THD—
DOMINANT NEW PARTIAL—
A/B playback is RMS-matched approximately so the contrast is mainly timbral, not simply louder versus quieter.
TRANSFER CURVEinput x → output y
ONE PERIODinput / output waveform
HARMONIC CONTENT OF OUTPUT
symmetric curve → mainly odd harmonics
x(t)=sin(2πf₀t) → a·sin(2πf₀t+φ)
The frequency content is not multiplied into a new harmonic family.
y(t)=tanh(gx(t))
Waveform shape changes. A symmetric odd nonlinearity produces mainly odd harmonics.
y(t)=tanh(g(x+b))−tanh(gb)
A bias breaks odd symmetry, so even harmonics can appear as well.
How can a device create frequencies that were not there before?nonlinearity
Feed a perfect sine into a linear system and the output can only contain that same frequency, with a different amplitude or phase.
A nonlinear system changes the shape of the wave itself. Once the waveform is no longer a sine, Fourier analysis needs extra harmonics to describe it.
y = tanh(g x)
For this symmetric waveshaper, odd harmonics become especially prominent as the gain increases.
In plain English: distortion is not merely “making the same sound dirtier”. It can manufacture new spectral ingredients.
MANY PARTS → ONE TEXTURE
Many separate sounds can merge into one texture.
If only a few lines are present, you can follow them individually. Add more lines or more events and the ear may stop tracking each one; you begin to hear density, colour or a cloud instead.
Many continuous lines
Define individual trajectories, then increase their number and overlap until tracking a single line becomes difficult.
fᵢ(t) = f̄ · 2^((δᵢ + mᵢ(t))/1200)
Many stochastic events
Define a probability law for event times and spectral placement, then listen to the population rather than the individual grain.
Δt ~ Exp(λ), E[N(T)] = λT
When does plurality become a field?
The answer is not one fixed threshold. Frequency spacing, temporal coherence, timbre, onset structure, level and attention all matter.
ENGINE A · CONTINUOUS TRAJECTORIES
Ligeti lens — from line to texture
A deliberately simplified bank of slowly moving sinusoids. Every line is deterministic. Density and overlap change what is easy to follow.
MEAN STATIC SPACING—
CENTRE220 Hz
MODELdeterministic line bank
The line bank is a controlled perceptual model, not a reconstruction of a Ligeti score.
FREQUENCY TRAJECTORIES · 5 s
18 deterministic lines
LESS DENSE
Individual trajectories remain easier to follow.
MORE DENSE
Perception can shift toward width, shimmer, colour and collective motion.
When do many parts become one texture?microstructure → macro-percept
With five clearly separated lines, you can often follow individual movements. With dozens of overlapping lines, the ear may stop tracking each trajectory and hear one shifting band or cloud instead.
The same perceptual question can arise from a different mechanism: many short events generated statistically can also accumulate into a mass.
Important distinction: Ligeti-like continuous polyphonic lines and Xenakis-like stochastic event populations are not the same compositional method. This lab puts them side by side because both let us hear how microscopic detail can become a macroscopic texture.
ENGINE B · STOCHASTIC EVENTS
Xenakis lens — from events to statistical mass
Event times follow a Poisson process. Each short grain receives a frequency drawn from a controlled log-frequency band. Increase λ and listen to event identity give way to density.
EXPECTED EVENTS · 4 s112
MEAN INTER-ARRIVAL35.7 ms
RANDOM SEED1729
Poisson timing is one transparent stochastic model. Xenakis used many probabilistic and algorithmic procedures; this is not an Achorripsis, Pithoprakta or Concret PH reconstruction.
ONE STOCHASTIC REALISATION · 4 s
—
LOW λ
The ear can attend to isolated events and gaps.
HIGH λ
The population is more naturally described by density, bandwidth and collective texture.
Why Poisson timing? a transparent stochastic baseline
For a homogeneous Poisson process with rate λ, the waiting time between events is exponentially distributed:
Δt = −ln(U)/λ, U ~ Uniform(0,1).
Over a duration T, the expected number of events is λT.
The realised count fluctuates from one random seed to another.
This gives us one clean way to separate two ideas: the rule governing the population and the unpredictable details of one realisation. That distinction is the point of the demonstration.
DO NOT COLLAPSE THEM
Two different roads to “mass”.
Many parts remain individually defined even when perception tends toward a fused field.
The process law can matter more than the identity of any one event.
What is written or generated microscopically is not necessarily what the listener names macroscopically.
SAME SOURCE · DIFFERENT PLACE
The same sound changes when you move through a room.
Move the listener and the path from source to ear changes. Different reflections arrive with different delays and phases, so the same source can become louder at some frequencies and weaker at others.
p(r,ω) ∝ e−ikr / r
In a simple 3-D free field, distance changes phase, delay and geometric spreading.
Yᵢ(f) = Hᵢ(f) X(f)
A room gives each receiver position its own complex transfer function.
yᵢ(t) = hᵢ(t) * x(t)
The same idea becomes convolution with an impulse response.
L = HLX, R = HRX
Direction can be encoded by different transfer functions to the two ears.
ENGINE A · OUR COMPUTATIONAL ROOM
Same C3. Move only the receiver.
This is the bridge back to the FEM research line. The source stays at (1.2, 2.0) m. Only the listening position changes.
source
receiver
selected
POSITIONsource signal
DIRECT DISTANCE—
GEOMETRIC DELAY—
WHAT YOU HEARunfiltered reference
These are the fine-mesh three-harmonic C3 response resyntheses from the research page, not full-band binaural room recordings.
Open the FEM / transfer-function research line ↗Why does moving the listener change the sound?the room filters each frequency differently
A musical source contains many frequencies. At one seat, reflections may reinforce 130 Hz and weaken 260 Hz; at another seat the pattern can be different. That changes colour even when the performer plays exactly the same thing.
Yi(f) = Hi(f) X(f)
Read this as: the source spectrum X passes through a position-dependent room filter Hi, producing what receiver i hears.
The later HRTF and virtual-room demos use different models. They are kept separate on purpose: room response, binaural direction and synthetic reverberation are related ideas, not the same calculation.
ENGINE B · DISTRIBUTED SOURCES
Frontality, islands, trajectory.
Headphones turn the browser into a small controlled sketchbook: first compare a frontal arrangement with four distributed sources, then move one source continuously around the listener.
NONO LENS
Front stage ↔ four islands
The same sparse four-event phrase is placed first across a narrow frontal arc, then around the listener. The tones are unchanged; only the spatial arrangement changes.
Generic browser HRTF rendering. This is a spatial-listening demonstration, not a reconstruction of Prometeo.
XENAKIS LENS
A sound can trace a path through the audience.
Move a bright synthetic source around the listener and watch the trajectory. Radius and angular speed are independent controls.
The browser uses a generic HRTF if available. The orbit is didactic; it does not model a specific hall or Xenakis score.
What is the browser actually doing? generic HRTF rendering, not our FEM solver
For headphone spatialisation, a direction-dependent filter can be written schematically as
L(f) = HL(θ,φ,f) X(f),
R(f) = HR(θ,φ,f) X(f).
The two ear signals differ in timing, level and spectral filtering.
The Web Audio PannerNode uses a generic HRTF model when the browser supports it.
This is different from the project's FEM room solver: our FEM computes a pressure field in a simplified 2-D cavity; the HRTF sketch approximates binaural directional listening around a head.
ENGINE C · CARRY A ROOM AS A FILTER
Neuwirth lens — an acoustic space can become compositional material.
A room can be measured or modelled, represented by an impulse response, then applied to another signal. Here the mechanism is simplified so the convolution remains audible and transparent.
MODELsynthetic impulse response
TAIL ENVELOPE−60 dB at RT60
OPERATIONconvolution
The generated impulse response is a transparent teaching model with a few early reflections and a seeded decaying tail. It is not the measured San Lorenzo impulse response.
SYNTHETIC IMPULSE RESPONSE
RT60 1.8 s · pre-delay 28 ms
dry phrase→h(t)→convolution→virtual acoustic
How can a microphone array capture “space”? directional field decomposition
A spherical microphone array samples the sound field over many directions. One common representation expands the directional field in spherical harmonics, which form an orthogonal basis over the sphere.
IRCAM's spatial-audio research describes using these representations to analyse and re-synthesise directional room impulse responses and to convert measured sound fields into binaural or higher-order Ambisonics formats.
Our browser experiment stops much earlier:
it demonstrates only the basic convolution idea y = h*x.
The historical/technical source shows how much richer a measured 3-D virtual acoustic can be.
MODEL BOUNDARY
Three spatial models. Do not confuse them.
Complex pressure field, receiver transfer functions, low-frequency proof of concept.
Directional binaural cues around a listener; browser-dependent and not individually measured.
A transparent impulse-response illustration, not a measured concert-hall reconstruction.
Switch between three private, level-matched excerpts from Furtwängler's 1944 Eroica, then remove the gross speed difference and listen again.
Open the comparison room ↗