023 — The Curve Was Already a Chord
agylövés: Lysarith
mathematics, verification, prose & build: GPT-5.6 Sol (Codex terminal)
2026-08-08
A polar curve already has two voices. Its horizontal coordinate is sent to the left ear; its vertical coordinate to the right. No notes are assigned to the picture. The point itself is played fast enough to hear. On a cosine rose, the petals disclose the exact pair of frequencies they were made from.
The petals are not decoration
For r(θ)=cos(kθ), multiplication by cos θ and sin θ splits the radial frequency into two sidebands. The left and right coordinate signals contain only k−1 and k+1 times the traversal frequency. Change the number of petals and watch both spectrum bars move to the bins predicted before the sound is made.
x(θ) = ½[cos((k−1)θ) + cos((k+1)θ)]
y(θ) = ½[sin((k+1)θ) − sin((k−1)θ)]
The geometry and the chord are not two interpretations of one formula. They are the same coordinate functions, run at different speeds.
How to listen
Wear headphones if you have them
The left ear receives x(θ); the right receives y(θ). Their phase difference makes the sound turn rather than merely alternate.
Start with k = 4
The rose has eight visible petals and coordinate harmonics at 3 and 5. At 55 traversals per second, the components sit at 165 Hz and 275 Hz. Move k; the picture and measured spectrum follow.
The other curves are controls, not the theorem
The limaçon, spiral and propeller are played by the same coordinate rule. Their spectra can be broader. Only the cosine rose is claimed to collapse exactly to two sidebands.
What the browser changes
It normalises peak amplitude and fades playback in and out. It does not quantise the result into a musical scale. The roughness belongs to the coordinate signal.
origin & verification ↗