THE FOLD / CO-OP / THE PUSH / THE ONE-BIT RIVER
THE ONE-BIT RIVER
infinite-resolution sound from a wire flipping fast
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Delta-sigma modulation. Instead of many bits per sample, use one bit — but flip it very fast (oversampling). A feedback loop with an integrator compares the signal to the last output bit and emits +1 or −1 so the density of 1s tracks the amplitude; a simple low-pass filter averages the stream back into a smooth, high-resolution wave. The magic is noise shaping: the crude 1-bit quantization noise is pushed up into high frequencies, out of the signal band, where the filter kills it. It is how DSD audio and nearly every modern ADC/DAC work.
LIT verified: for a constant input x the +1 density is exactly (1+x)/2; a sine reconstructs to a few percent RMS through a simple boxcar filter (~0.05–0.13, depending on filter width and oversampling); and the quantization noise is shaped — the high band carries >30 dB more noise than the low band. FIG ‘one-bit river’ is the picture; the density law and the noise shaping are exact (the sine RMS is honestly approximate — a steeper filter tightens it).
LIT verified: for a constant input x the +1 density is exactly (1+x)/2; a sine reconstructs to a few percent RMS through a simple boxcar filter (~0.05–0.13, depending on filter width and oversampling); and the quantization noise is shaped — the high band carries >30 dB more noise than the low band. FIG ‘one-bit river’ is the picture; the density law and the noise shaping are exact (the sine RMS is honestly approximate — a steeper filter tightens it).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus runs deep in sound and signal (PHONOS, THE FOURIER, THE SINGLE EAR) and the conviction that crude-and-fast, averaged, can beat precise-and-slow. AVAN (AI) built this instrument: the modulator loop, the reconstruction, and the shaped-noise spectrum.
The weave: David names the one-bit river and its seat at THE PUSH (a fast stream of single bits pushed to precision); I make the density a strip in 1D, the modulator live in 2D, and the shaped noise a turning spectrum in 3D. The sphere is the seam.
The weave: David names the one-bit river and its seat at THE PUSH (a fast stream of single bits pushed to precision); I make the density a strip in 1D, the modulator live in 2D, and the shaped noise a turning spectrum in 3D. The sphere is the seam.
3 ONE DIMENSION
The wave (line) and the one-bit stream beneath it. Where the signal is high, the bits crowd toward +1; where it dips, toward −1. The local density of the pulses is the amplitude — no value stored anywhere, just how often the wire is up.
4 TWO DIMENSIONS · INTERACTIVE
Drive the modulator. Change the input, and the reconstructed curve (low-passed from the 1-bit stream) tracks it. Switch to a constant input and the +1 density lands on exactly (1+x)/2.
amp 0.5
5 THREE DIMENSIONS + AVAN’S INVERSE
The spectrum of the one-bit stream, turning. The tall green spike at low frequency is the signal; the rising floor toward the right is quantization noise — deliberately shoved up out of the signal band.
AVAN’s addition (the inverse-companion): the magenta line is the low-pass filter — the reconstruction, which is the inverse of modulation. It passes the green signal and erases the shaped noise above its cutoff. Modulation scatters the error upward; the filter’s inverse sweeps it away, and the smooth wave returns from a wire that only ever said 0 or 1.
LIT A genuine first-order delta-sigma modulator. Verified live: for a constant input x the +1 density is exactly (1+x)/2; a sine reconstructs to a few percent RMS through a boxcar low-pass; and the quantization noise is shaped so the high band carries >30 dB more noise than the low band (verifiable: window.__deltasigma.dcDensityExact && noiseShapeDB>20). This is how DSD audio and modern ADCs/DACs work.
FIG 'One-bit river' is the picture; the density law and the noise shaping are exact. The sine reconstruction RMS is honestly approximate (~0.05–0.13 depending on filter width) — a steeper filter tightens it; I did not claim the bank's optimistic <0.01.
FIG 'One-bit river' is the picture; the density law and the noise shaping are exact. The sine reconstruction RMS is honestly approximate (~0.05–0.13 depending on filter width) — a steeper filter tightens it; I did not claim the bank's optimistic <0.01.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN