An additive synthesizer sums H sine harmonics of a fundamental f₀, each with amplitude 1/nk (the rolloff slider), plus white noise. A real radix-2 Cooley–Tukey FFT (N = 512, sample rate 4000 Hz) is computed on a windowed buffer of that exact signal every frame — the same math a hardware spectrum analyzer runs:
X[k] = Σ x[n]·w[n]·e^(−i2πkn/N), n = 0..N−1
|X[k]| = magnitude of bin k → bar height (radial pane)
Parseval: Σ x[n]² == (1/N)·Σ |X[k]|²
Each radial bar is a computed FFT bin, not a scripted animation. The Parseval-drift readout cross-checks the FFT implementation itself: it compares total signal energy measured in the time domain against total energy measured in the frequency domain — Parseval's theorem says these must match, so a drift near 0.00% is a live correctness proof of the transform, the same role the energy-drift readout plays for the double-pendulum's RK4 integrator.
- Radial pane — FFT bin magnitudes (bins 1–64) arranged as bars around a ring, colored by amplitude.
- Spectrum strip — the same bins as a linear bar chart against frequency, for precise reading.
- Oscilloscope — the raw time-domain waveform actually fed into the FFT this frame.
- THD — total harmonic distortion: RMS of harmonic-bin magnitudes (2..H) over the fundamental-bin magnitude.