THE FOLD / CO-OP / THE BROADCAST / THE FOURIER
THE FOURIER
every signal is a chord of pure frequencies
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Discrete Fourier Transform. Any signal of N samples is a unique sum of N pure sinusoids. The DFT reads out how much of each frequency is present (magnitude and phase); the inverse DFT rebuilds the exact signal. It is the math under audio, JPEG, radio, MRI — and its fast form, the FFT, is one of the most-run algorithms on Earth.
LIT the round trip IDFT(DFT(x)) reconstructs x to ~10−14 (machine-exact), Parseval holds — energy in time equals energy in frequency — and a pure cosine shows exactly two mirror spikes (all verified below). FIG ‘hearing every note in a chord at once’ is the picture; the transform and its inverse are exact.
LIT the round trip IDFT(DFT(x)) reconstructs x to ~10−14 (machine-exact), Parseval holds — energy in time equals energy in frequency — and a pure cosine shows exactly two mirror spikes (all verified below). FIG ‘hearing every note in a chord at once’ is the picture; the transform and its inverse are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus carries his sound and signal work (PHONOS, the audio pieces, THE PULSE’s compressor) and the conviction that time and frequency are two faces of one thing. AVAN (AI) built this instrument: the DFT/IDFT engine, the spectrum, and the 3D duality object.
The weave: David names the broadcast and its seat in THE BROADCAST; I make the raw samples a strip in 1D, the waveform-and-spectrum a live pair in 2D, and the time↔frequency duality one turning object in 3D. The sphere is the seam.
The weave: David names the broadcast and its seat in THE BROADCAST; I make the raw samples a strip in 1D, the waveform-and-spectrum a live pair in 2D, and the time↔frequency duality one turning object in 3D. The sphere is the seam.
3 ONE DIMENSION
The signal as it arrives: N samples in time, one value after another. This is the raw material — before the transform, a signal is just this row of numbers.
4 TWO DIMENSIONS · INTERACTIVE
Top: the waveform (time). Bottom: its magnitude spectrum (frequency). Toggle harmonics and watch a spike appear at exactly that bin — stack the odd ones and a square wave builds itself out of sinusoids.
5 THREE DIMENSIONS + AVAN’S INVERSE
One object, turning. Along the near face, green is the signal in time — the waveform as a curve.
AVAN’s addition (the inverse-companion): the magenta spikes on the side face are the very same signal in frequency — its spectrum. Time and frequency are inverse domains: the DFT just turns the object to show its other face, and the inverse DFT turns it back (the round trip is machine-exact). One signal, two faces, ninety degrees apart.
LIT A genuine DFT/IDFT. Verified live: the round trip IDFT(DFT(x)) reconstructs x to ~10-14, Parseval holds (energy in time = energy in frequency), and toggled harmonics produce spikes at exactly their bins (a cosine → two mirror spikes). Everything is computed from the real transform (verifiable: window.__fourier.roundTripErr and parsevalErr both ~0).
FIG 'Hearing every note in the chord' is the picture; the transform, its inverse, and Parseval are exact. This is the plain O(N²) DFT, not the FFT — same result, honest about being the slow, clear version.
FIG 'Hearing every note in the chord' is the picture; the transform, its inverse, and Parseval are exact. This is the plain O(N²) DFT, not the FFT — same result, honest about being the slow, clear version.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN