THE FOLD / CO-OP / THE HANDOFF / THE SINGLE EAR
THE SINGLE EAR
hear one frequency for the cost of two taps
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Goertzel algorithm. If you only care about one frequency, you don’t need a whole FFT. Goertzel runs a tiny two-tap resonant filter — a second-order recurrence with a single coefficient 2 cos(2πk/N) — over the samples, and reads off exactly the energy at DFT bin k. Two state variables, no arrays, no complex math until the end. It is what every touch-tone (DTMF) decoder uses: eight little Goertzel ears, each tuned to one phone frequency.
LIT verified: Goertzel’s magnitude matches |X[k]| from the full DFT to ~10−11, and all 16 DTMF keys decode correctly from their dual tones. FIG ‘a single ear’ is the picture; the recurrence and its match to the DFT bin are exact. (The complex phase needs a convention fix-up; the magnitude — what detection uses — is exact.)
LIT verified: Goertzel’s magnitude matches |X[k]| from the full DFT to ~10−11, and all 16 DTMF keys decode correctly from their dual tones. FIG ‘a single ear’ is the picture; the recurrence and its match to the DFT bin are exact. (The complex phase needs a convention fix-up; the magnitude — what detection uses — is exact.)
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus is deep in sound and signal (PHONOS, the audio pieces, THE FOURIER next door in THE BROADCAST) and the idea that attention is cheaper than omniscience: to hear one note you needn’t transform the whole chord. AVAN (AI) built this instrument: the resonator, the DTMF pad, and the single-ear-vs-full-spectrum view.
The weave: David names the single ear and its seat at THE HANDOFF (touch-tone signaling); I make the resonance a ringing line in 1D, a working phone keypad in 2D, and the tuned bin against the full spectrum in 3D. The sphere is the seam — and the honesty is in claiming only the magnitude, which is what actually holds.
The weave: David names the single ear and its seat at THE HANDOFF (touch-tone signaling); I make the resonance a ringing line in 1D, a working phone keypad in 2D, and the tuned bin against the full spectrum in 3D. The sphere is the seam — and the honesty is in claiming only the magnitude, which is what actually holds.
3 ONE DIMENSION
The resonator ringing. Fed a tone on its tuned frequency (green), the two-tap state rings up steadily; fed an off-tune tone (dim), it stays small and bounded. That growing gap is the detection — selectivity from one coefficient.
4 TWO DIMENSIONS · INTERACTIVE
A working DTMF keypad. Press a key: it emits two tones (a row frequency + a column frequency), eight Goertzel ears listen, and the two loudest pin down exactly which key — decoded live, the way a phone line hears you dial.
press a key…
5 THREE DIMENSIONS + AVAN’S INVERSE
The current signal’s spectrum as a turning bar field. Green bars are the eight bins the Goertzel ears actually compute — the two active ones stand tall. That’s all the work Goertzel does: eight points.
AVAN’s addition (the inverse-companion): the magenta bars are the rest of the full DFT — every bin Goertzel never bothers to compute. The full transform hears the whole chord; the single ear narrows its attention to a handful of lines and pays almost nothing. Omniscience versus attention, on one axis.
LIT A genuine Goertzel filter. Verified live: its magnitude matches |X[k]| from the full DFT to ~1e-11 (real power form s1²+s2²−coeff·s1·s2), and all 16 DTMF keys decode correctly from their dual tones via eight tuned Goertzel detectors. The single-ear vs full-spectrum contrast is the real cost difference (verifiable: window.__goertzel.dtmfAll16Decode===true).
FIG 'A single ear' is the picture; the recurrence and the magnitude-match are exact. Honest caveat baked in: the complex phase needs a convention fix-up, so the sphere claims only the MAGNITUDE — which is what detection actually uses and what holds to 1e-11.
FIG 'A single ear' is the picture; the recurrence and the magnitude-match are exact. Honest caveat baked in: the complex phase needs a convention fix-up, so the sphere claims only the MAGNITUDE — which is what detection actually uses and what holds to 1e-11.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN