The shape
ck = (1/N) Σn zn · e−2πi kn/N,
z(t) = Σ|k|≤K ck · ei k ω₀ t
- harmonics K
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- samples N
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- f₀ (one lap)
- —
- largest term
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- beats per lap
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- drift ε
- off
sound: left = Re z(t), right = Im z(t), traversed f₀ times per second. Fewer
harmonics round the curve, dull the tone and smooth away the corners the pen strikes the
drum on, in one move. Drift (S) turns one term k + ε times per lap: the curve re-forms,
the tail draws a rosette and that partial sits ε·f₀ off its harmonic.
Grains slide down the gradient of the time-averaged ⟨u²⟩ and settle where the membrane
stands still: the nodal lines. Sweep the frequency and the pattern re-forms.